Boundary Actuation Research Articles

Bistable Insect-Scale Jumpers with Tunable Energy Barriers for Multimodal Locomotion.

Drawing inspiration from the jumping mechanisms of insects (e.g., click beetles), bistable structures can convert slow deformations of soft actuating material into fast jumping motions (i.e., power amplification). However, bistable jumpers often encounter large energy barriers for energy release/re-storage, posing a challenge in achieving multimodal (i.e., height/distance) and continuous jumps at the insect scale (body length under 20mm). Here, a new offset-buckling bistable design is introduced that features antisymmetric equilibrium states and tunable energy barriers. Leveraging this design, a Boundary Actuation Tunable Energy-barrier (BATE) jumper (body length down to 15mm) is developed, and transform BATE jumper from height-jump mode (up to 12.7 body lengths) to distance-jump mode (up to 20 body lengths). BATE jumpers can perform agile continuous jumping (within 300ms for energy release/re-storage times) and real-time status detection is further demonstrated. This insect-level performance of the proposed BATE jumper showcases its potential toward future applications in exploration, search, and rescue.

Stabilization of Underactuated Linear Coupled Reaction–Diffusion PDEs via Distributed or Boundary Actuation

Stabilization of Underactuated Linear Coupled Reaction–Diffusion PDEs via Distributed or Boundary Actuation

Concurrent Learning Robust Adaptive Fault Tolerant Boundary Regulation of Hyperbolic Distributed Parameter Systems.

This article develops a robust adaptive boundary output regulation approach for a class of complex anticollocated hyperbolic partial differential equations subjected to multiplicative unknown faults in both the boundary sensor and actuator. The regulator design is based on the internal model principle, which amounts to stabilize a coupled cascade system, which consists of a finite-dimensional internal model driven by a hyperbolic distributed parameter system (DPS). To this end, a systematic sliding mode equipped with a backstepping approach is developed such that the robust state feedback control can be realized. Moreover, since the available information is a faulty boundary measurement at the right side point, state estimation is required. However, due to the presence of boundary unknown faults, we need to solve an issue of joint fault-state estimation. Restrictive persistent excitation conditions are usually required to guarantee the exact estimation of faults but are unrealistic in practice. To this end, a novel concurrent learning (CL) adaptive observer is proposed so that exponential convergence is obtained. It is the first time that the spirit of CL is introduced to the field of DPSs. Consequently, the observer-based adaptive boundary fault tolerant control scheme is developed, and rigorous theoretical analysis is given such that the exponential output regulation can be achieved. Finally, the effectiveness of the proposed methodology is demonstrated via comparative simulations.

Adaptive stabilization for a class of wave equations with uncertain nonlinear boundary actuator dynamics

This paper investigates the stabilization of a wave equation with uncertain boundary actuator dynamics. The remarkable characters of the system under investigation are revealed by the essential nonlinearities involved in the boundary actuator and the serious parametric unknowns coming from anti-damping coefficient at uncontrolled boundary and those in the actuator, which result into the incapability of the traditional control methods on this topic. For this, a novel adaptive control strategy is proposed in this paper. Specifically, a Riemann variables based state transformation is first introduced to change the original system into a new one whose stabilization implies that of the original system. Then, a state-feedback controller joint with some adaptive laws which are smartly chosen for the compensation of unknown parameters is explicitly designed by a finite-dimensional backstepping procedure. Owing to the compensation of boundary actuator dynamics, the performance analysis of the closed-loop system is more complicated than that of the related literature. For this, a rigorous recursive step is given to show the stability of the resulting closed-loop system. Finally, a numerical example is provided to verify the effectiveness of the proposed theoretical results.

Adaptive Neural Tracking Control of a Class of Hyperbolic PDE With Uncertain Actuator Dynamics.

This article investigates the adaptive neural tracking control problem for a class of hyperbolic PDE with boundary actuator dynamics described by a set of nonlinear ordinary differential equations (ODEs). Particularly, the control input appears in the ODE subsystem with unknown nonlinearities requiring to be estimated and compensated, which makes the control task rather difficult. It is the first time to consider tracking control of such a class of systems, rendering our contributions essentially different from the existing literature that merely focus on the stabilization problem. By formulating a virtual exosystem to generate a reference trajectory, we propose a novel design of the adaptive geometric controller for the considered system where neural networks (NNs) are employed to approximately estimate nonlinearities, and finite and infinite-dimensional backstepping techniques are leveraged. Moreover, rigorously theoretical proofs based on the Lyapunov theory are provided to analyze the stability of the closed-loop system. Finally, we illustrate the results through two numerical simulations.

Adaptive constrained tracking control for dynamical actuator driven linear 2 × 2 hyperbolic PDE systems with nonlinear uncertainties

The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation.

PDE modelling and switch strategy controller design for a flexible detection system with output constraint and actuator failure

A novel approach for modelling a mine detection robot system is presented in this paper using partial differential equations and Hamilton’s principle. The system comprises a rigid arm, a detecting sensor module, and a flexible string. The paper addresses the challenges of boundary deflection constraint and actuator fault. To overcome these challenges, a unique controller is developed using Barrier Lyapunov function and switching strategy. This controller enables the robot system to achieve the desired position and eliminate the vibration of flexible string. The superiority of the designed control law is demonstrated through simulation results. Furthermore, the asymptotic stability of the closed-loop flexible system is established by the use of extending LaSalle’s Invariance principle to an infinite dimensional system. This principle ensures that the system will eventually reach a stable state over time.

Tracking model predictive control and moving horizon estimation design of distributed parameter pipeline systems

This manuscript proposes moving horizon control and state/parameter estimation designs for pipeline networks modelled by partial differential equations (PDEs) with boundary actuation. The spatial–temporal pressure and velocity dynamics within the pipelines are described by a system of six coupled one-dimensional first-order nonlinear hyperbolic PDEs. To address the discrete-time modelling challenge and preserve the infinite-dimensional nature of the pipeline system, the Cayley–Tustin transformation is deployed for model time discretization without any spatial discretization or model reduction. Considering the lack of full state information across the entire pipeline manifold, unknown states and uncertain parameters are estimated using moving horizon estimation (MHE). Based on the estimated states and parameters, a tracking model predictive control (MPC) strategy for the discrete-time infinite-dimensional pipeline system is proposed, which enables specific operation while ensuring physical constraint satisfaction. The effectiveness of the proposed controller and estimator designs is demonstrated via numerical examples.

Disturbance Observer-Based Boundary Control for an Antistable Stochastic Heat Equation With Unknown Disturbance

In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.

Boundary Vibration Constraint Control Design of Motor-Driven Industrial Moving Belt Systems Subject to Actuator Input Saturation

This paper investigates the adaptive controller design problem for a class of motor-driven industrial belt systems with boundary vibration constraints and actuator input saturation. The operating equations of the considered system are obtained based on the Eulerian description and Hamilton’s principle, and an auxiliary system is desiigned to compensate for the input saturation effect of the actuator. In a practical engineering context, the system is not only affected by the time-varying boundary disturbances but also the key parameters of the whole closed-loop system are unknown. For this reason, a disturbance observer and the adaptive laws are designed to estimate the disturbances suffered by the system and to compensate for the parameter uncertainty of the system, respectively. With the application of our designed control strategy, the boundary vibration of the system are shown to satisfy the constraint, i.e., the safe operation of the motor-driven industrial belt system is ensured. Finally, the stability and effectiveness of the designed control scheme are demonstrated by examples of numerical simulations.

Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.

Output regulation for general heterodirectional linear hyperbolic PDEs coupled with nonlinear ODEs

This paper considers general heterodirectional linear hyperbolic PDEs with boundary actuation and collocated measurement, that are bidirectionally coupled with nonlinear ODEs at the unactuated boundary. An output feedback regulator is designed to have the control output track a reference in the presence of disturbances, with a nonlinear signal model generating the reference and the disturbances. This leads to new challenges for both the observer and controller design. The regulator design makes use of results from output regulation theory for nonlinear lumped-parameter and distributed-parameter systems. The derivation of the state feedback regulator requires solving a new type of regulator equations, which consist of a Cauchy problem for linear and semilinear hyperbolic PDEs. A key component of the novel observer design is a nonlinear retarded observer for a finite-dimensional subsystem. Both the controller and the observer design are systematic in nature and derived in several successive steps, that include backstepping transformations and, most notably, predictions of PDE and ODE states. The latter are shown to be the unique solution of general nonlinear Volterra integro-differential equations. Combining the observer with the state feedback regulator is proven to achieve output regulation for the closed-loop system. A numerical example illustrates and confirms the theoretical results.

Boundary state feedback control for semilinear fractional-order reaction diffusion systems

This paper investigates the stabilization results for a semilinear time fractional-order reaction diffusion partial differential equations using backstepping method. The parabolic system is addressed with time fractional-order of (0, 1) in the Caputo sense. More general type of actuation setup which can be expressed as Dirichlet, Neumann and Robin type boundary actuation's is considered. The main aim is to achieve the stabilization of the considered system using an invertible transformation through the stability of target system which is verified by Mittag-Leffler stability based on (Q, S, R) dissipativity theory and linear matrix inequality (LMI) technique. The explicit solutions of kernel functions are found by method of successive approximation and to be design the boundary control law for the closed loop system. Finally, the proposed results are validated through numerical examples.

Model predictive control of a second‐order hyperbolic transport‐reaction process

The model predictive controller (MPC) design is developed for a tubular chemical reactor, considering a second-order hyperbolic partial differential equation as the model of the transport-reaction process with boundary actuation. Without loss of generality, closed–closed boundary conditions and relaxed total flux are assumed. At the same time, the model is discretized in time by the Cayley–Tustin method, and, under the assumption that only the reactor's output is measurable, the observer design for the state reconstruction is addressed and integrated with the MPC design. The Luenberger observer gain is obtained by solving the operator Ricatti equation in the discrete-time setting, while the MPC accounts for constrained and optimal control. The simulations show that the output-based MPC design stabilizes the system under the input and output constraints satisfaction. In addition, to address the models' disparities, the results for both parabolic and hyperbolic equations are presented and discussed.

Explicit model predictive control for PDEs: The case of a heat equation

Explicit model predictive control design is carefully developed for discrete-time linear plants on Hilbert spaces, and we highlight the role of the so-called Slater condition in the reliable explicit solution of the MPC optimization. We then proceed to present an explicit MPC algorithm that accounts for the stabilization and input constraints satisfaction. We do structure preserving temporal discretization of the infinite-dimensional parabolic PDE system by application of the Cayley transformation. The salient feature of explicit MPC design is the realization of the region-free approach in explicit MPC design with identification of active constraint sets to realize optimal stabilization and constraints satisfaction. Finally, the resulting design is illustrated by the application to the PDE model given by an unstable heat equation with boundary actuation and Neumann boundary conditions. The example demonstrates simultaneous stabilization and input constraints satisfaction on the one hand, and on the ability to deal with a relatively high plant dimension and a long optimization horizon on the other hand.

Regional Stabilization of the 1-D Kuramoto–Sivashinsky Equation via Modal Decomposition

In this letter, we suggest regional stabilization of the semilinear 1D KSE under nonlocal or boundary actuation. We employ modal decomposition and derive regional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H^{1}$ </tex-math></inline-formula> stability conditions for the closed-loop system. Given a decay rate that defines the number of state modes in the controller, we provide LMIs for finding the the controller gain as well as a bound on the domain of attraction. In the case of boundary control, we suggest a dynamic extension with a novel internally stable dynamics. The latter allows to enlarge a bound on the domain of attraction. Numerical examples illustrate the efficiency of the method.

Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.

Event-triggered-based fixed-time adaptive prescribed tracking control for stochastic nonlinear systems with actuator faults

In this paper, the event-triggered fixed-time control scheme is developed for a class of stochastic nonlinear systems with prescribed boundary constraints and actuator faults. For the controlled systems with unknown nonlinear functions, the neural networks are employed to reestablish the system model. Based on the event-triggered control technology, an original adaptive fixed-time control strategy with prescribed performance and the fault state is proposed by using the backstepping technique. Under the developed adaptive controller, the tracking error satisfies the predefined boundary functions and all the closed-loop system signals are bounded in probability in a fixed time, and the convergence time is irrelevant to the initial states of the system. The practicability of the designed controller is illustrated by a simulation example.

Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditions

The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space.

Sampled-data finite-dimensional boundary control of 1D parabolic PDEs under point measurement via a novel ISS Halanay’s inequality

Recently, finite-dimensional observer-based controllers were introduced for 1D parabolic PDEs via the modal decomposition method. In the present paper we suggest a sampled-data implementation of a finite-dimensional boundary controller for 1D parabolic PDEs under discrete-time point measurement. We consider the heat equation under boundary actuation and point (either in-domain or boundary) measurement. In order to manage with point measurement, we employ dynamic extension and prove H1-stability. Due to dynamic extension, which leads to proportional–integral controller, we suggest a sampled-data implementation of the controller via a generalized hold device. We take into account the quantization effect that leads to a disturbed closed-loop system and input-to-state stability (ISS) analysis. We use Wirtinger-based piecewise continuous in time Lyapunov functionals which compensate sampling in the finite-dimensional state and lead to the simplest efficient stability conditions for ODEs. To compensate sampling in the infinite-dimensional tail, we introduce a novel form of Halanay’s inequality for ISS, which is appropriate for functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of our method.