[Show abstract][Hide abstract] ABSTRACT: H ighly resonant dynamics can severely degrade the performance of technological systems. Structural modes in machines and robots, ground and aerospace vehicles, and precision instrumentation, such as atomic force microscopes and optical systems, can limit the ability of control systems to achieve the desired performance. Consequently, control systems must be designed to suppress the effects of these dynamics, or at least avoid exciting them beyond open-loop levels. Open-loop techniques for highly resonant systems, such as input shaping , as well as closed-loop techniques, such as damping augmentation , , can be used for this purpose. Feedback Control of Negative-Imaginary Systems OCTOBER 2010 « IEEE CONTROL SYSTEMS MAGAZINE 55 Structural dynamics are often difficult to model with high precision due to sensitivity to boundary conditions as well as aging and environmental effects. Therefore, active damping augmentation to counteract the effects of external commands and disturbances must account for parametric uncertainty and unmodeled dynamics. This problem is simplified to some extent by using force actuators combined with colocated measurements of velocity, position, or acceleration, where colocated refers to the fact that the sensors and actuators have the same location and the same direction. Colocated control with velocity measurements, called negative-velocity feedback, can be used to directly increase the effective damping, thereby facilitating the design of controllers that guarantee closed-loop stability in the presence of plant parameter variations and unmodeled dynamics , . This guaranteed stability property can be established by using results on passive systems , . However, the theoretical properties of negative-velocity feedback are based on the idealized assumption of coloca-tion and require the availability of velocity sensors, which may be expensive. Also, the choice of measured variable may depend on whether the desired objective is shape control or damping augmentation. An alternative approach to negative-velocity feedback is positive-position feedback, where position sensors are used in place of velocity sensors. Although position sensors can facilitate the objective of shape control, it is less obvious how they can be used for damping augmentation. Nevertheless, it is shown in  and  that a positive-position feedback controller can be designed to increase the damping of the modes of a flexible structure. Furthermore , this controller is robust against uncertainty in the modal frequencies as well as unmodeled plant dynamics. As shown in –, the robustness properties of positive position feedback are similar to those of negative-velocity feedback. This article investigates the robustness of positive-position feedback control of flexible structures with colocated force actuators and position sensors. In particular, the theory of negative-imaginary systems ,  is used to reveal the robustness properties of multi-input, multi-output (MIMO) positive-position feedback controllers and related types of controllers for flexible structures , – . The negative-imaginary property of linear systems can be extended to nonlinear systems through the notion of counterclockwise input-output dynamics , . It is shown in  for the single-input, single-output (SISO) linear case that the results of  and  guarantee the stability of a positive-position feedback control system in the presence of unmodeled dynamics and parameter uncertainties that maintain the negative-imaginary property of the plant. Positive-position feedback can be regarded as one of the last areas of classical control theory to be encompassed by modern control theory. In this article, positive-position feedback, negative-imaginary systems, and related control methodologies are brought together with the underlying systems theory. Table 1 summarizes notation used in this article, while Table 2 lists acronyms.
[Show abstract][Hide abstract] ABSTRACT: This paper proposes a systematic synthesis methodology for a combined feedforward/feedback architecture to control multiple-input, multiple-output nanopositioning systems. Coprime factorization of the open loop model is used to design the reference and feedforward filters of the proposed control scheme to achieve enhanced tracking, eliminate the limitation of the feedback on tracking performance, and increase the bandwidth of the closed-loop system. Two types of coprime factorization, namely inner–outer factorization and normalized coprime factorization are discussed. A case study based on hardware experiment is presented to analyze the proposed control architecture and demonstrate its superiority over feedback-only control. In addition to the no-load case, the performance of the system is also tested with loads on the nanopositioning stage.
IEEE Transactions on Control Systems Technology 05/2015; 23(3):1003-1013. DOI:10.1109/TCST.2014.2360498 · 2.52 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition. Then, an algebraic Riccati equation method is given to determine if a system is negative imaginary. Also, a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property. The controller synthesis procedure is based on the proposed negative imaginary lemma. Using this procedure, the closed-loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty, such as in the case of unmodeled spill-over dynamics in a lightly damped flexible structure. A numerical example is presented to illustrate the usefulness of the results.
Systems & Control Letters 03/2015; 77:63-68. DOI:10.1016/j.sysconle.2015.01.008 · 1.89 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper studies the negative imaginary properties of descriptor linear systems based on state-space realizations. Under the assumption of a minimal realization, necessary and sufficient conditions are established to characterize the negative imaginary properties of descriptor systems in terms of linear matrix inequalities with equality constraints. In particular, a negative imaginary lemma, a strict negative imaginary lemma and a lossless negative imaginary lemma are developed. A multiple-input and multiple-output RLC circuit network is used as an illustrative example to validate the developed theory.
IEEE Transactions on Automatic Control 01/2015; DOI:10.1109/TAC.2015.2444233 · 3.17 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we lay the foundations of a not necessarily rational negative
imaginary systems theory and its relations with positive real systems theory
and, hence, with passivity. In analogy with the theory of positive real
functions, in our general framework, negative imaginary systems are defined in
terms of a domain of analyticity of the transfer function and of a sign
condition that must be satisfied in such domain. In this way, on the one hand,
our theory does not require to restrict the attention to systems with rational
transfer function and, on the other hand | just by suitably selecting the
domain of analyticity to be either the right half complex plane or the
complement of the unit disc in the complex plane | we particularize our theory
to both continuous-time and to discrete-time systems. Indeed, to the best of
our knowledge, this is first time that discrete-time negative imaginary systems
are studied in the literature. In this work, we also aim to provide a unitary
view of the different notions that have appeared so far in the literature
within the framework of positive real and in the more recent theory of negative
imaginary systems, and to show how these notions are characterized and linked
to each other.
A stability analysis result for the interconnection of discrete-time systems
is also derived.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we present characterisations of linear, shift-invariant, discrete-time systems that exhibit mixtures of small gain-type properties and positive real-type behaviours in a certain manner. These “mixed” systems are already fairly well characterised in the continuous-time domain, but the widespread adoption of digital controllers makes it necessary to verify whether commonly used discretisation procedures preserve the characteristic of “mixedness”. First, we analyse the effects of classical discretisation methods on the “mixed” property using Nyquist methods. A frequency domain feedback stability result is then presented. Finally, we develop a spectral-based characterisation of “mixed” discrete-time systems which provides a practical computational test that can also be applied to the MIMO case. Several examples validate the developed theory.
European Journal of Control 09/2014; 20(5). DOI:10.1016/j.ejcon.2014.07.002 · 0.79 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given a linear time-invariant plant, the search for a suitable multiplier over the class of Zames–Falb multipliers is a challenging problem which has been studied for several decades. Recently, a new linear matrix inequality search has been proposed over rational and causal Zames–Falb multipliers. This letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers. The addition of a Popov multiplier to the anticausal Zames–Falb multiplier is implemented by analogy with the causal search. As a result, a search over a noncausal subset of Zames–Falb multipliers is obtained. A comparison between all the search methods proposed in the literature is given.
Systems & Control Letters 08/2014; 70:17–22. DOI:10.1016/j.sysconle.2014.05.005 · 1.89 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper proposes a two degree of freedom control using a combined feedforward/feedback architecture for MIMO nanopositioning stages. The proposed control system provides higher bandwidth and better performance compared with a single degree of freedom feedback controller. The paper proposes a systematic synthesis methodology to design the controller based on closed loop performance. The results are verified via simulation and hardware experiment.
[Show abstract][Hide abstract] ABSTRACT: The negative imaginary (NI) property is exhibited by many systems such as flexible structures with force actuators and position sensors and can be used to prove the robust stability of flexible structure control systems. In this paper, we derive methods to check for the NI and strict negative imaginary (SNI) properties in both the single-input single-output as well as multi-input multi-output cases. The proposed methods are based on spectral conditions on a corresponding Hamiltonian matrix obtained for a given system transfer function matrix. Under certain conditions, a given transfer function matrix satisfies the NI property if and only if the corresponding Hamiltonian matrix has no pure imaginary eigenvalues with odd multiplicity. It is also shown that a given transfer function matrix satisfies the SNI property if and only if the corresponding Hamiltonian matrix has no eigenvalues on the imaginary axis, except at the origin. The results of this paper are applied to check the NI property in two nanopositioning applications.
[Show abstract][Hide abstract] ABSTRACT: Recent results in equivalence between classes of multipliers for slope-restricted nonlinearities are extended to multipliers for bounded and monotone nonlinearities. This extension requires a slightly modified version of the Zames–Falb theorem and a more general definition of phase-substitution. The results in this paper resolve apparent contradictions in the literature on classes of multipliers for bounded and monotone nonlinearities.
Systems & Control Letters 04/2014; 66:65–71. DOI:10.1016/j.sysconle.2014.01.003 · 1.89 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, the problem of designing a control law in case of rotor failure in quadrotor vehicles is addressed. First, a nonlinear mathematical model for a quadrotor vehicle is derived, which includes translational and rotational dynamics. Then a robust feedback linearization controller is developed, which sacrifices the controllability of the yaw state due to rotor failure to linearize the closed-loop system around a working point, where roll and pitch angles are zero and the angular speed around the vertical axis is a nonzero constant. An H∞ loop shaping technique is adopted to achieve regulation of these variables around the chosen working point. Finally, an outer loop is proposed for achieving control of the linear displacement under the assumption of small angles approximation for the pitch and roll angles. The proposed control strategy allows the vehicle to use the remaining three functional rotors to enter a constant angular speed around its vertical axis, granting stability and representing an effective way to deal with a rotor failure in quadrotor vehicles
Read More: http://arc.aiaa.org/doi/abs/10.2514/1.59869?journalCode=jgcd
Journal of Guidance Control and Dynamics 03/2014; 37(2):580-591. DOI:10.2514/1.59869 · 1.15 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper presents a survey of recent results on the theory of negative
imaginary systems. This theory can be applied to the robust control of large
flexible structures with colocated force actuators and position sensors.
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with the control design of nonlinear systems using feedback linearisation. The paper highlights the destabilisation effect of unmodelled actuator dynamics when applying feedback linearisation. To overcome this difficulty, a two stage feedback linearisation technique is proposed to compensate for actuator dynamics and subse- quently linearise nonlinear systems. A case study of a tri-rotor UAV is used to showcase the benefits of the proposed method in comparison with classical feedback linearisation. The paper is written from a UAV application's point of view, however, the proposed procedure is still valid for any input-affine invertible nonlinear system.
2013 IEEE 52nd Annual Conference on Decision and Control (CDC); 12/2013
[Show abstract][Hide abstract] ABSTRACT: Flexible structures with collocated force actuators and position sensors lead to negative imaginary dynamics. However, in some cases, the mathematical models obtained for these systems, for example, using system identification methods may not yield a negative imaginary system. This paper provides two methods for enforcing negative imaginary dynamics on such mathematical models, given that it is known that the underlying dynamics ought to belong to this system class. The first method is based on a study of the spectral properties of Hamiltonian matrices. A test for checking the negativity of the imaginary part of a corresponding transfer function matrix is first developed. If an associated Hamiltonian matrix has pure imaginary axis eigenvalues, the mathematical model loses the negative imaginary property in some frequency bands. In such cases, a first-order perturbation method is proposed for iteratively collapsing the frequency bands whose negative imaginary property is violated and finally displacing the eigenvalues of the Hamiltonian matrix away from the imaginary axis, thus restoring the negative imaginary dynamics. In the second method, direct spectral properties of the imaginary part of a transfer function are used to identify the frequency bands where the negative imaginary properties are violated. A pointwise-in-frequency scheme is then proposed to restore the negative imaginary system properties in the mathematical model.
International Journal of Control 07/2013; 86(7). DOI:10.1080/00207179.2013.804951 · 1.14 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Different classes of multipliers have been proposed in the literature for obtaining stability criteria using passivity theory, integral quadratic constraint (IQC) theory or Lyapunov theory. Some of these classes of multipliers can be applied with slope-restricted nonlinearities. In this paper the concept of phase-containment is defined and it is shown that several classes are phase-contained within the class of Zames–Falb multipliers. There are two main consequences: firstly it follows that the class of Zames–Falb multipliers remains, to date, the widest class of available multipliers for slope-restricted nonlinearities; secondly further restrictions may be avoided when exploiting the parametrization of the other classes of multipliers.