## Parametric robust H2 control design using LMI synthesis (2000)

Venue: | AIAA Journal Guidance, Control, and Dynamics |

Citations: | 6 - 6 self |

### BibTeX

@ARTICLE{Banjerdpongchai00parametricrobust,

author = {David Banjerdpongchai and Jonathan How Y},

title = {Parametric robust H2 control design using LMI synthesis},

journal = {AIAA Journal Guidance, Control, and Dynamics},

year = {2000}

}

### OpenURL

### Abstract

This paper presents a new, iterative algorithm for designing full order LTI controllers for systems with real parameter uncertainty. Robust stability isdetermined for these systems using the Popov analysis criterion and multiplier, and robust performance is investigated using a bound on the output energy. Control design to minimize the robust performance metric naturally leads to Bilinear Matrix Inequalities, which can be decoupled to a large extent. However, coupling remains in the problem since we simultaneously optimize the parameters of both the Popov stabilitymultiplier and the compensator. We present a heuristic, iterative algorithm to solve this design problem, and demonstrate that it works effectively on two numerical examples. In the process, we illustrate that the key advantages of this control design approach are the high reliability of the numerical techniques and the relative simplicity of implementing the algorithm. 1

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Citation Context ...rbance input, y : R+ ! R ny is the measured output and z : R+ ! R nz is the output. The perturbations i are nonlinearities that satisfy the sector bound [0; 1], i.e., 0 i( )= 1. As discussed in Refs. =-=[11, 7]-=-, we can use a loop transformation to handle the more general sector conditions i i( )= i. Our goal is to nd a strictly proper full order LTI controller, of the form _xc = Acxc + Bcy u = Ccxc where xc... |

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Citation Context .... Email: howjo@sun-valley.stanford.edu. Member AIAA. Copyright c 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. using a frequency dependent Popov multiplier =-=[2, 3]-=- or its generalizations [4]. A Lyapunov function representation of the Popov analysis test can be combined with an H2 cost function to provide a bound on the robust performance. Designing a controller... |

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Citation Context ...shown, controllers are developed in two main steps. The rst step does require an iteration, but in the process capitalizes on the very e cient design tools that are available for solving LMI problems =-=[14, 15]-=-. The resulting compensators are full-order, and cannot include architecture constraints. However, the solution procedure is very robust which reduces the user workload, and, is easily expandable to i... |

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Citation Context .... Email: howjo@sun-valley.stanford.edu. Member AIAA. Copyright c 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. using a frequency dependent Popov multiplier =-=[2, 3]-=- or its generalizations [4]. A Lyapunov function representation of the Popov analysis test can be combined with an H2 cost function to provide a bound on the robust performance. Designing a controller... |

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Citation Context ...sis problem. The direct extension of this LMI analysis to the controller synthesis problem results in Bilinear Matrix Inequalities (BMI's), which currently cannot be solved directly. El Ghaoui et al. =-=[8]-=- proposed a heuristic solution procedure for these BMI problems using a two-stage optimization process, called the V-K iteration. During each phase of this iteration, some of the design variables in t... |

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Citation Context ...shown, controllers are developed in two main steps. The rst step does require an iteration, but in the process capitalizes on the very e cient design tools that are available for solving LMI problems =-=[14, 15]-=-. The resulting compensators are full-order, and cannot include architecture constraints. However, the solution procedure is very robust which reduces the user workload, and, is easily expandable to i... |

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Citation Context ...ford.edu. Member AIAA. Copyright c 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. using a frequency dependent Popov multiplier [2, 3] or its generalizations =-=[4]-=-. A Lyapunov function representation of the Popov analysis test can be combined with an H2 cost function to provide a bound on the robust performance. Designing a controller to optimize this bound pro... |

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Citation Context ...n complicated objectives, such as control design to minimize an H2 cost function, has been found to converge very slowly, if at all. This synthesis algorithm was recently improved by El Ghaoui et al. =-=[9]-=- leading to a systematic design approach for systems with unstructured uncertainty. The main result of this paper is to extend the results in Ref. [9] todevelop synthesis algorithms for systems with p... |

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Citation Context ... the actual robust performance is quite asymmetric about the nominal sti ness. This asymmetry was one of the characteristics of Popov controllers identi ed during some of the earlier results in Refs. =-=[5, 18]-=-, and indicates that asymmetric sector bounds might further reduce conservatism. We continue this discussion by directly comparing controllers designed four di erent ways: Sensitivity weight LQG (SWLQ... |

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Citation Context ...recently attracted much attention in the controls community. One of the many proposed techniques is Popov Controller Synthesis which captures the system uncertainties as sector-bounded nonlinearities =-=[1]-=-. Additional information about the type and structure of the system uncertainties is captured in the analysis Durand 110, Electrical Engineering Department. Email: banjerd@isl.stanford.edu. y Durand 2... |

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Citation Context ...optimize this bound provides a synthesis tool that guarantees robust stability and performance. Several previous investigations have been performed to develop solutions to this control design problem =-=[1, 5, 6]-=-. However, until recently, these approaches were based on numerical optimizations using quasi-Newton search algorithms to solve the necessary conditions which must be analytically derived from the per... |

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Citation Context ... positive symmetric matrix satisfying: A T P + PA+ C T z Cz =0: In case of nonlinear systems, it is no longer relevant to discuss the performance in terms of an H2 norm. However, as discussed in Ref. =-=[12]-=-, it is still valid to use the second de nition. With these results, the performance of the closedloop system in Eq. (2) is de ned as follows. Let w =0,~x0;i;i =1;:::;nw be a basis of the input space ... |

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Citation Context ...llers to those obtained for two benchmark problems. The rst benchmark problem was selected from Ref. [6] as an external check of the synthesis algorithm itself. The second benchmark problem from Ref. =-=[17]-=- was selected to enable Table 1: Typical sdpsol program for Popov analysis. The simplicity of this user interface signi cantly reduces the overhead associated with developing and implementing the Popo... |

6 |
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Citation Context ...c sector bounds might further reduce conservatism. We continue this discussion by directly comparing controllers designed four di erent ways: Sensitivity weight LQG (SWLQG) [17], Maximum Entropy (ME) =-=[19]-=-, Multiple Model (MM) [20], and Popov controller synthesis (Popov) using LMI synthesis. Fig. 9 compares the robust performance of three controllers (analyzed in detail in Ref. [17]) that achieve appro... |

2 |
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Citation Context ...her reduce conservatism. We continue this discussion by directly comparing controllers designed four di erent ways: Sensitivity weight LQG (SWLQG) [17], Maximum Entropy (ME) [19], Multiple Model (MM) =-=[20]-=-, and Popov controller synthesis (Popov) using LMI synthesis. Fig. 9 compares the robust performance of three controllers (analyzed in detail in Ref. [17]) that achieve approximately the same levels o... |

1 | Using Linear Matrix Inequalities to Design Controllers for Robust - Yang, Livadas, et al. - 1996 |

1 |
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Citation Context ...t) go to zero. (2) Robust Performance: Consider an LTI system G fA; Bw;Cz; 0g, where A; Bw;Cz have the dimensions de ned earlier. The following de - nitions for H2 norm of the system G are equivalent =-=[12, 13]-=-. (a) Let e1;:::;enw be a basis of the input space (i.e., the ith element ofei equals to 1 and the rest are zero). Let zi(t) be the output of the system if we apply the impulse ei to the system and se... |