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In this report, we present a new Linear-Quadratic Semistabilizers (LQS) theory for linear network systems. This new semistable H2 control framework is developed to address the robust and optimal semistable control issues of network systems while preserving network topology subject to white noise. Two new notions of semistabilizability and semicontrollability are introduced as a means to connecting semistability with the Lyapunov equation based technique. With these new notions, we first develop a semistable H2 control theory for network systems by exploiting the properties of semistability. A new series of necessary and sufficient conditions for semistability of the closed-loop system have been derived in terms of the Lyapunov equation. Based on these results, we propose a constrained optimization technique to solve the semistable H2 network-topology-preserving control design for network systems over an admissible set. Then optimization analysis and the development of numerical algorithms for the obtained constrained optimization problem are conducted. We establish the existence of optimal solutions for the obtained nonconvex optimization problem over some admissible set. Next, we propose a heuristic swarm optimization based numerical algorithm towards efficiently solving this nonconvex, nonlinear optimization problem. Finally, several numerical examples will be provided.
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Optimal actuator design for a vibration control problem is calculated. The actuator shape is optimized according to the closed-loop performance of the resulting linear-quadratic regulator and a penalty on the actuator size. The optimal actuator shape is found by means of shape calculus and a topological derivative of the linear-quadratic regulator (LQR) performance index. An abstract framework is proposed based on the theory for infinite-dimensional optimization of both the actuator shape and the associated control problem. A numerical realization of the optimality condition is presented for the actuator shape using a level-set method for topological derivatives. A Numerical example illustrating the design of actuator for Euler-Bernoulli beam model is provided.
This paper proposes an algorithmic technique for a class of optimal control problems where it is easy to compute a pointwise minimizer of the Hamiltonian associated with every applied control. The algorithm operates in the space of relaxed controls and projects the final result into the space of ordinary controls. It is based on the descent direction from a given relaxed control towards a pointwise minimizer of the Hamiltonian. This direction comprises a form of gradient projection and for some systems, is argued to have computational advantages over direct gradient directions. The algorithm is shown to be applicable to a class of hybrid optimal control problems. The theoretical results, concerning convergence of the algorithm, are corroborated by simulation examples on switched-mode hybrid systems as well as on a problem of balancing transmission- and motion energy in a mobile robotic system.
The robustness of quantum control in the presence of uncertainties is important for practical applications but their quantum nature poses many challenges for traditional robust control. In addition to uncertainties in the system and control Hamiltonians and initial state preparation, there is uncertainty about interactions with the environment leading to decoherence. This paper investigates the robust performance of control schemes for open quantum systems subject to such uncertainties. A general formalism is developed, where performance is measured based on the transmission of a dynamic perturbation or initial state preparation error to a final density operator error. This formulation makes it possible to apply tools from classical robust control, especially structured singular value analysis, to assess robust performance of controlled, open quantum systems. However, there are additional difficulties that must be overcome, especially at low frequency ($sapprox0$). For example, at $s=0$, the Bloch equations for the density operator are singular, and this causes lack of continuity of the structured singular value. We address this issue by analyzing the dynamics on invariant subspaces and defining a pseudo-inverse that enables us to formulate a specialized version of the matrix inversion lemma. The concepts are demonstrated with an example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. In addition, a new performance index is introduced for this system. Instead of the tracking or transfer fidelity error, performance is measured by the steady-steady entanglement generated, which is quantified by a non-linear function of the system state called concurrence. Simulations show that there is no conflict between this performance index, its log-sensitivity and stability margin under decoherence, unlike for conventional control problems [...].
In stochastic dynamic environments, team stochastic games have emerged as a versatile paradigm for studying sequential decision-making problems of fully cooperative multi-agent systems. However, the optimality of the derived policies is usually sensitive to the model parameters, which are typically unknown and required to be estimated from noisy data in practice. To mitigate the sensitivity of the optimal policy to these uncertain parameters, in this paper, we propose a model of robust team stochastic games, where players utilize a robust optimization approach to make decisions. This model extends team stochastic games to the scenario of incomplete information and meanwhile provides an alternative solution concept of robust team optimality. To seek such a solution, we develop a learning algorithm in the form of a Gauss-Seidel modified policy iteration and prove its convergence. This algorithm, compared with robust dynamic programming, not only possesses a faster convergence rate, but also allows for using approximation calculations to alleviate the curse of dimensionality. Moreover, some numerical simulations are presented to demonstrate the effectiveness of the algorithm by generalizing the game model of social dilemmas to sequential robust scenarios.
This paper concerns a first-order algorithmic technique for a class of optimal control problems defined on switched-mode hybrid systems. The salient feature of the algorithm is that it avoids the computation of Frechet or G^ateaux derivatives of the cost functional, which can be time consuming, but rather moves in a projected-gradient direction that is easily computable (for a class of problems) and does not require any explicit derivatives. The algorithm is applicable to a class of problems where a pointwise minimizer of the Hamiltonian is computable by a simple formula, and this includes many problems that arise in theory and applications. The natural setting for the algorithm is the space of continuous-time relaxed controls, whose special structure renders the analysis simpler than the setting of ordinary controls. While the space of relaxed controls has theoretical advantages, its elements are abstract entities that may not be amenable to computation. Therefore, a key feature of the algorithm is that it computes adequate approximations to relaxed controls without loosing its theoretical convergence properties. Simulation results, including cpu times, support the theoretical developments.
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