Russian Academy of Sciences
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
ห้องสมุดไป่ตู้
Abstract
1
1 Introduction
Controlling nonlinear systems onto unstable periodic orbits contained in an attractor is the subject of substantial and well developed research 1, 2, 3, 4]. As these periodic orbits are solutions of the dynamical equations for the nonlinear system, it is possible to stabilize them with only small perturbations to various parameters. Many of the control algorithms have been based on the work of Ott, Grebogi, and Yorke (OGY) 3, 4]. This technique provides a powerful method for controlling unstable periodic orbits when working on a Poincare section. The time to reach the desired periodic orbit from an arbitrary starting location can be very long, and this time grows exponentially as the maximum amplitude allowed for the control parameter decreases. To reduce this time a \targeting" technique was proposed 5]. This procedure is computationally intensive and can be di cult to implement in practical situations and in high dimensional phase spaces as it requires the tracing of stable and unstable manifolds of the system through the phase space. We consider here an alternative approach based on optimal control theory. Optimal control has a long history but has rarely 6, 7, 8] been part of the discussion of control for chaotic systems. The methods we use are a local application of the Pontryagin maximum principle 9]. We derive a simple exact one step control formulae for directing the system towards a selected set of points in phase space, and then keeping it there. This sequence of target points will typically lie on an unstable periodic orbit of the uncontrolled system. The control rule requires that only small values of the control parameters near the nominal uncontrolled state can be applied. This means small perturbations to the system may be used to control it accurately and e ciently. It is important to distinguish what we do here from the traditional signal tracking problem. We do not have a pre-speci ed reference signal and implied time origin in mind, though once the target set is reached we will be tracking it in the conventional meaning, if it is in fact an orbit of the system. A new aspect which we bring to discussions of optimal control is the use of methods for reconstructing a system phase space from observations of a single scalar variable 2, 10] and then modeling the dynamics in that space using properties of the system attractor 2]. With these models we are able to make predictions of the evolution of an orbit, and then to apply a control to alter that outcome|all without knowing the equations of motion for the system. Our control method, for known or reconstructed systems, uses short control pulses of small amplitude, and so takes advantage of the sensitivity of chaotic systems to perturbations. The system instabilities are exploited to get to the target rather than struggled against. In this broad sense, our techniques are like the OGY method. They di er from that method in two important ways: a potentially long time before control is applied is not required nor is detailed calculation of stable and unstable manifolds for the control formula or for targeting. At the heart of our results is an exact formula for `one step' control when the control parameter remains close to the nominal operating value. The `one step' could be the evolution of the dynamical system through one observed time step or the composition of many time1 maps taking the system from the present state to a state many steps ahead along the 2
Optimal Control of Nonlinear Systems to Given Orbits
Henry D. I. Abarbanel,1 Department of Physics and Marine Physical Laboratory Scripps Institution of Oceanography University of California, San Diego Mail Code 0402 La Jolla, CA 92093-0402 hdia@ Lev Korzinov, korz@ and Alistair I. Mees,2 alistair@ Institute for Nonlinear Science University of California, San Diego La Jolla, CA 92093-0402 and Igor M. Starobinets, Institute for Applied Physics Russian Academy of Sciences Nizhny Novgorod, Russia 603600
1 2
Institute for Nonlinear Science Permanent Address: Department of Mathematics, University of Western Australia, Nedlands, Perth
Using optimal control techniques we derive and demonstrate the use of an exact single step control method for directing a nonlinear system to a target orbit and keeping it there. We require that values of the control parameters remain near the uncontrolled settings. The full nonlinearity of the problem in state space variables is retained. The `one step' of the control is typically a composition of known or learned maps over (a) the time required to learn the state, (b) the time to compute the control, and (c) the time to apply the control. No special targeting is required, yet the time to control is quite rapid. Working with the dynamics of a well studied nonlinear electrical circuit, we show how this method works e ciently and accurately in two situations: when the known circuit equations are used, and when control is performed only on a Poincare section of the reconstructed phase space. In each case, because the control rule is known analytically, the control strategy is computationally e cient while retaining high accuracy. The target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics, and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system.
Abstract
1
1 Introduction
Controlling nonlinear systems onto unstable periodic orbits contained in an attractor is the subject of substantial and well developed research 1, 2, 3, 4]. As these periodic orbits are solutions of the dynamical equations for the nonlinear system, it is possible to stabilize them with only small perturbations to various parameters. Many of the control algorithms have been based on the work of Ott, Grebogi, and Yorke (OGY) 3, 4]. This technique provides a powerful method for controlling unstable periodic orbits when working on a Poincare section. The time to reach the desired periodic orbit from an arbitrary starting location can be very long, and this time grows exponentially as the maximum amplitude allowed for the control parameter decreases. To reduce this time a \targeting" technique was proposed 5]. This procedure is computationally intensive and can be di cult to implement in practical situations and in high dimensional phase spaces as it requires the tracing of stable and unstable manifolds of the system through the phase space. We consider here an alternative approach based on optimal control theory. Optimal control has a long history but has rarely 6, 7, 8] been part of the discussion of control for chaotic systems. The methods we use are a local application of the Pontryagin maximum principle 9]. We derive a simple exact one step control formulae for directing the system towards a selected set of points in phase space, and then keeping it there. This sequence of target points will typically lie on an unstable periodic orbit of the uncontrolled system. The control rule requires that only small values of the control parameters near the nominal uncontrolled state can be applied. This means small perturbations to the system may be used to control it accurately and e ciently. It is important to distinguish what we do here from the traditional signal tracking problem. We do not have a pre-speci ed reference signal and implied time origin in mind, though once the target set is reached we will be tracking it in the conventional meaning, if it is in fact an orbit of the system. A new aspect which we bring to discussions of optimal control is the use of methods for reconstructing a system phase space from observations of a single scalar variable 2, 10] and then modeling the dynamics in that space using properties of the system attractor 2]. With these models we are able to make predictions of the evolution of an orbit, and then to apply a control to alter that outcome|all without knowing the equations of motion for the system. Our control method, for known or reconstructed systems, uses short control pulses of small amplitude, and so takes advantage of the sensitivity of chaotic systems to perturbations. The system instabilities are exploited to get to the target rather than struggled against. In this broad sense, our techniques are like the OGY method. They di er from that method in two important ways: a potentially long time before control is applied is not required nor is detailed calculation of stable and unstable manifolds for the control formula or for targeting. At the heart of our results is an exact formula for `one step' control when the control parameter remains close to the nominal operating value. The `one step' could be the evolution of the dynamical system through one observed time step or the composition of many time1 maps taking the system from the present state to a state many steps ahead along the 2
Optimal Control of Nonlinear Systems to Given Orbits
Henry D. I. Abarbanel,1 Department of Physics and Marine Physical Laboratory Scripps Institution of Oceanography University of California, San Diego Mail Code 0402 La Jolla, CA 92093-0402 hdia@ Lev Korzinov, korz@ and Alistair I. Mees,2 alistair@ Institute for Nonlinear Science University of California, San Diego La Jolla, CA 92093-0402 and Igor M. Starobinets, Institute for Applied Physics Russian Academy of Sciences Nizhny Novgorod, Russia 603600
1 2
Institute for Nonlinear Science Permanent Address: Department of Mathematics, University of Western Australia, Nedlands, Perth
Using optimal control techniques we derive and demonstrate the use of an exact single step control method for directing a nonlinear system to a target orbit and keeping it there. We require that values of the control parameters remain near the uncontrolled settings. The full nonlinearity of the problem in state space variables is retained. The `one step' of the control is typically a composition of known or learned maps over (a) the time required to learn the state, (b) the time to compute the control, and (c) the time to apply the control. No special targeting is required, yet the time to control is quite rapid. Working with the dynamics of a well studied nonlinear electrical circuit, we show how this method works e ciently and accurately in two situations: when the known circuit equations are used, and when control is performed only on a Poincare section of the reconstructed phase space. In each case, because the control rule is known analytically, the control strategy is computationally e cient while retaining high accuracy. The target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics, and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system.