Andrey Smyshlyaev

Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations

In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equations well posedness and the kernels smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

Backstepping observers for a class of parabolic PDEs

In this paper we design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.

Adaptive Boundary Control for Unstable Parabolic PDEs—Part I: Lyapunov Design

We develop adaptive controllers for parabolic partial differential equations (PDEs) controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. These are the first adaptive controllers for unstable PDEs without relative degree limitations, open-loop stability assumptions, or domain-wide actuation. It is the first necessary step towards developing adaptive controllers for physical systems such as fluid, thermal, and chemical dynamics, where actuation can be only applied non-intrusively, the dynamics are unstable, and the parameters, such as the Reynolds, Rayleigh, Prandtl, or Peclet numbers are unknown because they vary with operating conditions. Our method builds upon our explicitly parametrized control formulae to avoid solving Riccati or Bezout equations at each time step. Most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree.

On control design for PDEs with space-dependent diffusivity or time-dependent reactivity

In this paper the recently introduced backstepping method for boundary control of linear partial differential equations (PDEs) is extended to plants with non-constant diffusivity/thermal conductivity and time-varying coefficients. The boundary stabilization problem is converted to a problem of solving a specific Klein-Gordon-type linear hyperbolic PDE. This PDE is then solved for a family of system parameters resulting in closed-form boundary controllers. The results of a numerical simulation are presented for the case when an explicit solution is not available.

Boundary Stabilization of a 1-D Wave Equation with In-Domain Antidamping

We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.

Arbitrary Decay Rate for Euler-Bernoulli Beam by Backstepping Boundary Feedback

We consider a problem of stabilization of the Euler-Bernoulli beam. The beam is controlled at one end (using position and moment actuators) and has the ldquoslidingrdquo boundary condition at the opposite end. We design the controllers that achieve any prescribed decay rate of the closed loop system, removing a long-standing limitation of classical ldquoboundary damperrdquo controllers. The idea of the control design is to use the well-known representation of the Euler-Bernoulli beam model through the Schrodinger equation, and then adapt recently developed backstepping designs for the latter in order to stabilize the beam. We derive the explicit integral transformation (and its inverse) of the closed-loop system into an exponentially stable target system. The transformation is of a novel Volterra/Fredholm type. The design is illustrated with simulations.

Boundary Controllers and Observers for the Linearized Schrödinger Equation

We consider a problem of stabilization of the linearized Schrodinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. How- ever, the decay rate of the closed-loop system cannot be prescribed. The second, full-state feedback boundary control design, achieves an arbitrary decay rate. We formally view the Schrodinger equa- tion as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction-advection-diffusion equations. The resulting controller is then sup- plied with the backstepping observer to obtain an output-feedback compensator. The designs are illustrated with simulations.

Rejection of sinusoidal disturbance of unknown frequency for linear system with input delay

We present a new approach for rejection of a sinusoidal disturbance of unknown frequency, bias, amplitude, and phase for a linear unstable plant with a delay in the control channel. To solve the problem, we combine the well-known predictor feedback approach with the adaptive scheme that identifies the frequency of the disturbance. Compared to the existing results, the dynamic order of our adaptive scheme is low (equal to three) and the approach applies to plants that are unstable and have an arbitrary relative degree. The results are illustrated by the numerical example.

Control of a Tip-Force Destabilized Shear Beam by Observer-Based Boundary Feedback

We consider a model of the undamped shear beam with a destabilizing boundary condition. The motivation for this model comes from atomic force microscopy, where the tip of the cantilever beam is destabilized by van der Waals forces acting between the tip and the material surface. Previous research efforts relied on collocated actuation and sensing at the tip, exploiting the passivity property between the corresponding input and output in the beam model. In this paper we design a stabilizing output-feedback controller in a noncollocated setting, with measurements at the free end (tip) of the beam and actuation at the beam base. Our control design is a novel combination of the classical “damping boundary feedback” idea with a recently developed backstepping approach. A change of variables is constructed which converts the beam model into a wave equation (for a very short string) with boundary damping. This approach is physically intuitive and allows both an elegant stability analysis and an easy selection of design parameters for achieving desired performance. Our observer design is a dual of the similar ideas, combining the damping feedback with backstepping, adapted to the observer error system. Both stability and well-posedness of the closed-loop system are proved. The simulation results are presented.

Boundary Control of the Linearized Ginzburg--Landau Model of Vortex Shedding

In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg--Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg--Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: (1) it deals with two coupled partial differential equations, and (2) under certain circumstances handles equations defined on a semi-infinite domain.