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

A robust momentum management and attitude control system for the space station

A game theoretic controller is synthesized for momentum management and attitude control of the Space Station in the presence of uncertainties in the moments of inertia. Full state information is assumed since attitude rates are assumed to be very assurately measured. By an input-output decomposition of the uncertainty in the system matrices, the parameter uncertainties in the dynamic system are represented as an unknown gain associated with an internal feedback loop (IFL). The input and output matrices associated with the IFL form directions through which the uncertain parameters affect system response. If the quadratic form of the IFL output augments the cost criterion, then enhanced parameter robustness is anticipated. By considering the input and the input disturbance from the IFL as two noncooperative players, a linear-quadratic differential game is constructed. The solution in the form of a linear controller is used for synthesis. Inclusion of the external disturbance torques results in a dynamic feedback controller which consists of conventional PID (proportional integral derivative) control and cyclic disturbance rejection filters. It is shown that the game theoretic design allows large variations in the inertias in directions of importance

Robust control of systems with real parameter uncertainty and unmodelled dynamics

Two significant contributions have been made during this research period in the research 'Robust Control of Systems with Real Parameter Uncertainty and Unmodelled Dynamics' under NASA Research Grant NAG-1-1102. They are: (1) a fast algorithm for computing the optimal H(sub infinity) norm for the four-block, the two block, or the one-block optimal H(sub infinity) optimization problem; and (2) a construction of an optimal H infinity controller without numerical difficulty. In using GD (Glover and Doyle) or DGKF (Doyle, Glover, Khargonekar, and Francis) approach to solve the standard H infinity norm which required bisection search. In this research period, we developed a very fast iterative algorithm for this computation. Our algorithm was developed based on hyperbolic interpolations which is much faster than any existing algorithm. The lower bound of the parameter, gamma, in the H infinity Riccati equation for solution existence is shown to be the square root of the supremum over all frequencies of the maximum eigenvalue of a given transfer matrix which can be computed easily. The lower band of gamma such that the H infinity Riccati equation has positive semidefinite solution can be also obtained by hyperbolic interpolation search. Another significant result in this research period is the elimination of the numerical difficulties arising in the construction of an optimal H infinity controller by directly applying the Glover and Doyle's state-space formulas. With the fast iterative algorithm for the computation of the optimal H infinity norm and the reliable construction of an optimal H infinity controller, we are ready to apply these tools in the design of robust controllers for the systems with unmodelled uncertainties. These tools will be also very useful when we consider systems with structured uncertainties

A Characterization of all Solutions to the Four Block General Distance Problem

All solutions to the four block general distance problem which arises in H^∞ optimal control are characterized. The procedure is to embed the original problem in an all-pass matrix which is constructed. It is then shown that part of this all-pass matrix acts as a generator of all solutions. Special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all-pass transfer functions. As an application, necessary and sufficient conditions are found for the existence of an H^∞ optimal controller. Following that, a descriptor representation of all solutions is derived