An explicit solution to the optimal LQG problem for flexible structures with collocated rate sensors

We present a class of compensators in explicit form (not requiring numerical computer calculations) for stabilizing flexible structures with collocated rate sensors. They are based on the explicit solution, valid for both Continuum and FEM Models, of the LQG problem for minimizing mean square rate. They are robust with respect to system stability (will not destabilize modes even with mismatch of parameters), can be instrumented in state space form suitable for digital controllers, and can be specified directly from the structure modes and mode 'signature' (displacement vectors at sensor locations). Some simulation results are presented for the NASA LaRC Phase-Zero Evolutionary Model - a modal Trust model with 86 modes - showing damping ratios attainable as a function of compensator design parameters and complexity

Minimum attainable RMS attitude error using co-located rate sensors

A closed form analytical expression for the minimum attainable attitude error (as well as the error rate) in a flexible beam by feedback control using co-located rate sensors is announced. For simplicity, researchers consider a beam clamped at one end with an offset mass (antenna) at the other end where the controls and sensors are located. Both control moment generators and force actuators are provided. The results apply to any beam-like lattice-type truss, and provide the kind of performance criteria needed under CSI - Controls-Stuctures-Integrated optimization

Combined structures-controls optimization of lattice trusses

The role that distributed parameter model can play in CSI is demonstrated, in particular in combined structures controls optimization problems of importance in preliminary design. Closed form solutions can be obtained for performance criteria such as rms attitude error, making possible analytical solutions of the optimization problem. This is in contrast to the need for numerical computer solution involving the inversion of large matrices in traditional finite element model (FEM) use. Another advantage of the analytic solution is that it can provide much needed insight into phenomena that can otherwise be obscured or difficult to discern from numerical computer results. As a compromise in level of complexity between a toy lab model and a real space structure, the lattice truss used in the EPS (Earth Pointing Satellite) was chosen. The optimization problem chosen is a generic one: of minimizing the structure mass subject to a specified stability margin and to a specified upper bond on the rms attitude error, using a co-located controller and sensors. Standard FEM treating each bar as a truss element is used, while the continuum model is anisotropic Timoshenko beam model. Performance criteria are derived for each model, except that for the distributed parameter model, explicit closed form solutions was obtained. Numerical results obtained by the two model show complete agreement

Active stability augmentation of large space structures: A stochastic control problem

A problem in SCOLE is that of slewing an offset antenna on a long flexible beam-like truss attached to the space shuttle, with rather stringent pointing accuracy requirements. The relevant methodology aspects in robust feedback-control design for stability augmentation of the beam using on-board sensors is examined. It is framed as a stochastic control problem, boundary control of a distributed parameter system described by partial differential equations. While the framework is mathematical, the emphasis is still on an engineering solution. An abstract mathematical formulation is developed as a nonlinear wave equation in a Hilbert space. That the system is controllable is shown and a feedback control law that is robust in the sense that it does not require quantitative knowledge of system parameters is developed. The stochastic control problem that arises in instrumenting this law using appropriate sensors is treated. Using an engineering first approximation which is valid for small damping, formulas for optimal choice of the control gain are developed

Some nonlinear damping models in flexible structures

A class of nonlinear damping models is introduced with application to flexible flight structures characterized by low damping. Approximate solutions of engineering interest are obtained for the model using the classical averaging technique of Krylov and Bogoliubov. The results should be considered preliminary pending further investigation

Application of optical distributed sensing and computation to control of large space structures

A real time holographic sensing technique is introduced and its advantages are investigated from the filtering and control point of view. A feature of holographic sensing is its capability to make distributed measurements of the position and velocity of moving objects, such as a vibrating flexible space structure. This work is based upon the distributed parameter models of linear time invariant systems, particularly including the linear oscillator equations describing the vibration of large flexible space structures. The general conclusion is that application of optical distributed sensors bring gains in the situation where Kalman filtering is necessary for state estimation. In this case, both steady state and transient filtering error covariance become smaller. This in turn results in smaller cost in the LQG problem

Reduction of boundary value problem to Possio integral equation in theoretical aeroelasticity

The present paper is the first in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. We provide a rigorous reduction of the initial boundary value problem involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions to a singular integral equation, the Possio equation. The question of its solvability will be addressed in our forthcoming work. Copyright (C) 2008 A. V. Balakrishnan and M. A. Shubov

Constrained Stabilization of Discrete-Time Systems

Based on the growth rate of the set of states reachable with unit-energy inputs, we show that a discrete-time controllable linear system is globally controllable to the origin with constrained inputs if and only if all its eigenvalues lie in the closed unit disk. These results imply that the constrained Infinite-Horizon Model Predictive Control algorithm is globally stabilizing for a sufficiently large number of control moves if and only if the controlled system is controllable and all its eigenvalues lie in the closed unit disk. In the second part of the paper, we propose an implementable Model Predictive Control algorithm and show that with this scheme a discrete-time linear system with n poles on the unit disk (with any multiplicity) can be globally stabilized if the number of control moves is larger than n. For pure integrator systems, this condition is also necessary. Moreover, we show that global asymptotic stability is preserved for any asymptotically constant disturbance entering at the plant input

Efficient load measurements using singular value decomposition

Various basic research was performed on efficient load measurement estimation techniques for aircraft structure analysis. An overview is presented of the load measurement problem. Two basic equivalent approaches to load measurement evaluations were considered. Under approach 1, the load values are modeled as depending linearly on the measured values. Under approach 2, the measured values depend linearly on the load values. By using the modern Singular Value Decomposition method, it was shown that under all conditions of the number of loads and number of gages, approach 1 is equivalent to approach 2. By using the conventional normal equation (linear regression) approach, approach 1 is only valid when the number of loads is equal to or greater than the number of gages, while approach 2 is the reverse. Furthermore, except for the case of the number of loads equals the number of gages, the load prediction formulas under the two approaches are not equivalent