Dejan M. Bošković

Boundary control of an unstable heat equation via measurement of domain-averaged temperature

In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium (stabilizing) but also the heat generation inside the rod (destabilizing). The heat generation adds a destabilizing linear term on the right-hand side of the equation. The boundary control law designed is in the form of an integral operator with a known, continuous kernel function but can be interpreted as a backstepping control law. This interpretation provides a Lyapunov function for proving stability of the system. The control is applied by insulating one end of the rod and applying either Dirichlet or Neumann boundary actuation on the other.

Backstepping control of chemical tubular reactors

Abstract In this paper, a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. A model that assumes no radial velocity and concentration gradients in the reactor, the temperature gradient described by use of a proper value of the effective radial conductivity, a homogeneous reaction, the properties of the reaction mixture characterized by average values, the mechanism of axial mixing described by a single parameter model, and the kinetics of the first order is considered. Depending on the values of the nondimensional Peclet numbers, Damkohler number, the dimensionless adiabatic temperature rise, and the dimensionless activation energy, the coupled PDE equations for the temperature and concentration can have multiple equilibria that can be either stable or unstable. The objective is to stabilize an unstable steady state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l2-norm with appropriate homogeneous boundary conditions. In the real system, the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration. The control design is accompanied by a simulation study that shows the feedback control law designed with sensing only on a very coarse grid (using just a few measurements of the temperature and concentration fields) can successfully stabilize the actual system for a variety of different simulation settings (on a fine grid).

Backstepping in Infinite Dimension for a Class of Parabolic Distributed Parameter Systems

Abstract. In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop unstable eigenvalues and is not restricted to a particular type of boundary actuation. Stabilization is achieved through the design of coordinate transformations that have the form of recursive relationships. The fundamental difficulty of such transformations is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how to design coordinate transformations such that they are sufficiently regular (not continuous but L∞). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.

Global attitude/position regulation for underwater vehicles

A nonlinear controller is designed for a 6 DOF model of an unmanned underwater vehicle (UUV) which includes both the kinematics and the dynamics. It is shown how the use of a Lyapunov function consisting of a quadratic term in the velocity (both linear and angular), a quadratic term in the position and a logarithmic term in the attitude leads to a design of a control law that achieves global asymptotic stabilization to an arbitrary set point in position/attitude. The control law is made linearly bounded by avoiding cancellation of some of the quadratic nonlinearities in the model. No information about the inertia matrix, the damping, and the Coriolis/centripetal parameters is used in the controller, endowing it with a certain amount of parametric robustness. The control law is given in terms of the modified Rodrigues parameters. An extensive simulation study shows that the proposed control law achieves excellent tracking for slowly changing trajectories, even though it is designed only for set point regulation. The nonlinear controller dramatically outperforms a linear controller.

Stabilization of a solid propellant rocket instability

A globally stabilizing feedback boundary control law for an arbitrarily fine discretization of a one-dimensional nonlinear PDE model of unstable burning in solid propellant rockets is presented. The PDE has a destabilizing boundary condition imposed on one part of the boundary. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs, properly modified to accommodate the imposed destabilizing nonlinear boundary condition at the burning end, we transform the original system into a target system that is asymptotically stable in l/sup 2/-norm with the same type of boundary condition at the burning end, and homogeneous Dirichlet boundary condition at the control end. The control design is accompanied by a simulation study that shows that the feedback control law designed using only one step of backstepping (using just two temperature measurements) can successfully stabilize the actual system for a variety of different simulation settings.

Global stabilization of a thermal convection loop

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop is presented. The loop consists of a viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled. Stability analysis of the thermal convection loop shows that the no-motion steady state for the uncontrolled case is unstable for the values of the non-dimensional Rayleigh number R/sub /spl alpha//>1. The objective is to stabilize the unstable no-motion steady state using boundary control of velocity and temperature on the outer cylinder. In our controller design we start by discretizing the original PDE model in space using a finite difference method which gives a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we obtain a discretized coordinate transformation that transforms the original coupled system into two uncoupled systems that are asymptotically stable in 12-norm with homogeneous Dirichlet boundary conditions. Using the property that the discretized coordinate transformation is smoothly invertible for an arbitrary grid choice, we conclude that the discretized version of the original system is globally asymptotically, stable and obtain nonlinear feedback boundary control laws for velocity and temperature in the original set of coordinates. The control design is accompanied by an extensive simulation study. Numerical results show that the feedback control law designed on a very coarse grid can successfully stabilize the system for a very wide range of the Rayleigh number. This means that an excellent closed loop performance is achieved using just a few measurements of the flow and temperature fields implying that the proposed backstepping design has a potential to be successfully applied in a real experimental setting.

BOUNDARY CONTROL OF CHEMICAL TUBULAR REACTORS

Abstract In this paper a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. The objective is to stabilize an unstable steady-state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l 2 –norm with appropriate homogeneous boundary conditions. In the real system the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration.

BOUNDARY CONTROL OF A CLASS OF UNSTABLE PARABOLIC PDESVIA BACKSTEPPING

Abstract In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop unstable eigenvalues and is not restricted to a particular type of boundary actuation. The stabilization is achieved through the design of coordinate transformations that have the form of recursive relationships. The fundamental difficulty of such transformations is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as grid becomes infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how to design coordinate transformations such that they are sufficiently regular (not continuous but L ∞ ). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.