Observer-based boundary control of semi-linear parabolic PDEs with non-collocated distributed event-triggered observation

Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

“…That is, for the case of m ≥ n, the closed-loop coupled PDEs described by (8) and (9) with (3) and (4) is exponentially stable in the sense of ⋅ 2 if the space algebraic LMIs (10) and (11) are feasible. From (26), we get (12). The proof is complete.…”

“…then there exists an observer-based output feedback compensator (5) with (6), where the elements l^j, j ∈ N in the observer gain L are given by (12), such that the resulting closed-loop coupled PDEs described by (8) and (9) with (3) and (4) is exponentially stable in the norm ⋅ 2 .…”

“…In [9,10], static feedback compensator design has been reported for boundary control of semi-linear parabolic PDEs with collocated boundary observation. In [8,10,12], observer-based dynamic feedback compensator design has been proposed for boundary control of parabolic PDEs with collocated/non-collocated boundary observation. The above results are only applicable for the singleinput-single-output case.…”

Takagi–Sugeno fuzzy‐model‐based event‐triggered point control for semilinear partial differential equation systems using collocated pointwise measurements

“…More recently, some scholars successfully combined T–S fuzzy model and incomplete control/measurements for nonlinear PDE systems. To mention a few, Reference 23 considered the problem of guaranteed cost state estimation based on pointwise measurements, Reference 24 investigated the problem of exponential stabilization using sampled‐data piecewise fuzzy control approach, Reference 25 proposed a pointwise stabilization approach, and Reference 26 discussed boundary control synthesis based on noncollocated observation. Even though a great deal of energy has been devoted to the study of pointwise control and point measurements, the research of fuzzy control design for nonlinear PDE systems remains an open topic.…”

Observer Design for Lipschitz Nonlinear Parabolic PDE Systems With Unknown Input

“…∂s , a Luenberger-type PDE observer was initial developed to exponentially track the state of linear distributed parameter system in [21]; boundary control for a kind of semi-linear parabolic PDE systems in [22]; the design of robust adaptive neural observer was investigated for parabolic PDE systems which contain unknown nonlinearities and bounded disturbances by the technique of modal decomposition in [23]; for more researches, we recommend our readers to see [24], [25]), or design the observer directly (such as, based on the description of ODE (ordinary differential equation)-PDE model, adaptive observer which relies on the constraints of a first order hyperbolic that without parameter uncertainty of the PDE was presented for diffusion parabolic PDE system in [26]; [27] presented online estimates strategy for the state vector of a finite-dimensional ODE which is nonlinear with the structure of strict-feedback and the infinite-dimensional state of a linear parabolic PDE with the technic of boundary measurement sampling; Wang et al [28] investigated the H ∞ state estimation for the T-S fuzzy model of a class of nonlinear PDE system with the technic of spatially local averaged measurements, which can guarantee the exponential stability and satisfy an H ∞ performance for the estimation error fuzzy PDE system; and [29] proposed an interval parabolic PDE observer with the constraint of nonnegative values of boundary and initial conditions, estimator-based H ∞ sampled-data fuzzy control and observer-based fuzzy fault-tolerant control can be respectively seen in [30], [31] for nonlinear parabolic PDE systems; for more researches, one can refer to [32], [33]). The above findings are significant in the field of observer design and inspire other scholars to new insight and perspective in the more in-depth studies.…”