Classical Linear Optimal Control Research Articles

A note on the minimum‐norm in dual space approach to some classical linear optimal control problems

AbstractSome classical optimal control problems of linear systems can be characterized as finding minimum‐norm vectors from within linear varieties in appropriate dual spaces. Then, the solution to such an optimal control problem can be derived from the alignment between the optimal vector in the dual space and the optimal vector in the primal space, and the dual maximization problem of the minimum‐norm problem. This note presents a detailed introduction to this minimum‐norm in dual space approach by examples of minimum‐supremum‐norm, minimum‐energy, and minimum‐time optimal control problems of linear systems. Connections and differences between these problems in light of the introduced approach are discussed.

Study of seismic-responses and semi-active control of Taizhou Bridge under multi-support excitation

PurposeIn order to make sure of the safety of a long-span suspension bridge under earthquake action, this paper aims to study the traveling wave effect of the bridge under multi-support excitation and optimize the semi-active control schemes based on magneto-rheological (MR) dampers considering reference index as well as economical efficiency.Design/methodology/approachThe finite element model of the long-span suspension bridge is established in MATLAB and ANSYS software, which includes different input currents and semi-active control conditions. Six apparent wave velocities are used to conduct non-linear time history analysis in order to consider the seismic response influence in primary members under traveling wave effect. The parameters α and β, which are key parameters of classical linear optimal control algorithm, are optimized and analyzed taking into account five different combinations to obtain the optimal control scheme.FindingsWhen the apparent wave velocity is relatively small, the influence on the structural response is oscillatory. Along with the increase of the apparent wave velocity, the structural response is gradually approaching the response under uniform excitation. Semi-active control strategy based on MR dampers not only restrains the top displacement of main towers and relative displacement between towers and girders, but also affects the control effect of internal forces. For classical linear optimal control algorithm, the values of two parameters (α and β) are 100 and 8 × 10–6 considering the optimal control effect and economical efficiency.Originality/valueThe emphasis of this study is the traveling wave effect of the triple-tower suspension bridge under multi-support excitation. Meanwhile, the optimized parameters of semi-active control schemes using MR dampers have been obtained, providing relevant references in improving the seismic performance of three-tower suspension bridge.

A novel approach of active control of structures based on the critically damped condition

In this research, the active control of structures based on the critically damped condition was proposed for reducing displacements and drifts of structures under seismic loading. In this approach, the second-order dynamic equation is converted into a first-order dynamic equation and a first-order derivative operator with the same constants, which satisfy critically damped conditions of the system. Based on this concept and using the defined performance index, the feedback gain matrix of the structural velocity is obtained. In the method of active control of structures based on the critically damped condition, the control force, which is the linear definition of the feedback gain matrix and the structural velocity, is exerted on the structure in each time step. Four types of numerical examples are examined to the comparison verification of active control of structures based on the critically damped condition with the classical linear optimal control, standard model predictive control, and other methods proposed in several references. Programming and calculations are done by Mathcad prime mathematical software.

Experimental Investigation on Semi-Active Control of Base Isolation System Using Magnetorheological Dampers for Concrete Frame Structure

The traditional passive base isolation is the most widely used method in the engineering practice for structural control, however, it has the shortcoming that the optimal control frequency band is significantly limited and narrow. For the seismic isolation system designed specifically for large earthquakes, the structural acceleration response may be enlarged under small earthquakes. If the design requirements under small earthquakes are satisfied, the deformation in the isolation layer may become too large to be accepted. Occasionally, it may be destroyed under large earthquakes. In the isolation control system combined with rubber bearing and magnetorheological (MR) damper, the MR damper can provide instantaneous variable damping force to effectively control the structural response at different input magnitudes. In this paper, the control effect of semi-active control and quasi-passive control for the isolation control system is verified by the shaking table test. In regard to semi-active control, the linear quadratic regulator (LQR) classical linear optimal control algorithm by continuous control and switch control strategies are used to control the structural vibration response. Numerical simulation analysis and shaking table test results indicate that isolation control system can effectively overcome the shortcoming due to narrow optimum control band of the passive isolation system, and thus to provide optimal control for different seismic excitations in a wider frequency range. It shows that, even under super large earthquakes, the structure still exhibits the ability to maintain overall stability performance.

A new active control performance index for vibration control of three-dimensional structures

Remarkable results have been reported about conventional active control algorithms during the last 30years, but nearly all the existing literature considers 2-dimensional in plane structures to implement and verify the active control algorithms. To simulate the behavior of real buildings more accurately, more realistic and complex models should be used in the performance evaluation and design of controllers and their control algorithms. This paper presents a new performance index for active vibration control of three-dimensional structures. To analytically validate the proposed performance index, a six story three-dimensional structure is considered as an example with a fully active tendon controller system implemented in one direction of the building. Tier building formulation is used for three-dimensional dynamic analysis. The building is modeled as a structure composed of members connected by a rigid floor diaphragm such that it has three degrees of freedom at each floor, i.e., lateral displacements in two perpendicular directions and a rotation with respect to a vertical axis for the third dimension. The performance of the building with the active tendons controlled using a classical linear optimal control algorithm is compared to the performance of the proposed control algorithm under several far-fault and near-fault earthquakes using several performance measures. Comparison between the computational results shows that the proposed algorithm outperforms the performance of the classical linear optimal control algorithm for the actively controlled building.

Investigating the behavior of smart thin beams with piezoelectric actuators under dynamic loads

In this paper, the constitutive equation of motion for an Euler–Bernoulli beam in which a number of piezoelectric patches are bonded to the bottom and top surfaces of it, and arbitrary boundary conditions, is derived by employing Hamilton's principle. Assuming a number of linear springs with high stiffness as intermediate supports, the motion equation of a multi-span smart beam could be found. Classical linear optimal control algorithm with displacement–velocity and velocity–acceleration feedbacks is used. Utilizing eigenfunction expansion method, the equation of motion is decoupled into a number of ordinary differential equations. All the numerical examples are presented for the simple boundary conditions. The applied dynamic excitations are a rectangular impulse, moving load and the moving mass. Parametric studies on the capability of the control system in vibration suppression of the beams under these dynamic loads are achieved. The obtained results reveal the efficiency of the proposed control system in reducing the response of the beam structures to the required levels.

Vibration Suppression in Smart Thin Beams with Piezoelectric Actuators under a moving Load/Mass Accounting for Large Deflections of the Base Structure

In this article, the motion equations of a single span Euler–Bernoulli beam with geometrically nonlinear behavior under an arbitrary dynamic loading are derived via the Hamilton’s Principle. In order to actively control the response of the structure, piezoceramic patches bonded on the lower surface of the beam are utilized. Employing the Eigen Function Expansion Method and considering the first vibrational mode, an equivalent linear control algorithm based on the well–known classical linear optimal control algorithm with displacement–velocity feedback is proposed. Numerical examples for a simply supported beam with immovable–immovable and immovable–movable axial boundary conditions are presented under a moving load and mass excitations. By using a single piezoceramic patch bonded symmetrically at the beam mid–span, the deflection of the beam is decreased into any required levels for both linear and nonlinear behavior of the base beam and therefore, the good performance of the proposed control algorithm is proved.

Vibration Suppression in Smart Thin Beams with Piezoelectric Actuators under a moving Load/Mass Accounting for Large Deflections of the Base Structure.

In this article, the motion equations of a single span Euler–Bernoulli beam with geometrically nonlinear behavior under an arbitrary dynamic loading are derived via the Hamilton’s Principle. In order to actively control the response of the structure, piezoceramic patches bonded on the lower surface of the beam are utilized. Employing the Eigen Function Expansion Method and considering the first vibrational mode, an equivalent linear control algorithm based on the well–known classical linear optimal control algorithm with displacement–velocity feedback is proposed. Numerical examples for a simply supported beam with immovable–immovable and immovable–movable axial boundary conditions are presented under a moving load and mass excitations. By using a single piezoceramic patch bonded symmetrically at the beam mid–span, the deflection of the beam is decreased into any required levels for both linear and nonlinear behavior of the base beam and therefore, the good performance of the proposed control algorithm is proved.

Control of Structural Response Under Earthquake Excitation

Abstract: This study firstly proposes some representative simple methods to obtain the suboptimal passive damping and stiffness parameters from the optimal control gain matrix since it is not possible to add the exact optimal damping and stiffness parameters to the structure in practice. It is shown numerically that modifying the structural damping and the stiffness in the proposed suboptimal ways may suppress the uncontrolled vibrations while the performance levels depend on the seismic inputs. Since the proposed approach is intrinsically passive and has no adaptive property against changing dynamic effects, this study secondly proposes a new performance index so that the mechanical energy of the structure, control and the seismic energies are considered simultaneously in the minimization procedure. The implementation of the resulting closed‐loop control algorithm does not require both a priori knowledge of the seismic excitation and the solution of the nonlinear matrix Riccati equation. The performance of the proposed approach is investigated, e.g., structures subjected to three seismic inputs and compared to the performance of the uncontrolled, the classical linear optimal control, and the passive cases. It is shown by the numerical simulation results that the proposed algorithm is capable of suppressing the uncontrolled seismic structural displacements and the absolute accelerations simultaneously and performs almost as well as the classical linear optimal control in reducing the displacements with comparable control effort and performs better than the classical linear optimal control in reducing the absolute accelerations. The results show that while the proposed active approach has similar performance to the classical linear optimal control, the classical linear optimal control increases the absolute accelerations slightly compared to the proposed active approach in regulating displacements, while the proposed active approach regulates and reduces both displacements and absolute accelerations. The proposed approach is promising in protecting both the structural and non‐structural members from the seismic forces since a simultaneous reduction both in the displacements and the absolute accelerations is achieved.

A Simple Active Control Algorithm for Earthquake Excited Structures

Abstract: A simple integral type quadratic functional is proposed as the performance index so that the optimal control policy is derived based on the minimization of the proposed performance index between the successive control instants by using the method of calculus of variations. The resulting optimal control law is applied to seismically excited linear buildings modeled as lumped mass shear frame structures. Active tendon actuators are considered as control devices. The performance of the proposed control (PC) is investigated when the example structure is subjected to three different seismic inputs and compared to the uncontrolled case and the classical linear optimal control (CLOC), which requires the solution of nonlinear matrix Riccati equation. It is shown by numerical simulation results that the PC is capable of suppressing the uncontrolled seismic vibrations of the linear structures and performs better than the CLOC.

A Computational Approach to Optimal Damping Controller Design for a GCSC

This paper presents a computational approach for an offline measurement-based design of an optimal damping controller using adaptive critics for a new type of flexible ac transmission system device-the gate-controlled series capacitor (GCSC). Remote measurements are provided to the controller to damp out system modes. The optimal controller is developed based on the heuristic dynamic programming (HDP) approach. Three multilayer-perceptron neural networks are used in the design-the identifier/model network to identify the dynamics of the power system, the critic network to evaluate the performance of the damping controller, and the controller network to provide optimal damping. This measurement-based technique provides an alternative to the classical linear model-based optimal control design. The eigenvalue analysis of the closed-loop system is performed with time-domain responses using the Prony method. An analysis of the simulation results shows potential of the HDP-based optimal damping controller on a GCSC for enhancing the stability of the power system.

Seismic response of torsionally coupled structures with active control device

AbstractA study is carried out for investigating the effectiveness of active control systems for one‐storey torsionally coupled structures. The governing equations of motion of the system are derived. The control forces are obtained using the classical linear optimal control algorithm. The stochastic seismic response of a torsionally coupled controlled system is investigated under important parametric variations such as: uncoupled lateral time period, system eccentricity and ratio of uncoupled torsional to lateral frequency of the system. In order to investigate the effectiveness of the control device, the response of the system with control device is compared with the corresponding system without control device. Further, the controlled response of the torsionally coupled system is compared with the corresponding response of the uncoupled system. It is shown that the control mechanism is also equally effective in reducing the seismic response of a torsionally coupled system. In addition, it is to be noted that the effectiveness of the active control device is overestimated when it is designed without considering the effects of torsional coupling.

VIBRATION SUPPRESSION USING A LASER VIBROMETER AND PIEZOCERAMIC PATCHES

A new vibration suppression technique is investigated that uses a scanning laser Doppler vibrometer to measure structural velocities for feedback in a control system. Piezoceramic patches are used for control actuators and to measure strains for feedback in the control system. Simulations using a finite-element model of a cantilever beam and laser sensing showed that if the laser can be scanned faster than the highest natural frequency of the beam, and all the velocity states are measured, the performance of classical linear optimal control can be approached. To further verify the technique, an experiment using a cantilever beam structure was built and the laser sensor was tested along with other types of sensors. In the experiments, only the first vibration mode of the cantilever beam was controlled because of a limitation in the speed of the scanning mirror used. The testing showed that a hybrid-sensing technique in which the laser and a piezoceramic patch are used simultaneously for sensing, and separate piezoceramic patches are used for actuation, was a very effective approach for vibration suppression. Although laser sensing requires expensive components, the technique proposed can be used for the control of structures that are large, inaccessible, require non-contact sensors, or where a large number of coordinates must be measured.

Measurement, filtering and control in quantum open dynamical systems

A Markovian model for a quantum automata, i.e. an open quantum dynamical system with input and output channels and a feedback is described. A multi-stage version of the theory of quantum measurement and statistical decisions applied to the optimal control problem for quantum dynamical discrete-time objects is developed. Quantum analogies of Stratonovich nonstationary filtering and Bellman quantum dynamical programming for the discrete time are obtained. The Gaussian case of quantum one-dimensional linear Markovian dynamical system with a quantum linear transmission line is studied. The optimal quantum multi-stage decision rule consisting of the classical linear optimal control strategy and quantum optimal filtering procedure is found. The latter contains the optimal quantum coherent measurement on the output of the line and the recursive processing by Kalman-Busy filter. All the results are illustrated by an example of the optimal control problem for a quantum open oscillator at the input of a quantum wave transmission line.