## Optimal Asymptotic Identification Under Bounded Disturbances (0)

Venue: | IEEE Trans. Automat. Contr |

Citations: | 14 - 3 self |

### BibTeX

@ARTICLE{Tse_optimalasymptotic,

author = {David N. C. Tse and Munther A. Dahleh and John N. Tsitsiklis},

title = {Optimal Asymptotic Identification Under Bounded Disturbances},

journal = {IEEE Trans. Automat. Contr},

year = {},

volume = {38},

pages = {1176--1190}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper investigates the intrinsic limitation of worst-case identification of LTI systems using data corrupted by bounded disturbances, when the unknown plant is known to belong to a given model set. This is done by analyzing the optimal worst-case asymptotic error achievable by performing experiments using any bounded inputs and estimating the plant using any identification algorithm. First, it is shown that under some topological conditions on the model set, there is an identification algorithm which is asymptotically optimal for any input. Characterization of the optimal asymptotic error as a function of the inputs is also obtained. These results hold for any error metric and disturbance norm. Second, these general results are applied to three specific identification problems: identification of stable systems in the ` 1 norm, identification of stable rational systems in the H1 norm, and identification of unstable rational systems in the gap metric. For each of these problems, the...

### Citations

238 | Information-Based Complexity
- Traub, Wasilkowski, et al.
- 1988
(Show Context)
Citation Context ...uestions give a characterization of the difficulty of identification using a given model set. A natural framework to study worst-case identification is provided by information-based complexity theory =-=[17, 28, 29]-=-. This theory provides a general mathematical framework for analyzing the optimal error achievable in solving a problem using a given amount of possibly inaccurate and partial information. Information... |

182 |
Analysis of feedback systems with structured uncertainties
- Doyle
- 1982
(Show Context)
Citation Context ... plants of arbitrary order. The significance of these error metrics is that if the worst-case error is small in these metrics, methods exist for synthesizing controllers to achieve robust performance =-=[2, 3]-=-. The results show that accurate identification is possible in the worst case for a specific choice of inputs depending on the model set. For identification in the ` 1 norm, algorithms for computing e... |

63 |
A General Theory of Optimal Algorithms
- Traub, Wozniakowski
- 1980
(Show Context)
Citation Context ...uestions give a characterization of the difficulty of identification using a given model set. A natural framework to study worst-case identification is provided by information-based complexity theory =-=[17, 28, 29]-=-. This theory provides a general mathematical framework for analyzing the optimal error achievable in solving a problem using a given amount of possibly inaccurate and partial information. Information... |

60 |
Optimal robustness in the gap metric
- Georgiou, Smith
- 1990
(Show Context)
Citation Context ...n of unstable systems. The question of whether one can design closed-loop experiments to achieve such conditions is left open. An appropriate error metric to use for unstable plants is the gap metric =-=[5, 27, 34]-=-. The important property of the gap metric is that it generates the graph topology [32], which is the weakest topology in which closed-loop stability is a robust property, or in which the closed-loop ... |

39 |
A survey of optimal recovery
- Micchelli, Rivlin
- 1977
(Show Context)
Citation Context ...uestions give a characterization of the difficulty of identification using a given model set. A natural framework to study worst-case identification is provided by information-based complexity theory =-=[17, 28, 29]-=-. This theory provides a general mathematical framework for analyzing the optimal error achievable in solving a problem using a given amount of possibly inaccurate and partial information. Information... |

37 |
Estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors: linear families of models
- Milanese, Belforte
- 1982
(Show Context)
Citation Context ...ompletely new. Although mainstream system identification research adopts stochastic models for the noise, there is a line of work which deals with worst-case identification under bounded disturbances =-=[4, 13, 18, 19, 20, 24, 26]-=-. More recently, specific identification algorithms are proposed in [7, 8, 9, 22] for worst-case identification in the H1 metric from noisy frequency response data and in [11, 21] for identification i... |

34 |
Theory of Functions of a Complex Variable. Vols
- Markushevich
- 1985
(Show Context)
Citation Context ...e nor the non-existence of a bounded input having both the properties required by Corollary 4.15 has been established. However, bounded inputs which excite at all frequencies do exist. In fact, Lusin =-=[15]-=- has constructed a sequence which excites at all frequencies despite the fact that the sequence actually tends to 0. Stability testing is a necessary property the inputs must satisfy in order to have ... |

33 |
On the Value of Information in System Identification-Bounded Noise
- Fogel, Huang
- 1982
(Show Context)
Citation Context ...ompletely new. Although mainstream system identification research adopts stochastic models for the noise, there is a line of work which deals with worst-case identification under bounded disturbances =-=[4, 13, 18, 19, 20, 24, 26]-=-. More recently, specific identification algorithms are proposed in [7, 8, 9, 22] for worst-case identification in the H1 metric from noisy frequency response data and in [11, 21] for identification i... |

32 |
Topology: A First
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Citation Context ...finite-horizon diameter of information, ae(g; h)sD(u; M; ffi). 2 The desired topological condition involves the topology of component-wise convergence of sequences, or the so-called product topology. =-=[23]-=-. Lemma 3.8 Fix the inputs u 2 Bl N 1 and ffi ? 0. Let A ae M \Theta M be compact in the product topology, and suppose Tu;ffi (g; h) is finite for every (g; h) 2 A. Then sup (g;h)2A Tu;ffi (g; h) is a... |

32 |
On the connection between the complexity and credibility of inferred models
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(Show Context)
Citation Context ...istics and pattern recognition. It is interesting to note that this same principle of Occam's Razor has also been applied to guarantee convergence in distribution-free probabilistic learning problems =-=[1, 25]-=-. In contrast to the oe-compactness condition that guarantees convergence, a stronger compactness condition guarantees uniform convergence. Proposition 3.9 Suppose convergence in the ae-topology on M ... |

30 |
Control System Synthesis
- Vidyasagar
- 1985
(Show Context)
Citation Context ...ve such conditions is left open. An appropriate error metric to use for unstable plants is the gap metric [5, 27, 34]. The important property of the gap metric is that it generates the graph topology =-=[32]-=-, which is the weakest topology in which closed-loop stability is a robust property, or in which the closed-loop system varies continuously as a function of the open-loop system. Intuitively, this mea... |

29 |
Optimal algorithms theory for robust estimation and prediction
- Milanese, Tempo
- 1985
(Show Context)
Citation Context ...ompletely new. Although mainstream system identification research adopts stochastic models for the noise, there is a line of work which deals with worst-case identification under bounded disturbances =-=[4, 13, 18, 19, 20, 24, 26]-=-. More recently, specific identification algorithms are proposed in [7, 8, 9, 22] for worst-case identification in the H1 metric from noisy frequency response data and in [11, 21] for identification i... |

28 |
Optimal input signals for parameter estimation in dynamic systems - survey and new results
- Mehra
- 1974
(Show Context)
Citation Context ...neral aspects of optimal worst-case asymptotic identification in a general error metric. Moreover, the issue of optimal experiment design, although considered in stochastic system identification (eg. =-=[6, 16, 35]-=-), has not been satisfactorily addressed in the worst-case setting. The contributions of this paper are two-folded. At a more general level, it introduces a framework for the analysis of optimal worst... |

16 |
Control-oriented system identification: A worst-case/deterministic approach in H
- Helmicki, Jacobson, et al.
- 1991
(Show Context)
Citation Context ...e experiment length required to guarantee that any plant in the model set can be identified to a prescribed accuracy. It is the notion of convergence considered by Helmicki et. al. in their framework =-=[10]-=-. Demanding uniform convergence is too restrictive a formulation for a general theory of fundamental limitations of worst-case identification. Although such uniform convergence is certainly desirable,... |

16 |
Robust approximation and identification
- Mäkilä, Partington
- 1991
(Show Context)
Citation Context ...een a growing line of work with the common theme that system identification should be performed so that the worst-case error of the resulting model is small in a metric compatible with robust control =-=[7, 8, 9, 22, 30]-=-. This paper addresses the questions of asymptotically optimal identification algorithms and experiment designs from this point of view. Our emphasis is less on finding efficient algorithms and more o... |

16 |
The gap metric: robustness of stabilization of feedback systems
- El-Sakkary
- 1985
(Show Context)
Citation Context ...n of unstable systems. The question of whether one can design closed-loop experiments to achieve such conditions is left open. An appropriate error metric to use for unstable plants is the gap metric =-=[5, 27, 34]-=-. The important property of the gap metric is that it generates the graph topology [32], which is the weakest topology in which closed-loop stability is a robust property, or in which the closed-loop ... |

12 |
Robust identification and Galois sequences
- Mäkilä
- 1991
(Show Context)
Citation Context ...s [4, 13, 18, 19, 20, 24, 26]. More recently, specific identification algorithms are proposed in [7, 8, 9, 22] for worst-case identification in the H1 metric from noisy frequency response data and in =-=[11, 21]-=- for identification in the ` 1 metric from time series data. In contrast to these works, we deal with general aspects of optimal worst-case asymptotic identification in a general error metric. Moreove... |

12 |
Identification and application of bounded-parameter models
- Norton
- 1987
(Show Context)
Citation Context |

5 |
A class of algorithms for identification in H1
- Gu, Khargonekar
- 1992
(Show Context)
Citation Context ...een a growing line of work with the common theme that system identification should be performed so that the worst-case error of the resulting model is small in a metric compatible with robust control =-=[7, 8, 9, 22, 30]-=-. This paper addresses the questions of asymptotically optimal identification algorithms and experiment designs from this point of view. Our emphasis is less on finding efficient algorithms and more o... |

5 |
Experiment design in a bounded-error context: comparison with D-optimality
- Pronzato, Walter
- 1989
(Show Context)
Citation Context |

4 | Identification in H1 : linear algorithms - Helmicki, Jacobson, et al. - 1990 |

4 |
Reformulation of the parameter identification problem for systems with bounded disturbances
- Lozano-Leal, Ortega
- 1987
(Show Context)
Citation Context |

4 |
Optimal Experimental Design for Dynamic System Identification
- Zarrop
- 1979
(Show Context)
Citation Context ...neral aspects of optimal worst-case asymptotic identification in a general error metric. Moreover, the issue of optimal experiment design, although considered in stochastic system identification (eg. =-=[6, 16, 35]-=-), has not been satisfactorily addressed in the worst-case setting. The contributions of this paper are two-folded. At a more general level, it introduces a framework for the analysis of optimal worst... |

3 |
Experiment design for system identification
- Goodwin
- 1987
(Show Context)
Citation Context ...neral aspects of optimal worst-case asymptotic identification in a general error metric. Moreover, the issue of optimal experiment design, although considered in stochastic system identification (eg. =-=[6, 16, 35]-=-), has not been satisfactorily addressed in the worst-case setting. The contributions of this paper are two-folded. At a more general level, it introduces a framework for the analysis of optimal worst... |

3 |
Identification in H1 : A robust convergent nonlinear algorithm
- Helmicki, Jacobson, et al.
- 1989
(Show Context)
Citation Context ...een a growing line of work with the common theme that system identification should be performed so that the worst-case error of the resulting model is small in a metric compatible with robust control =-=[7, 8, 9, 22, 30]-=-. This paper addresses the questions of asymptotically optimal identification algorithms and experiment designs from this point of view. Our emphasis is less on finding efficient algorithms and more o... |

3 |
Robust Performance of Adaptive Controllers with General Uncertainty Structure
- Krause, Stein, et al.
- 1990
(Show Context)
Citation Context ...ealing to the law of large numbers. As far as we know, this issue has not been considered in an unknown-but-bounded noise setting; in fact, it has been taken for granted that consistency always holds =-=[12]-=-. Instead, it will now be shown that a compactness condition on the model set will guarantee consistency. The following theorem shows that, under a oe-compactness assumption on M, D(u; M; ffi) is an u... |

3 |
Parameter Set Estimation of Systems with Uncertain Nonparametric Dynamics and Disturbances
- Lau, Kosut, et al.
- 1990
(Show Context)
Citation Context ... is available for computing estimates, and infinite-horizon experiments, where the entire infinite data record is available. The question is when the latter can be viewed as a limit of the former. In =-=[14]-=-, such a consistency result is established by placing a stationarity assumption on the noise and then appealing to the law of large numbers. As far as we know, this issue has not been considered in an... |

3 |
Estimation theory and prediction in the presence of unknown and bounded uncertainty: a survey
- Milanese
- 1989
(Show Context)
Citation Context |

3 |
Optimal and Robust Identification in the ` 1 norm
- Tse, Dahleh, et al.
- 1991
(Show Context)
Citation Context |

3 |
On the metric complexity of casual linear systems: ffl-entropy and ffl-dimension for continuous-time
- Zames
- 1979
(Show Context)
Citation Context ...on accuracy achievable by any identification algorithm in the limit of observing more and more data corrupted by non-stochastic noise. Thus, this work is in the flavor of the questions posed by Zames =-=[33]-=-. We will deal exclusively with discrete-time, single-input-single-output linear time-invariant systems. In this formulation, the unknown plant is a priori known to be in a certain subset M of the spa... |

3 |
Uncertainty in unstable systems: the gap metric
- Zames, El-Sakkary
- 1982
(Show Context)
Citation Context ...n of unstable systems. The question of whether one can design closed-loop experiments to achieve such conditions is left open. An appropriate error metric to use for unstable plants is the gap metric =-=[5, 27, 34]-=-. The important property of the gap metric is that it generates the graph topology [32], which is the weakest topology in which closed-loop stability is a robust property, or in which the closed-loop ... |

2 |
Occam's Razor
- Haussler, Warmuth
- 1987
(Show Context)
Citation Context ...istics and pattern recognition. It is interesting to note that this same principle of Occam's Razor has also been applied to guarantee convergence in distribution-free probabilistic learning problems =-=[1, 25]-=-. In contrast to the oe-compactness condition that guarantees convergence, a stronger compactness condition guarantees uniform convergence. Proposition 3.9 Suppose convergence in the ae-topology on M ... |

2 |
Controller Design in the Presence of Structured Uncertainty
- Dahleh, Khammash
(Show Context)
Citation Context ... plants of arbitrary order. The significance of these error metrics is that if the worst-case error is small in these metrics, methods exist for synthesizing controllers to achieve robust performance =-=[2, 3]-=-. The results show that accurate identification is possible in the worst case for a specific choice of inputs depending on the model set. For identification in the ` 1 norm, algorithms for computing e... |

2 |
Worst-case system identification in ` 1 : Optimal algorithms and error bounds
- Jacobson, Nett
- 1991
(Show Context)
Citation Context ...s [4, 13, 18, 19, 20, 24, 26]. More recently, specific identification algorithms are proposed in [7, 8, 9, 22] for worst-case identification in the H1 metric from noisy frequency response data and in =-=[11, 21]-=- for identification in the ` 1 metric from time series data. In contrast to these works, we deal with general aspects of optimal worst-case asymptotic identification in a general error metric. Moreove... |

2 |
Optimal and Robust Identification Under Bounded Disturbances
- Tse
- 1991
(Show Context)
Citation Context ... Mitter and George Zames for the very helpful discussions. A Proof of Theorem 4.14 To prove this result, we need the following lemma, the proof of which is elementary but tedious, and can be found in =-=[31]-=-. Lemma A.1 Let u 2 Bl1 and let h be a complex-valued impulse response (i.e. the sequence values can be complex) with a strictly proper rational transfer function H(z) = P M \Gamma1 i=0 ff i z i (z \G... |

1 |
A Lower and Upper Bound for the Gap Metric
- Zhu, Hautus, et al.
- 1989
(Show Context)
Citation Context ...ve this result, it suffices to show that the infinite-horizon gap diameter of information satisfies D gap (u; M fd ; ffi)s2 ffi p 1 + ffi 2 We make use of the following lower bound for the gap metric =-=[36]-=-: ffi(h; 0)skhkH1 q 1 + khk 2 H1 Now, D gap (u; M fd ; ffi) = sup h2M fd sup kdk1ffi diam gap S1 (M fd ; u; hsu + d; ffi)sdiam gap S1 (M fd ; u; 0; ffi) = sup g2M fd ;kg uk 1ffi 2ffi(g; 0) since ffi(g... |