Constant Gain Output Feedback Research Articles

Transition control of the blasius boundary layer using linear robust control theory

The paper considers control system design for linearized three-dimensional perturbations about a nominal laminar boundary layer over a flat plate (the Blasius profile). The objective is prevention of the laminar to turbulent transition using appropriate inputs, outputs, and feedback controllers. They are synthesized with a view to reducing transient energy growth, a known precursor to important transition scenarios. The linearized Navier–Stokes equations are reduced to the Orr–Sommerfeld and Squire equations with wall-normal velocity actuation entering through the boundary conditions on the wall. The sensor output is taken to be the wall-normal derivative of the wall-normal vorticity measured on the plate. Several multivariable output controllers are examined, including simple constant gain output feedback, loop transfer recovery, and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\infty }$$\\end{document} loop shaping. Reduced order compensators are developed using balanced truncation and analyzed for robustness using the gap metric between reduced order models and full order models. It is demonstrated that the level of minimum transient energy growth that can be achieved is similar for these diverse controller methodologies but falls short of that which can be achieved using optimal state feedback.

Laminar–Turbulent Transition Control Using Passivity Analysis of the Orr–Sommerfeld Equation

The control problem for linearized two-dimensional wall-bounded parallel shear flows is considered as a means of preventing the laminar-to-turbulent transition. The linearized Navier–Stokes equations are reduced to the Orr–Sommerfeld equation with wall-normal velocity actuation entering through the boundary conditions on the wall. An analysis of the work-energy balance is used to identify appropriate sensor outputs that could lead to a passive closed-loop system using simple feedback laws. These sensor outputs correspond to the second spatial derivative (normal direction) of the streamwise velocity at the wall, and it is demonstrated that they can be realized by pressure measurements made at appropriate locations along the wall. Spatial discretization of the Orr–Sommerfeld equation in the wall-normal direction is accomplished using Hermite cubic finite elements, and the resulting spectrum is shown to agree with literature values for both plane Poiseuille flow and the Blasius boundary layer. Both cases are shown to be stabilized by simple constant-gain output feedback using the special choice of sensor measurement. In each case, this holds for a single gain and a wide range of Reynolds numbers and wave numbers. However, the closed-loop system produced in the Poiseuille case is shown to be passive (i.e., it has a positive-real transfer function), whereas in the Blasius case, it is nonminimum-phase and hence is not passive.

A behavioral approach to the control of discrete linear repetitive processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next, a notion of eigenvalues is defined for the repetitive process under an action of a closed-loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed-loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on these processes.

Output Feedback Control of the Nonlinear Aeroelastic Response of a Slender Wing

This paper presents the design of optimal constant gain output feedback based controllers for a nonlinear aeroelastic system. Controllers are designed for flutter suppression as well as gust-load alleviation. This controller architecture is one of the simplest, using direct feedback of the sensor outputs, but its performance is highly dependent on sensor selection and placement. Also, optimal design of such controllers require an accurate knowledge of the expected disturbance mode and gust spectrum. This paper presents results pertaining to the performance of SOF controllers for aeroelastic control (linear and nonlinear) and compares it to that of LQR and LQG controllers. Controllers are designed for various sensor placement. The gain and phase margins of the various controllers are also presented to understand the robustness characteristics. For optimal sensor placement and with knowledge of the disturbance, the constant gain output feedback controller performance and robustness was found to be equivalent to that of linear quadratic regulator and linear quadratic Gaussian controllers for the example considered. These controllers were also shown to be very easy to alter and combine with other controllers/filters for better overall system response.

Robust parallel compensator design for output feedback stabilization of plants with structured uncertainty

This paper deals with a design problem of robust parallel compensator which guarantees the existence of a constant gain output feedback controller for plants with structured uncertainty. First, robust design conditions of such a compensator are derived by using Kharitonov’s Theorem. Then, based on conditions provided, the parallel compensator is designed by utilizing the maximal range of interval perturbations of the plant. A numerical example is given to demonstrate the validity of the results.

Control of Sound Transmission through a Convected Fluid Loaded Plate with Piezoelectric Sensoriactuators

The use of piezoceramic transducers to control sound transmission through a convected fluid loaded plate is analytically investigated. The investigated system consists of an elastic plate subjected to convected fluid loading (i.e., aeroelasticity) on one side and coupled to an otherwise rigid acoustic enclosure on the other side. The control of sound transmission from the convected fluid side to the cavity is achieved by applying constant gain output feedback using piezoceramic transducers as sensoriactuators. A sensoriactuator is a single transducer which simultaneously acts as a sensor and actuator, thus providing significant stability and robustness advantages. Results demonstrate that such a control system can significantly reduce the transmitted sound. However, variations in the velocity of the convected fluid can reduce the effectiveness of such a control system.

Approximate recovery of H-infinity loop shapes using fixed-order dynamic compensation

This paper presents a method for designing fixed-order dynamic compensators that approximate the Hm full state feedback, closed-loop transfer function properties from disturbance inputs to controlled outputs. This formulation uses an observer canonical form to represent the dynamic compensator which allows the design to be treated as a modified constant gain output feedback problem. The approximate recovery is accomplished through a unique selection of the quadratic performance index weighting matrices. This design procedure is demonstrated by two design examples. The first example is a simple fourth-order model used to demonstrate the procedure. The second example is a longitudinal flight controller for the F-18/HARV Supermaneuverable aircraft. This flight controller is compared to both a full- and reduced-order H^ compensator.

Robust eigenstructure assignment with structured state-space uncertainty and unmodelled dynamics

New time domain stability robustness results have been obtained for a non-strictly proper linear time-invariant plant which is subject to simultaneous time-varying structured state-space uncertainly and norm bounded unmodelled dynamics. Sufficient conditions for robust stability are obtained both for a constant gain output feedback controller and an output feedback dynamic compensator. The robustness conditions are explicitly in terms of the closed-loop eigenvalues and eigenvectors so that these new results are especially well suited for a robust eigenstructure assignment design method. An example is presented of a robust eigenstructure assignment autopilot using dynamic compensation for the yaw/roll dynamics of a bank-to-turn missile.

Optimal output feedback for linear time-periodic systems

An approach is developed for applying optimal output feedback control theory to the design of fixed gain controllers for time-periodic systems. Constant feedback gains based on plant outputs are calculated by minimizing a linear quadratic performance cost functional. A solution algorithm is developed using Floquet- Lyapunov theory and a generalized harmonic expansion technique. The theory is applied to the control of a helicopter rotor blade in forward flight through individual blade control. It is shown that constant gain output feedback can be used to augment the stability of the resulting time-periodic system, with the potential for greatly reducing the implementation complexity. HE primary purpose of this paper is to present a general framework for applying output feedback techniques to linear time-periodic systems. The motivation for such an ap- proach is, in part, to reduce the control implementation re- quirements. The use of constant gain optimal output feedback in designing active feedback controllers provides many advan- tages over existing linear quadratic Gaussian (LQG) design methods. In particular, an on-line observer is not required, and the method can be extended to include compensator de- sign wherein the order of the compensator may be prespecified during the design process. Unneeded feedback channels may also be eliminated. 1 The number of sensors may be reduced while eliminating the need to estimate all of the unmeasurable states in real time. Recently, efficient compensator design formulations have been developed for linear time-invariant systems including robustness considerations for both unstruc- tured uncertainty and parametric uncertainty.2'3 The potential advantages of designing a fixed-order linear time-invariant controller are especially appealing for systems whose equations of motion are characterized by linear peri- odic coefficient dynamics. Aerospace applications include he- licopter forward flight stability and control and satellite or- bital stability problems. For helicopter aeromechanical and aeroelastic stability and control applications, the number of states required to model the system accurately continues to increase as research in structural elasticity, dynamic inflow, and aeroelasticity progresses. However, the feasibility and cost effectiveness of measuring/est imating all of the higher frequency structural elastic and dynamic inflow positions and rates for helicopter rotors in real time in an operational envi- ronment is questionable. This also drives the need for poten- tial alternatives to the full state feedback solution. An output feedback approach provides additional flexibility in that higher fidelity state space physical models may be used for the

Pole assignment for multi-input multi-output systems using output feedback

In this paper the problem of pole assignment using constant gain output feedback is studied for MIMO system with system order n > m + l − 1, where m and l are the number of inputs and outputs, respectively. A new procedure is presented to design a constant gain output feedback matrix which assigns ( m + l − 2) poles exactly to the desired locations and shifts all the unassigned poles to suitable locations using root locus techniques.

Pole assignment with constraints using output feedback

In this paper the pole assignment with constant gain output feedback is studied for a SIMO System. A reduced order system with the set of unassigned poles satisfying the constraint conditions, obtained by removing the polynomial with the set of assigned poles, simplifies the design procedure. The root locus method is adopted for the proper fixation of the feedback gain.

A General Design Approach for Optimal Output Feedback Control

Abstract This paper considers the problem of designing an optimal constant gain output feedback and optimal observers for linear, time-invariant control systems. A general optimization criterion for output feedback control is introduced, which is symmetric with respect to the system input u and the output y . This symmetry can be achieved by introducing measurement disturbances in form of delta-impulses. The first step in developing this procedure is the definition of the criterion in a deterministic form. Furthermore, one defines a corresponding stochastic form, which leads to the same optimal regulator. As an application of the new general feedback problem a design procedure for an optimal observer in analogy to the Kalman filter can be shown. The results of this paper will be illustrated by an example.

Fresh look into the design and computation of optimal output feedback controls for linear multivariable systems

This paper presents as its main result a new and efficient method for the design of optimal dynamic compensators for linear multivariable control systems. The discussion starts with a fresh look into the design of optimal constant gain output feedback. In this context, three new ideas are introduced that prove their usefulness particularly in the dynamic compensator case: (1) deterministic measurement disturbances are included in the formulation of the underlying optimization problem, (2) the relationship between a deterministic and a stochastic optimal control problem is shown, (3) by proper modifications of the quadratic cost function, the effort involved in the numerical solution of the optimal output feedback problem is considerably reduced. Furthermore, a canonical form is introduced for the dynamic compensator and an algorithm is developed for the numerical computation of its free parameters. It is based on quasi-newtonian gradient methods and an efficient line search procedure. Some features of the new design method are demonstrated by means of a non-trivial benchmark taken from the literature.

Two-time scale stabilization of systems with output feedback

The problem of constant gain output feedback regulator design for linear systems with ill-conditioned dynamics is considered in the context of singular perturbation theory. A design approach is developed in which gains can be separately calculated to stabilize reduced-order slow and fast subsystem models. By employing the notion of combined control and observation spillover suppression, conditions are derived assuring that these gains will stabilize the full-order system, assuming sufficient frequency separation between the slow and fast subsystems. An LQ design procedure is described in which the spillover suppression conditions are satisfied by adjoining penalty functions to the subsystem performance indices. The theory is demonstrated in a controller design for a flexible space structure.

Robust stabilization of singularly perturbed systems

We discuss the problem of designing stabilizing controllers for singularly perturbed systems on the basis of simplified models. In [1], it was shown that a constant gain output feedback controller designed on the basis of the simplified model need not stabilize the ‘true’ system containing both fast and slow modes. This phenomenon was then expanded to include the case where the simplified system is strictly proper in [2]. The objectives of this note are threefold: (i) to show that, given any proper system and any stabilizing controller for it that is proper but not strictly proper, there exists a singular perturbation of the system that is destabilized by that controller, (ii) to show that any strictly proper controller for a singularly perturbed system designed on the basis of a reduced order model will stabilize the true system for sufficiently small values of the fast dynamics parameter, and (iii) to provide a characterization, in the same spirit as [3,4], of the set of all strictly proper controllers that stabilize a given proper plant. By combining these results, it is possible to generate the class of all robustly stabilizing controllers for a given singularly perturbed system.

Stabilizability of multivariable systems and the Ljusternik—Šnirel'mann category of real Grassmannians

In this paper a numeric criterion, k( m, p) ⩾ n, is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs, n states, and p outputs. This criterion is defined in terms of a topological invariant of the space of gains, both finite and infinite, arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space. This invariant, originally considered by Ljusternik and Šnirel'mann in the calculus of variations, can be estimated from below by using a theorem of Eilenberg together with the methods of the Schubert calculus, thus leading to some very explicit corollaries concerning generic stabilizability. As far as I am aware, these corollaries cannot be derived from existing results concerning pole-assignability. After passing to a problem on coverings of the (compactified) gain space, the main technical problem which remains in a high gain stability analysis — stated here as a High Gain Lemma — which appears to be of independent interest. That is, the topological argument implies that the closed-loop system is ‘stable’ for some gain, possibly infinite. If the root-locus map were defined and continuous at all infinite gains, then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument. However, it is known that if mp > n then the root-locus map has points of discontinuity at infinity, and, since mp ⩾ n is now known to be necessary for generic stabilizability, in most cases of interest one requires a far more subtle argument than one would give, for example, in the scalar case.

Eigenvector assignment using output feedback

The application of constant-gain output feedback to arbitrarily assign eigenvectors for a system with a given set of eigenvalues is investigated. Necessary and sufficient. conditions for the existence of an output-feedback gain matrix for a given eigen-structure are developed. The allowable eigenspaces are thus characterized and the design procedure is illustrated through numerical examples.

Scalar Output Feedback in Linear Multivariable Systems

A complete new resolution is presented to the question of what changes occur to the individual transfer matrix elements of a linear multivariable system under local, scalar output feedback. In particular, it is first shown what poles become controllable and observable via any input/output pair when constant gain output feedback is applied between any (i-th) output, Yi and any (j-th) input, uj . The changes which simultaneously occur to the numerator elements of the transfer matrix are then determined through the employment of some new relationships derived from any appropriate relatively right prime factorization, R(s)P(s)-1, of T(s), the system transfer matrix. Certain extensions of this work are discussed along with its relevance to decentralized control.

Output feedback decoupling in the frequency domain†

For plants with invertible transfer matrices, now necessary and sufficient conditions are given for decoupling by constant gain output feedback and by dynamic output feedback. If the latter are not met, a dynamic pre-compensator is required. Design techniques associated with the approach are discussed.

Output feedback decoupling of multivariable systems

Necessary and sufficient conditions for output feedback decoupilng of linear time-invariant multivariable systems into single input-multioutput subsystems are derived in frequency domain. The conditions imply a new constructive method for determining a constant gain output feedback matrix and some of its elements can be freely chosen to reassign the closed-loop poles.