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Register a new userA model reference adaptive system approach for nonlinear online parameter identification
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Dynamical systems, for instance in model predictive control, often contain unknown parameters, which must be determined during system operation. Online or on-the-fly parameter identification methods are therefore necessary. The challenge of online methods is that one must continuously estimate parameters as experimental data becomes available. The existing techniques in the context of time-dependent partial differential equations exclude the case where the system depends nonlinearly on the parameters.Based on a model reference adaptive system approach, we present an online parameter identification method for nonlinear infinite-dimensional evolutionary system.
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We consider the problem of asymptotic reconstruction of the state and parameter values in systems of ordinary differential equations. A solution to this problem is proposed for a class of systems of which the unknowns are allowed to be nonlinearly parameterized functions of state and time. Reconstruction of state and parameter values is based on the concepts of weakly attracting sets and non-uniform convergence and is subjected to persistency of excitation conditions. In absence of nonlinear parametrization the resulting observers reduce to standard estimation schemes. In this respect, the proposed method constitutes a generalization of the conventional canonical adaptive observer design.
During the last decades, significant advances have been made in the area of power system stability and control. Nevertheless, when this analysis is carried out by means of decentralized conditions in a general network, it has been based on conservative assumptions such as the adoption of lossless networks. In the current paper, we present a novel approach for decentralized stability analysis and control of power grids through the transformation of both the network and the bus dynamics into the system reference frame. In particular, the aforementioned transformation allows us to formulate the network model as an input-output system that is shown to be passive even if the networks lossy nature is taken into account. We then introduce a broad class of bus dynamics that are viewed as multivariable input/output systems compatible with the network formulation, and appropriate passivity conditions are imposed on those that guarantee stability of the power network. We discuss the opportunities and advantages offered by this approach while explaining how this allows the inclusion of advanced models for both generation and power flows. Our analysis is verified through applications to the Two Area Kundur and the IEEE 68-bus test systems with both primary frequency and voltage regulation mechanisms included.
We develop an asymptotical control theory for one of the simplest distributed oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a dry-friction like feedback control, which turns out to be asymptotically optimal. We prove the existence of motion under the control using a rather explicit solution of a nonlinear wave equation. Remarkably, the solution is determined via purely algebraic operations. The main result is a proof of asymptotic optimality of the control thus constructed.
This supplement illustrates application of adaptive observer design from (Tyukin et al, 2013) for systems which are not uniquely identifiable. It also provides an example of adaptive observer design for a magnetic bearings benchmark system (Lin, Knospe, 2000).
We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solve a new adjoint PDE at each optimization iteration, which can be computationally costly. In contrast, an online adjoint algorithm updates the design variables in continuous-time and thus constantly makes progress towards minimizing the objective function. The online adjoint algorithm we consider is similar in spirit to the pseudo-time-stepping, one-shot method which has been previously proposed. Motivated by the application of such methods to engineering problems, we mathematically study the convergence of the online adjoint algorithm. The online adjoint algorithm relies upon a time-relaxed adjoint PDE which provides an estimate of the direction of steepest descent. The algorithm updates this estimate continuously in time, and it asymptotically converges to the exact direction of steepest descent as $t rightarrow infty$. We rigorously prove that the online adjoint algorithm converges to a critical point of the objective function for optimizing the PDE. Under appropriate technical conditions, we also prove a convergence rate for the algorithm. A crucial step in the convergence proof is a multi-scale analysis of the coupled system for the forward PDE, adjoint PDE, and the gradient descent ODE for the design variables.
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