2023
|
Lanza, Lukas; Dennstädt, Dario; Worthmann, Karl; Trenn, Stephan; Schaller, Manuel Sampled-data funnel control with zero-order hold Unpublished 2023, (submitted). @unpublished{LanzDann23pp,
title = {Sampled-data funnel control with zero-order hold},
author = {Lukas Lanza and Dario Dennstädt and Karl Worthmann and Stephan Trenn and Manuel Schaller},
url = {https://stephantrenn.net/wp-content/uploads/2023/03/Preprint-LDWTS230301.pdf, Preprint
https://arxiv.org/abs/2303.00523, arXiv},
year = {2023},
date = {2023-03-01},
abstract = {Output reference tracking for nonlinear systems with arbitrary relative degree via sampled-data feedback control with zero-order hold is studied. We propose a novel sample-and-hold feedback controller, which achieves output reference tracking with prescribed transient behaviour of the tracking error. Furthermore, we derive explicit bounds on the maximal required input signal and the global sampling time such that the proposed controller is feasible for all times. The results are illustrated with a numerical example.},
note = {submitted},
keywords = {discrete-time, funnel-control, input-constraints, relative-degree},
pubstate = {published},
tppubtype = {unpublished}
}
Output reference tracking for nonlinear systems with arbitrary relative degree via sampled-data feedback control with zero-order hold is studied. We propose a novel sample-and-hold feedback controller, which achieves output reference tracking with prescribed transient behaviour of the tracking error. Furthermore, we derive explicit bounds on the maximal required input signal and the global sampling time such that the proposed controller is feasible for all times. The results are illustrated with a numerical example. |
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin A novel two stages funnel controller limiting the error derivative Unpublished 2023, (submitted). @unpublished{HuTren23pp,
title = {A novel two stages funnel controller limiting the error derivative},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2023/02/Preprint-HTZ230227.pdf, Preprint},
year = {2023},
date = {2023-02-27},
urldate = {2022-11-30},
abstract = {As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme in this paper. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel.},
note = {submitted},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {unpublished}
}
As a powerful adaptive control method for the output tracking problem, funnel control has attracted considerable attention in theoretical research and engineering practice. The funnel control strategy can guarantee both transient behavior and arbitrary good accuracy. A noticeable shortcoming is however that the derivative of the tracking error may become unnecessarily large resulting in a bouncing behavior of the tracking error between the funnel boundaries. To avoid this phenomenon, we present a novel two stages funnel control scheme in this paper. This new scheme defines the control input in terms of a desired error derivative while still ensuring that the tracking error evolves within the prescribed funnel. In particular, we can quantify the range of the error derivative with a derivative funnel in terms of the known bounds of the system dynamics. Furthermore, we extend our approach to the situation where input saturations are present and extend the control law outside the funnel to ensure well-defined behavior in case the input saturations are too restrictive to keep the error within the funnel. |
Lee, Jin Gyu; Berger, Thomas; Trenn, Stephan; Shim, Hyungbo Edge-wise funnel output synchronization of heterogeneous agents with relative degree one Unpublished 2023, (submitted). @unpublished{LeeBerg23pp,
title = {Edge-wise funnel output synchronization of heterogeneous agents with relative degree one},
author = {Jin Gyu Lee and Thomas Berger and Stephan Trenn and Hyungbo Shim},
url = {https://stephantrenn.net/wp-content/uploads/2023/01/Preprint-LBTS230116.pdf, Preprint
https://arxiv.org/abs/2110.05330, ArXiV},
year = {2023},
date = {2023-01-16},
urldate = {2022-06-29},
abstract = {When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics.
This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes.},
note = {submitted},
keywords = {funnel-control, networks, nonlinear, relative-degree, synchronization},
pubstate = {published},
tppubtype = {unpublished}
}
When a group of heterogeneous node dynamics are diffusively coupled with a high coupling gain, the group exhibits a collective emergent behavior which is governed by a simple algebraic average of the node dynamics called the blended dynamics.
This finding has been utilized for designing heterogeneous multi-agent systems by building the desired blended dynamics first and then splitting it into the node dynamics. However, to compute the magnitude of the coupling gain, each agent needs to know global information such as the number of participating nodes, the graph structure, and so on, which prevents a fully decentralized design of the node dynamics in conjunction with the coupling laws. To resolve this issue, the idea of funnel control, which is a method for adaptive gain selection, can be exploited for a node-wise coupling, but the price to pay is that the collective emergent behavior is no longer governed by a simple average of the node dynamics. Our analysis reveals that this drawback can be avoided by an edge-wise design premise, which is the idea that we present in this paper. After all, we gain benefits such as a fully decentralized design without global information, collective emergent behavior being governed by the blended dynamics, and the plug-and-play operation based on edge-wise handshaking between two nodes. |
2022
|
Hu, Jiaming; Trenn, Stephan; Zhu, Xiaojin Funnel control for relative degree one nonlinear systems with input saturation Inproceedings In: Proceedings of the 2022 European Control Conference (ECC), pp. 227-232, London, 2022. @inproceedings{HuTren22,
title = {Funnel control for relative degree one nonlinear systems with input saturation},
author = {Jiaming Hu and Stephan Trenn and Xiaojin Zhu},
url = {https://stephantrenn.net/wp-content/uploads/2022/03/Preprint-HTZ220329.pdf, Preprint},
doi = {10.23919/ECC55457.2022.9837979},
year = {2022},
date = {2022-07-12},
urldate = {2022-07-12},
booktitle = {Proceedings of the 2022 European Control Conference (ECC)},
pages = {227-232},
address = {London},
abstract = {The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
The dilemma between transient behavior and accuracy in tracking control arises in both theoretical research and engineering practice and funnel control has shown great potential in solving that problem. Apart from the controlled system, the performance of funnel control strongly depends on the reference signal and the choice of prescribed funnel boundary. In this paper, we will present a new form of funnel controller for systems with control saturation. Compared to former research, the new controller is more reliable, and the closed-loop system can even achieve asymptotic tracking. Besides that, a new concept called constrained funnel boundary is introduced. Together with the new controller and the constrained funnel boundary, the application range of funnel control is extended significantly. |
2021
|
Chen, Yahao; Trenn, Stephan The differentiation index of nonlinear differential-algebraic equations versus the relative degree of nonlinear control systems Inproceedings In: PAMM · Proc. Appl. Math. Mech. 2020, pp. e202000162, Wiley-VCH GmbH, 2021, (Open Access.). @inproceedings{ChenTren21a,
title = {The differentiation index of nonlinear differential-algebraic equations versus the relative degree of nonlinear control systems},
author = {Yahao Chen and Stephan Trenn},
url = {https://stephantrenn.net/wp-content/uploads/2021/01/pamm.202000162.pdf, Paper},
doi = {10.1002/pamm.202000162},
year = {2021},
date = {2021-01-25},
booktitle = {PAMM · Proc. Appl. Math. Mech. 2020},
volume = {20},
number = {1},
pages = {e202000162},
publisher = {Wiley-VCH GmbH},
abstract = {It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differen- tiation index for nonlinear differential-algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems.},
note = {Open Access.},
keywords = {DAEs, nonlinear, normal-forms, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
It is claimed in [1] that the notion of the relative degree in nonlinear control theory is closely related to that of the differen- tiation index for nonlinear differential-algebraic equations (DAEs). In this paper, we give more insights on this claim via a recent proposed concept (see [2]) called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems. |
2013
|
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree Journal Article In: IEEE Trans. Autom. Control, vol. 58, no. 12, pp. 3126–3141, 2013. @article{LibeTren13b,
title = {The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130702.pdf, Preprint},
doi = {10.1109/TAC.2013.2277631},
year = {2013},
date = {2013-08-16},
journal = {IEEE Trans. Autom. Control},
volume = {58},
number = {12},
pages = {3126--3141},
abstract = {The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions.},
keywords = {funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound-a funnel-around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works under slightly adjusted feasibility assumptions. |
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller: time delays and case study Inproceedings In: Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland, pp. 1669–1674, 2013. @inproceedings{LibeTren13a,
title = {The bang-bang funnel controller: time delays and case study},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT130320.pdf, Preprint
http://ieeexplore.ieee.org/document/6669120, IEEE Xplore Article Number 6669120},
year = {2013},
date = {2013-07-01},
booktitle = {Proc. 12th European Control Conf. (ECC) 2013, Zurich, Switzerland},
pages = {1669--1674},
abstract = {We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered. |
Hackl, Christoph M.; Hopfe, Norman; Ilchmann, Achim; Mueller, Markus; Trenn, Stephan Funnel control for systems with relative degree two Journal Article In: SIAM J. Control Optim., vol. 51, no. 2, pp. 965–995, 2013. @article{HackHopf13,
title = {Funnel control for systems with relative degree two},
author = {Christoph M. Hackl and Norman Hopfe and Achim Ilchmann and Markus Mueller and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/HackHopf13.pdf, Paper},
doi = {10.1137/100799903 },
year = {2013},
date = {2013-03-19},
journal = {SIAM J. Control Optim.},
volume = {51},
number = {2},
pages = {965--995},
abstract = {Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {article}
}
Tracking of reference signals y_ref(.) by the output y(.) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e=y-y_ref and its derivative e' within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k_0(t)^2 e(t) - k_1(t) e'(t), where the simple proportional error feedback has gain functions k_0 and k_1 designed in such a way to preclude contact of e and e' with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. |
2012
|
Hackl, Christoph M.; Trenn, Stephan The bang-bang funnel controller: An experimental verification Inproceedings In: PAMM - Proc. Appl. Math. Mech., pp. 735–736, GAMM Annual Meeting 2012, Darmstadt Wiley-VCH Verlag GmbH, Weinheim, 2012. @inproceedings{HackTren12,
title = {The bang-bang funnel controller: An experimental verification},
author = {Christoph M. Hackl and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-HT120427.pdf, Preprint},
doi = {10.1002/pamm.201210356},
year = {2012},
date = {2012-03-01},
booktitle = {PAMM - Proc. Appl. Math. Mech.},
volume = {12},
number = {1},
pages = {735--736},
publisher = {Wiley-VCH Verlag GmbH},
address = {Weinheim},
organization = {GAMM Annual Meeting 2012, Darmstadt},
abstract = {We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive.},
keywords = {application, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
We adjust the newly developed bang-bang funnel controller such that it is more applicable for real world scenarios. The main idea is to introduce a third “neutral” input value to account for the situation when the error is already small enough and no control action is necessary. We present experimental results to illustrate the effectiveness of our new approach in the case of position control of an electrical drive. |
2010
|
Liberzon, Daniel; Trenn, Stephan The bang-bang funnel controller Inproceedings In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 690–695, 2010. @inproceedings{LibeTren10,
title = {The bang-bang funnel controller},
author = {Daniel Liberzon and Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806.pdf, Preprint
http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-LT100806longVersion.pdf, Preprint (long version)},
doi = {10.1109/CDC.2010.5717742},
year = {2010},
date = {2010-12-15},
booktitle = {Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA},
pages = {690--695},
abstract = {A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough.},
keywords = {CDC, funnel-control, input-constraints, nonlinear, relative-degree},
pubstate = {published},
tppubtype = {inproceedings}
}
A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. |
2009
|
Trenn, Stephan A normal form for pure differential algebraic systems Journal Article In: Linear Algebra Appl., vol. 430, no. 4, pp. 1070 – 1084, 2009. @article{Tren09a,
title = {A normal form for pure differential algebraic systems},
author = {Stephan Trenn},
url = {http://stephantrenn.net/wp-content/uploads/2017/09/Preprint-Tre081215.pdf, Preprint},
doi = {10.1016/j.laa.2008.10.004},
year = {2009},
date = {2009-01-01},
journal = {Linear Algebra Appl.},
volume = {430},
number = {4},
pages = {1070 -- 1084},
abstract = {In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability.},
keywords = {controllability, DAEs, normal-forms, observability, relative-degree},
pubstate = {published},
tppubtype = {article}
}
In this paper linear time-invariant differential algebraic equations (DAEs) are studied; the focus is on pure DAEs which are DAEs without an ordinary differential equation (ODE) part. A normal form for pure DAEs is given which is similar to the Byrnes–Isidori normal form for ODEs. Furthermore, the normal form exhibits a Kalman-like decomposition into impulse-controllable- and impulse-observable states. This leads to a characterization of impulse-controllability and observability. |
