[Show abstract][Hide abstract] ABSTRACT: Bicoprime factorisations can be considered a generalisation of the well known coprime factorisations. This paper deals with the possible applications of such factorisations in control theory. The concept of minimal dimension bicoprime factorisations is introduced and shown to be especially beneficial in the case of normal rank deficient and redundant control systems. Some methods of obtaining a bicoprime factorisation for a plant are given based on state-space data, right or left coprime factorisations and left or right standard factorisations. Finally, bicoprime factor uncertainty is characterised and shown to have an interesting structure closely resembling the standard four-block uncertainty.
[Show abstract][Hide abstract] ABSTRACT: This paper presents a consensus-based robust cooperative control framework for a wide class of linear time-invariant (LTI) systems, namely Negative-Imaginary (NI) systems. Output feedback, dynamic, Strictly Negative-Imaginary (SNI) controllers are applied in positive feedback to heterogeneous multi-input-multi-output (MIMO) plants through the network topology to achieve robust output feedback consensus. Robustness to external disturbances and model uncertainty is guaranteed via NI system theory. Cooperative tracking control of networked NI systems is presented as a corollary of the derived results by adapting the proposed consensus algorithm. Numerical examples are also given to demonstrate the effectiveness of proposed robust cooperative control framework.
[Show abstract][Hide abstract] ABSTRACT: A robust output feedback consensus problem for networked homogeneous negative-imaginary (NI) systems is investigated in this technical note. By virtue of NI systems theory, a set of reasonable yet elegant conditions are derived for output consensus under external disturbances as well as NI model uncertainty. As a byproduct, this technical note also reaffirms a previous result by Li et al. which shows the robustness of networked systems is always worse than that of single agent systems. Furthermore, the eventual convergence sets are also characterized for several special NI systems that are commonly studied in the literature. It is shown how the results in this work embed and generalize earlier results for these classes of systems. We show that the natural convergence set boils down to the centroid of the initial pattern when the initial conditions of the controllers are zero. Numerical examples are given to showcase the main results.
Full-text · Article · Sep 2015 · IEEE Transactions on Automatic Control
[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.
Full-text · Article · May 2015 · IEEE Transactions on Control Systems Technology
[Show abstract][Hide abstract] ABSTRACT: The controller synthesis problem of uncertain negative-imaginary systems has important engineering applications in, for example, lightly damped flexible structures with colocated position sensors and force actuators. This paper provides a systematic controller synthesis procedure via an LMI approach to construct a state-feedback internally stabilizing controller such that the nominal closed-loop system satisfies negative-imaginary properties and a DC gain condition. As a result of this, the closed-loop system can then be guaranteed to be robustly stable against uncertainties that are stable strictly negative-imaginary (e.g. unmodeled spill-over dynamics in a lightly damped flexible structure). An numerical example is given to show the usefulness of the proposed results.
[Show abstract][Hide abstract] ABSTRACT: Abstract—This paper proposes a systematic method to syn- thesise H1 loop shaping weights that would optimise perfor- mance while maintaining a pre-specified level of optimal robust stability margin. This objective is captured by two optimisation problems expressed in the form of Linear Matrix Inequalities (LMI), which are solved by means of an iterative algorithm. This scheme will aid the designer to gauge the achievable performance of a plant for a specified level of robustness and to adjust pre-designed weights to further enhance performance. Index Terms—H1 loop-shaping, weight synthesis, perfor- mance optimisation, robust stability margin, robust control.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we solve a class of two-block "- synthesis problems by showing that these problems can be converted to H1 synthesis problems. The "-synthesis problems considered correspond to controller design problems where plant model uncertainty and closed-loop performance are captured using the same type of dynamic uncertainty. We show that we can solve two-block "-synthesis problems when both blocks correspond to additive, inverse additive or coprime factor uncertainty (left and right). Index Terms—"-synthesis, H1 control, additive uncertainty, inverse additive uncertainty, coprime factor uncertainty.
[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.
Full-text · Article · Mar 2015 · Systems & Control Letters
[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.
No preview · Article · Jan 2015 · IEEE Transactions on Automatic Control
[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: The negative imaginary (NI) property occurs in many important applications. For instance, flexible structure systems with collocated force actuators and position sensors can be modeled as negative imaginary systems. Obtaining a mathematical model for this class of systems using system identification methods may result into inaccurate models that poorly reflect the negative imaginary property. In this paper, a modified subspace system identification algorithm that ensures the negative imaginary property is presented. As an application of these results, an example of modeling a flexible system with a piezoelectric actuator and position sensor is presented.
[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.
No preview · Article · Sep 2014 · European Journal of Control
[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.
Full-text · Article · Aug 2014 · Systems & Control Letters
[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.
No preview · Article · Jun 2014 · IEEE/ASME Transactions on Mechatronics
[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
Full-text · Article · Mar 2014 · Journal of Guidance Control and Dynamics
[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.