Recently a constructive method was introduced for finite-dimensional observer-based control of 1D parabolic PDEs. In this paper we present an improved method in terms of the reduced-order LMIs (that significantly shorten the computation time) and introduce predictors to manage with larger delays. We treat the case of a 1D heat equation under Neumann actuation and non-local measurement, that has not been studied yet. We apply modal decomposition and prove $L^2$ exponential stability by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension $N$ and resulting decay rate. The LMI dimension does not grow with $N$. The LMI is always feasible for large $N$, and feasibility for $N$ implies feasibility for $N+1$. For the first time we manage with delayed implementation of the controller in the presence of fast-varying (without any constraints on the delay-derivative) input and output delays. To manage with larger delays, we construct classical observer-based predictors. For the known input delay, the LMIs dimension does not grow with $N$, whereas for unknown one the LMIs dimension grows, but it is ssentially smaller than in the existing results. A numerical example demonstrates the efficiency of our method.
Problem of an adaptive state observer design for nonlinear system with unknown time-varying parameters and under condition of delayed measurements is considered. State observation problem was raised by many researchers (see for example Sanx et al. (2019)). In this paper the results proposed in Bobtsov et al. (2021b), Bobtsov et al. (2021a), Bobtsov et al. (2022a), Bobtsov et al. (2022b) are developed. The problem is solved under assumption that the state matrix can be represented as sum of known and unknown parts. The output vector is measured with a known constant delay. An adaptive observer …
Global Exponential Sampled-Data Observers for Nonlinear Systems with Delayed Measurements
Laboratoire GREYC CNRS-ENSICAEN（GREYC实验室）；Technical University of Crete（克里特理工大学）；Centre National de la Recherche Scientifique, CNRS-EECI SUPELEC（法国国家科学研究中心）
This paper presents results on the solvability of the observer design problem for general nonlinear triangular systems with inputs, under weak observability assumptions. The local state estimation is exhibited by means of a delayed time-varying Luenberger-type system. In order to achieve the global estimation, a switching sequence of observers is designed.
This paper is concerned with the stabilization of parallel-flow heat exchangers equation with boundary control and arbitrarily delayed observation. The observer and predictor-based method, developed by [B.Z. Guo, K.Y. Yang, Automatica, 2009, 45(6): 1468-1475], is used to stabilize equation. The exponentially stabilization of such system by estimated state feedback control law is proved.
This paper is concerned with the output feedback stabilization of a reaction-diffusion equation by means of bounded control inputs in the presence of saturations. Using a finite-dimensional controller composed of an observer coupled with a finite-dimensional state-feedback, we derive a set of conditions ensuring the stability of the closed-loop plant while estimating the associated domain of attraction in the presence of saturations. This set of conditions is shown to be always feasible for an order of the observer selected large enough. The stability analysis relies on Lyapunov functionals al…
This paper studies the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition while the system output is selected as a either Dirichlet or Neumann boundary trace. The control strategy takes the form of a finite-dimensional observer-based controller with feedback and observer gains that are computed in order to dominate the state delayed term. For any arbitrarily given value of the state delay, we show the exponential stability of the resulting closed-loop s…
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 …
We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation. We consider Neumann actuation with nonlocal measurement and employ modal decomposition with $N+1$ modes in the observer. We introduce a chain of $M$ sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay $r$, we present LMI stability conditions for finding $M$ and $N$ and the resulting exponential decay rate and prove that the LMIs are always feasible for any $r$. We also consider a classical observer-…
This paper deals with the problem of predicting the future state of discrete-time input-delayed systems in the presence of unknown disturbances that can affect both the state and the output equations of the plant. Since the disturbance is unknown, computing an exact prediction of the future plant states is not possible. To circumvent this problem, we propose using a high-order extended Luenberger-type observer for the plant states, disturbances, and their finite difference variables, combined with a new equation for computing the prediction based on Newton's series from the calculus of finite …
Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded. In this paper, for the first time, we manage with such controllers for the 1D heat equation with both operators being unbounded. We consider Dirichlet actuation and point measurement and use a modal decomposition approach via dynamic extension. We suggest a direct Lyapunov approach to the full-order closed-loop system, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion…
This paper provides observer-based sampled-data and event-triggered boundary control strategies for a class of reaction-diffusion PDEs with collocated sensing and Robin actuation. Infinite-dimensional backstepping design is used as the underlying control approach. It is shown that the continuous-time output feedback boundary control applied in a sample-and-hold fashion ensures global closed-loop exponential stability, provided that the sampling period is sufficiently small. Further, robustness to perturbations of the sampling schedule is guaranteed. For the event-triggered implementation of th…
We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a const…
This paper addresses the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs with delayed boundary measurement. The output takes the form of a either Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component which is employed to obtain the control input to be applied at current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided th…
In this paper, we consider infinite-dimensional port-Hamiltonian systems with in-domain actuation by means of an approach based on Stokes-Dirac structures as well as in a framework that exploits an underlying jet-bundle structure. In both frameworks, a dynamic controller based on the energy-Casimir method is derived in order to stabilise certain equilibrias. Moreover, we propose distributed-parameter observers deduced by exploiting damping injection for the observer error. Finally, we compare the approaches by means of an in-domain actuated vibrating string and show the equivalence of the cont…
This paper presents a dynamic state observer design for discrete-time linear time-varying systems that robustly achieves equalized recovery despite delayed or missing observations, where the set of all temporal patterns for the missing or delayed data is modeled by a finite-length language. By introducing a mapping of the language onto a reduced event-based language, we design a state estimator that adapts based on the history of available data at each step, and satisfies equalized recovery for all patterns in the reduced language. In contrast to existing equalized recovery estimators, the pro…
We present designs for exponential stabilization of an ODE-heat PDE-ODE coupled system where the control actuation only acts in one ODE. The combination of PDE backstepping and ODE backstepping is employed in a state-feedback control law and in an observer that estimates PDE and two ODE states only using one PDE boundary measurement. Based on the state-feedback control law and the observer, the output-feedback control law is then proposed. The exponential stability of the closed-loop system and the boundedness and exponential convergence of the control law are proved via Lyapunov analysis. Fin…
This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input del…
In this paper, a fractional Lotka-Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time-delay $τ$ due to the dead-time in obtaining the measurement of biomass $x(t)$. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay $τ$ as bifurcation parameter are obtained. Under the …
Finite-dimensional observer-based controller design for PDEs is a challenging problem. Recently, such controllers were introduced for the 1D heat equation, under the assumption that one of the observation or control operators is bounded. This paper suggests a constructive method for such controllers for 1D parabolic PDEs with both (observation and control) operators being unbounded. We consider the Kuramoto-Sivashinsky equation (KSE) under either boundary or in-domain point measurement and boundary actuation. We employ a modal decomposition approach via dynamic extension, using eigenfunctions …
In under-balanced drilling, the bottom-hole pressure is kept below pore pressure, causing pressure dependent influx of reservoir gas into the wellbore that makes the system unstable at low drawdowns. In this paper we propose a feedback controller which stabilizes the system around an arbitrary pressure setpoint, using only topside measurement, and assuming unknown reservoir parameters. A particular challenge with this problem is the distributed and highly nonlinear nature of the system dynamics. As the control model we use the ``reduced Drift Flux Model'' which models gas-liquid flow as a nonl…
Global Stabilization of Nonlinear Delay Systems With a Compact Absorbing Set
National Technical University of Athens（雅典国立技术大学）；University of California, San Diego（加州大学圣地亚哥分校）；Laboratoire GREYC CNRS-ENSICAEN（GREYC国家科学研究中心-ENSICAEN实验室）；Centre National de la Recherche Scientifique, CNRS-EECI SUPELEC（法国国家科学研究中心，CNRS-EECI苏佩莱克）
