A polynomial approach to the realization of J-lossless behaviours

In this paper, a class of behaviours known as J-lossless behaviours is introduced, where J is a symmetric two-variable polynomial matrix. For a certain J, it is shown that the resulting set of J-lossless behaviours are SISO behaviours such that for each of such behaviours, there exists a quadratic differential form which is positive for nonzero trajectories of the behaviour and whose derivative is equal to the product of the input variable and the derivative of the output variable. Earlier, Van der Schaft and Oeloff had considered a specific form of realization for such behaviours that plays an important role in their model reduction procedure. In our paper, we give a method of computation of a state space realization from a transfer function of such a behaviour in the same form as considered by Van der Schaft and Oeloff, using polynomial algebraic methods. Apart from being useful in enlarging the scope of the model reduction procedure of Van der Schaft and Oeloff, we show that our method of realization also has application in the synthesis of lossless mechanical systems with given transfer functions using springs and masses

A polynomial approach to the realization of J-lossless behaviors

We consider the problem of realizing lossless behaviours with respect to the supply rate equal to the scalar product of the input and of the derivative of the output variables. Using polynomial algebraic method we devise a realization procedure which, starting from an image representation, yields the same state representation used by van der Schaft and Oeloff in the context of model reduction. We also apply the insights derived from this realization procedure to the synthesis of lossless mechanical systems with given transfer functions using springs and masses

Dissipative systems: uncontrollability, observability and RLC realizability

The theory of dissipativity has been primarily developed for controllable systems/behaviors. For various reasons, in the context of uncontrollable systems/behaviors, a more appropriate definition of dissipativity is in terms of the dissipation inequality, namely the {\em existence} of a storage function. A storage function is a function such that along every system trajectory, the rate of increase of the storage function is at most the power supplied. While the power supplied is always expressed in terms of only the external variables, whether or not the storage function should be allowed to depend on unobservable/hidden variables also has various consequences on the notion of dissipativity: this paper thoroughly investigates the key aspects of both cases, and also proposes another intuitive definition of dissipativity. We first assume that the storage function can be expressed in terms of the external variables and their derivatives only and prove our first main result that, assuming the uncontrollable poles are unmixed, i.e. no pair of uncontrollable poles add to zero, and assuming a strictness of dissipativity at the infinity frequency, the dissipativities of a system and its controllable part are equivalent. We also show that the storage function in this case is a static state function. We then investigate the utility of unobservable/hidden variables in the definition of storage function: we prove that lossless autonomous behaviors require storage function to be unobservable from external variables. We next propose another intuitive definition: a behavior is called dissipative if it can be embedded in a controllable dissipative {\em super-behavior}. We show that this definition imposes a constraint on the number of inputs and thus explains unintuitive examples from the literature in the context of lossless/orthogonal behaviors.Comment: 26 pages, one figure. Partial results appeared in an IFAC conference (World Congress, Milan, Italy, 2011

Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion

Negative imaginary (NI) systems play an important role in the robust control of highly resonant flexible structures. In this paper, a generalized NI system framework is presented. A new NI system definition is given, which allows for flexible structure systems with colocated force actuators and position sensors, and with free body motion. This definition extends the existing definitions of NI systems. Also, necessary and sufficient conditions are provided for the stability of positive feedback control systems where the plant is NI according to the new definition and the controller is strictly negative imaginary. The stability conditions in this paper are given purely in terms of properties of the plant and controller transfer function matrices, although the proofs rely on state space techniques. Furthermore, the stability conditions given are independent of the plant and controller system order. As an application of these results, a case study involving the control of a flexible robotic arm with a piezo-electric actuator and sensor is presented

A theory of passive linear systems with no assumptions

This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordWe present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.This research was conducted in part during a Fellowship supported by the Cambridge Philosophical Society , http://www.cambridgephilosophicalsociety.org

Stability analysis and vibration control of a class of negative imaginary systems

This paper presents stability analysis and vibration control of a class of negative imaginary systems. A flexible manipulator that moves in a horizontal plane is considered and is modelled using the finite element method. The system with two poles at the origin is shown to possess negative imaginary properties. Subsequently, an integral resonant controller (IRC) which is a strictly negative imaginary controller is designed for the position and vibration control of the system. Using the IRC, the closed-loop system is observed to be internally stable and simuation results show that satisfactory hub angle response is achieved. Furthermore, vibration magnitudes at the resonance modes are suppressed by 48 dB