Research on Optimal Placement of Actuators of High-Rise Buildings Considering the Influence of Seismic Excitation on Structural Modes

Presently, most of the common placement methods of actuators are based on the structural response and system energy to select the optimal locations. In these methods, the contribution of controllability and the energy of seismic excitations to each mode of the structure are not considered, and a large number of cases need to be calculated. To solve this problem, the Clough–Penzien spectral model is combined with the Luenberger observable normal form of the system to calculate the energy of each state. The modal disturbance degree, considering modal energy and controllability, is defined by using the controllability gramian matrix and PBH system controllability index, and the modes are divided into the main disturbance modes (MDMs) and the secondary disturbance modes (SDMs). A novel optimal placement method of actuators based on modal controllability degree is proposed, which uses MDMs as the main control modes. The optimal placement of actuators and the vibration control simulation of a 20-story building model are carried out. The results show that the vibration reduction effect of the proposed placement method is significantly better than that of the method of uniformly distributed actuators (Uniform method) and the classical placement method of actuators based on the system controllability gramian matrix (Classical method).

Controllability Gramian and Kalman rank condition for mean-field control systems

This paper is concerned with the exact controllability of linear mean-field stochastic systems with deterministic coefficients. With the help of the theory of mean-field backward stochastic differential equations (MF-BSDEs, for short) and some delicate analysis, we obtain a mean-field version of the Gramian matrix criterion for the general time-variant case, and a mean-field version of the Kalman rank condition for the special time-invariant case.

A New Method for Determining the Degree of Controllability of State Variables for the LQR Problem Using the Duality Theorem

The control performance of a dynamic system can be checked by the degree of controllability. In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. We carried out the calculation of the degree of controllability for the linear quadratic optimal control (LQR) problem using a duality theorem. Compared with the traditional measures of controllability such as determinant, trace, and maximal eigenvalue of the inverse controllability Gramian, the proposed degree of controllability was developed for each state variable and takes into account both the controllability Gramian and the cost function. The new method is convenient to apply to LQR problem. In the numerical simulation, we determined the influence of the model parameters on the degree of controllability. Besides that, we analyzed the degree of controllability, which gives an insight into the relationship between the system model design and the control performance.

On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

This paper formulates the properties of point reachability and approximate point reachability of either a targeted state or output values in a general dynamic system which possess a linear time-varying dynamics with respect to a given reference nominal one and, eventually, an unknown structured nonlinear dynamics. Such a dynamics is upper-bounded by a function of the state and input. The results are obtained for the case when the time-invariant nominal dynamics is perfectly known while its time-varying deviations together with the nonlinear dynamics are not precisely known and also for the case when only the nonlinear dynamics is not precisely known. Either the controllability gramian of the nominal linearized system with constant linear parameterization or that of the current linearized system (which includes the time-varying linear dynamics) are assumed to be non-singular. Also, some further results are obtained for the case when the control input is eventually saturated and for the case when the controllability gramians of the linear parts are singular. Examples of the derived theoretical results for some epidemic models are also discussed.