Control Canonical Forms and Eigenvalue Assignment by Feedback for a Class of Linear Hyperbolic Systems

In this paper, it is shown that by exploiting the explicit para- metric state feedback solution, it is feasible to obtain the ultimate solution to minimum sensitivity problem. A numerical algorithm for construction of a robust state feedback in eigenvalue assignment problem for a control- lable linear system is presented. By using a generalized parametric vector companion form, the problem of eigenvalue assignment with minimum sensitivity is re-formulated as an unconstrained minimization problem. The derived explicit expressions of the solutions allow minimization of the sensitivity problem by using a powerful search technique.

Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions

This article considers the design of finite-dimensional compensators for distributed-parameter systems using eigenvalue assignment. The proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs. Since the additional outputs can be asymptotically reconstructed, the compensator can be designed using the separation principle, i.e. the closed-loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control. In order to solve the corresponding eigenvalue assignment problem, the parametric approach for the design of static output feedback controllers in finite-dimensions is extended to distributed-parameter systems. By using a parameter optimisation it is possible to assign all closed-loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance. A heat conductor is used to demonstrate the proposed design procedure.

The partial eigenvalue assignment problem concerns reassigning afew of undesired eigenvalues of a linear system to suitably chosenlocations and keeping the other large number of eigenvalues andeigenvectors unchanged (no spill-over). This paper considers thepartial eigenvalue assignment problem with time delay robustness.A time delay robustness measure is presented by analyzing thesensitivity of the assigned eigenvalues with respect to timedelay. The problem is formulated as an unconstrained minimizationproblem with the cost function involving the time delay robustnessmeasure. A numerical algorithm with analytical formulation of thegradient for the cost function is provided. A numerical example isincluded to show the effectiveness of the proposed method.

Numerical algorithms are described for solving the eigenvalue assignment (EVA) problem for multi-input systems by means of state feedback. The multi-input EVA problem is first reduced to one or more single-input EVA problems where the single-input systems are in "upper Hessenberg form." An algorithm based on the well-known implicitly shifted QR algorithm is then described for solving the single-input EVA problem. Some variations on the use of these algorithms are also proposed to handle certain special cases. Numerical properties of the algorithms are discussed and examples are given to illustrate their numerical performance.

A Block Algorithm for Multi-Input Eigenvalue Assignment

Eigenvalue assignment problem of a linear scalar system with a single discrete delay is analytically and exactly solved. The existence condition of the desired eigenvalue is established when the current and delay states are present in the feedback loop. Design of the feedback controller is then followed. Furthermore, eigenvalue assignment for the input-delay system is also obtained as well. Numerical examples illustrate the procedure of assigning the desired eigenvalue.

Rotational Motion Control by Feedback with Minimum L1-Norm

In this paper a numerical efficient approach to the problem of eigenvalue assignment by constant output feedback is presented. It improves the well known Kimura’s condition by 2, i. e., it is shown that if m + p ≥ n − 1 m+p\ge n-1 generically a solution to this design problem exists where n , m n,m and p denote the dimensions of the system states, inputs and outputs, respectively. The algorithm is based on a cascaded control scheme with up to three design steps. The first two steps merely require standard methods from linear algebra while the last step only in case of m + p = n − 1 m+p=n-1 demands for the numerical solution of a system of three polynomial equations each of order two. The design procedure explicitly embodies all degrees of freedom beyond eigenvalue assignment. Thus, they can be used to account for other design it is shown goals, e. g., to minimize the spectral condition number of the closed-loop system or a norm of the feedback gain as it is shown by numerical examples from literature.

The problem of reassigning a part of the open-loop spectrum of a linear system by feedback control, leaving the rest of the spectrum invariant, is called the partial eigenvalue assignment problem. In this paper, a projection method based on a direct use of the implicitly restarted Arnoldi method for the partial eigenvalue assignment problem in linear control systems is presented. The method is of interest in common situations where the system is very large and only a few of its eigenvalues need to be assigned. Numerical examples are given to display the performance and the accuracy of the proposed algorithm.

This paper develops boundary feedback control to reduce stop-and-go oscillations in congested traffic regime. The macroscopic traffic dynamics are governed by Aw-Rascle-Zhang(ARZ) model, consisting of second-order, nonlinear partial differential equations(PDEs). The criterion to distinguish free and congested regimes is proposed for the ARZ traffic model, leading to the study of hetero-directional hyperbolic PDE control of congested traffic regime. To stabilize the oscillations of traffic density and speed in a freeway segment, the boundary input through ramp metering is considered. We solve boundary control problems for freeway segment upstream and downstream of the ramp. A general boundary control model is proposed for both control problems. We then design a straight-forward boundary control law to stabilize the downstream of the ramp metering. For the more challenging upstream control problem, we develop a full-state feedback control law through backstepping transformation. The exponential stability in $L^{2}$ sense and finite time convergence to equilibrium are achieved and validated with simulation. The key novelty of this paper is to design both upstream and downstream ramp metering boundary control for congested traffic regime.

In this paper, we propose a measurement nonlinear feedback control scheme to generate Wigner-function negativity in an optical cavity having dynamics described as a linear quantum system. In general, linear optical quantum systems can be easily constructed with reliable devices; therefore, the idea of constructing the entire system with such an optical system and nonlinear feedback is reasonable for generating Wigner-function negativity. However, existing studies have insufficiently examined the realizability or actual implementation of feedback control, which essentially requires fast responses from the sensors and actuators. In order to solve this problem, we consider the realizable feedback control of the optical phase of a pumping beam supplied to a cavity by using electro-optical modulation, which can be utilized as a fast control actuator. Then, we introduce mathematical models of the feedback-controlled system and evaluate its effect on the generation of the Wigner-function negativity by using numerical simulation. Through various numerical simulations, we show that the proposed feedback control can effectively generate the negativity of the Wigner function.

The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.

A systems theory framework is presented for the linear stabilization of two-dimensional laminar plane Poiseuille flow. The governing linearized Navier-Stokes equations are converted to control-theoretic models using a numerical discretization scheme. Fluid system poles, which are closely related to Orr-Sommerfeld eigenvalues, and fluid system zeros are computed using the control-theoretic models. It is shown that the location of system zeros, in addition to the well-studied system eigenvalues, are important in linear stability control. The location of system zeros determines the effect of feedback control on both stable and unstable eigenvalues. In addition, system zeros can be used to determine sensor locations that lead to simple feedback control schemes. Feedback controllers are designed that make a new fluid-actuator-sensorcontroller system linearly stable. Feedback control is shown to be robust to a wide range of Reynolds numbers. The systems theory concepts of modal controllability and observability are used to show that feedback control can lead to short periods of highamplitude transients that are unseen at the output. These transients may invalidate the linear model, stimulate nonlinear effects, and/or form a path of ‘bypass’ transition in a controlled system. Numerical simulations are presented to validate the stabilization of both single-wavenumber and multiple-wavenumber instabilities. Finally, it is shown that a controller designed upon linear theory also has a strong stabilizing effect on two-dimensional finite-amplitude disturbances. As a result, secondary instabilities due to infinitesimal three-dimensional disturbances in the presence of a finite-amplitude two-dimensional disturbance cease to exist.

We address the problem of adaptive outputfeedback stabilization for flow-induced vibrations of a membrane at high Mach numbers. The aeroelastic instability is compensated by using boundary control and anti-collocated measurement. The membrane is infinite in spanwise and the streamwise vibrations are modeled by a one-dimensional wave Partial Differential Equation (PDE) with an aerodynamic forcing term. Based on Piston theory, the term is represented by an anti-damping term multiplying a constant coefficient and a convective term multiplying an unknown constant coefficient. We then transform the wave PDE to a one-dimensional 2 × 2 first-order hyperbolic PDEs with constant coupling coefficients. Before introducing an adaptive output feedback controller for the hyperbolic PDEs, we present a nonadaptive explicit state feedback controller. To deal with the absence of both full-state measurements and parameter knowledge, the adaptive outputfeedback design is used; the design is based on an observer canonical form which is obtained through a change of variables and a backstepping transformation. The form enables us to design an explicit state observer. We employ gradient-based parameter estimators. For the closed loop system, we achieve convergence of the state of the wave PDE to zero. We validate our result with a simulation.