Adaptive boundary control for unstable parabolic PDEs—Part III: Output feedback examples with swapping identifiers

We develop output-feedback adaptive controllers for two benchmark parabolic PDEs motivated by a model of thermal instability in solid propellant rockets. Both benchmark plants are unstable, have infinite relative degree, and are controlled from the boundary. One plant has an unknown parameter in the PDE and the other in the boundary condition. In both cases the unknown parameter multiplies the measured output of the system, which is obtained with a boundary sensor located on the “opposite side” of the domain from the actuator. In comparison with the Lyapunov output-feedback adaptive controllers in Krstic and Smyshlyaev [(2005). Adaptive boundary control for unstable parabolic PDEs—Part I: Lyapunov design. IEEE Transactions on Automatic Control, submitted for publication], the controllers presented here employ much simpler update laws and do not require a priori knowledge about the unknown parameters. We show how our two benchmarks examples can be combined and illustrate the adaptive stabilization design by simulation.     

Adaptive control of PDEs

This paper presents several recently developed techniques for adaptive control of PDE systems. Three different design methods are employed—the Lyapunov design, the passivity-based design, and the swapping design. The basic ideas for each design are introduced through benchmark plants with constant unknown coefficients. It is then shown how to extend the designs to reaction-advection-diffusion PDEs in 2D. Finally, the PDEs with unknown spatially varying coefficients and with boundary sensing are considered, making the adaptive designs applicable to PDE systems with an infinite relative degree, infinitely many unknown parameters, and open loop unstable.     

Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.     

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.     

Output-feedback adaptive control of a wave PDE with boundary anti-damping

We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation on one boundary and with an unknown anti-damping term on the opposite boundary. This model is representative of a torsional stick–slip instability in drillstrings in deep oil drilling, as well as of various acoustic instabilities. The key feature of the proposed controller is that it requires only the measurements of boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements, with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ODE. This enables us to employ a prediction-based design for systems with input delays, suitably converted to the adaptive output-feedback setting. The result’s relevance is illustrated with simulation example.     

Control of 1D parabolic PDEs with Volterra nonlinearities, Part II: Analysis

For a class of stabilizing boundary controllers for nonlinear 1D parabolic PDEs introduced in a companion paper, we derive bounds for the gain kernels of our nonlinear Volterra controllers, prove the convergence of the series in the feedback laws, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation. Using the inverse, we show L² and H¹ exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example.     

Stabilization of uncertain boundary control systems

We consider a class of linear infinite-dimensional systems subject to boundary uncertainties, which are linear and time-invariant. Using a deterministic approach to the design of stabilizing feedback controls, uniform exponential stability of the zero state is achieved. Lyapunov techniques are combined with Datko or Ichikawa theory. Three illustrative examples are presented.     

Adaptive boundary control of a flexible marine installation system

In this paper, boundary control of a marine installation system is developed to position the subsea payload to the desired set-point and suppress the cable’s vibration. Using Hamilton’s principle, the flexible cable coupled with vessel and payload dynamics is described as a distributed parameter system with one partial differential equation (PDE) and two ordinary differential equations (ODEs). Adaptive boundary control is proposed at the top and bottom boundaries of the cable, based on Lyapunov’s direct method. Considering the system parametric uncertainty, the boundary control schemes developed achieve uniform boundedness of the steady state error between the boundary payload and the desired position. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. Simulations are provided to illustrate the applicability and effectiveness of the proposed control.     

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.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.     

Adaptive identification and control algorithms for nonlinear bacterial growth systems

This paper suggests how nonlinear adaptive control of nonlinear bacterial growth systems could be performed. The process is described by a time-varying nonlinear model obtained from material balance equations. Two different control problems are considered: substrate concentration control and production rate control. For each of these cases, an adaptive minimum variance control algorithm is proposed and its effectiveness is shown by simulation experiments. A theoretical proof of convergence of the substrate control algorithm is given. A further advantage of the nonlinear approach of this paper is that the identified parameters (namely the growth rate and a yield coefficient) have a clear physical meaning and can give, in real time, a useful information on the state of the biomass.     

The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control

The stabilization of a one-dimensional wave equation with non-collocated observation at its unstable free end and control at another end is considered. The controller comprises a state estimator which is designed in the case where the velocity is not available. The method of “backstepping” is adopted in our design of the feedback law. We use the theory of C₀-semigroups and Lyapunov functionals to prove the strong stability of the resulting closed-loop system.     

A boundary control for motion synchronization of a two-manipulator system with a flexible beam

A novel synchronization motion control method is proposed in this paper for the system in which two manipulators are constrained by a flexible beam. Different from the general synchronization control method, the coupling dynamics among various actuators is considered as the shear force, which results from the synchronization errors. Then a simple boundary control is introduced to realize the synchronization motion of actuators by suppressing the shear force. In order to avoid the drawbacks of assumed modes model, the dynamic model of flexible beam is described by a distributed parameter model in this paper. A Riesz basis method is used to prove that the proposed control law can guarantee the synchronization system to be exponential stability. Simulation results demonstrate that the proposed method can effectively improve the performance of synchronization motion compared with other methods.     

Unification of discrete time explicit model reference adaptive control designs

Various discrete time explicit MRAC designs based on stability considerations are presented from a unified point of view. A general structure and a design procedure are given, which contain as particular cases various possible designs. Some of these designs have been already given in the literature but some are new. The general structure considered removes some of the restrictions encountered in previous designs and solves the important problem of independent specification of tracking and regulation objectives. The proof of boundedness of the control variables is also given. Simulations are included in order to evaluate the various designs in a deterministic environment.     

Local stabilization of semilinear parabolic system by boundary control

This paper addresses local stabilization of a semilinear parabolic system by boundary control. Under a certain assumption on the nonlinearity of the parabolic equation, a linear boundary feedback control law is applied to control the semilinear system. Based on the operator theories and relations of different norms, locally exponential stabilization of the closed loop system is established. Simulations support the established stability result.     

MHD channel flow control in 2D: Mixing enhancement by boundary feedback

A nonlinear Lyapunov-based boundary feedback control law is proposed for mixing enhancement in a 2D magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting, incompressible, and subject to an external transverse magnetic field. The MHD model is a combination of the Navier-Stokes PDE and the Magnetic Induction PDE, which is derived from the Maxwell equations. Pressure sensors, magnetic field sensors, and micro-jets embedded into the walls of the flow domain are employed for mixing enhancement feedback. The proposed control law, designed using passivity ideas, is optimal in the sense that it maximizes a measure related to mixing (which incorporates stretching and folding of material elements), while at the same time minimizing the control and sensing efforts. A DNS code is developed, based on a hybrid Fourier pseudospectral-finite difference discretization and the fractional step technique, to numerically assess the controller.     

Robust adaptive boundary control of a flexible marine riser with vessel dynamics

In this paper, robust adaptive boundary control for a flexible marine riser with vessel dynamics is developed to suppress the riser’s vibration. To provide an accurate and concise representation of the riser’s dynamic behavior, the flexible marine riser with vessel dynamics is described by a distributed parameter system with a partial differential equation (PDE) and four ordinary differential equations (ODEs). Boundary control is proposed at the top boundary of the riser based on Lyapunov’s direct method to regulate the riser’s vibration. Adaptive control is designed when the system parametric uncertainty exists. With the proposed robust adaptive boundary control, uniform boundedness under the ocean current disturbance can be achieved. The proposed control is implementable with actual instrumentation since all the required signals in the control can be measured by sensors or calculated by a backward difference algorithm. The state of the system is proven to converge to a small neighborhood of zero by appropriately choosing design parameters. Simulations are provided to illustrate the applicability and effectiveness of the proposed control.     

Adaptive suboptimal second-order sliding mode control for microgrids

ABSTRACT

This paper deals with the design of adaptive suboptimal second-order sliding mode (ASSOSM) control laws for grid-connected microgrids. Due to the presence of the inverter, of unpredicted load changes, of switching among different renewable energy sources, and of electrical parameters variations, the microgrid model is usually affected by uncertain terms which are bounded, but with unknown upper bounds. To theoretically frame the control problem, the class of second-order systems in Brunovsky canonical form, characterised by the presence of matched uncertain terms with unknown bounds, is first considered. Four adaptive strategies are designed, analysed and compared to select the most effective ones to be applied to the microgrid case study. In the first two strategies, the control amplitude is continuously adjusted, so as to arrive at dominating the effect of the uncertainty on the controlled system. When a suitable control amplitude is attained, the origin of the state space of the auxiliary system becomes attractive. In the other two strategies, a suitable blend between two components, one mainly working during the reaching phase, the other being the predominant one in a vicinity of the sliding manifold, is generated, so as to reduce the control amplitude in steady state. The microgrid system in a grid-connected operation mode, controlled via the selected ASSOSM control strategies, exhibits appreciable stability properties, as proved theoretically and shown in simulation.