Discrete‐time State Feedback Control

Investigations are made in order to predict the performance capabilities of preview controlled active vehicle suspensions using limited-bandwidth actuators. Mathematical tools for the implementation of preview control in discrete time, which were established in earlier work, are refined. This allows the use of multiple preview-inputs and reduces the system-order of a discrete-time preview system significantly with respect to former methods, while describing the dynamic system in an exact form and yielding an equivalent control law. This method is applied to a limited-bandwidth quarter-car system, which incorporates an actuator in series with a passive spring, the combination being in parallel with a passive damper. The control design is carried out using discrete-time linear quadratic regulator (LQR) theory. The optimal preview control law working in a physical sense, is worked out regarding the feedforward gain matrix. The performance indices and their component parts are considered when varying the actuator-bandwidth and preview time.

The stability properties of the linear quadratic deterministic optimal regulator are investigated in the discrete-time case. Guaranteed phase- and gain- margins are obtained for the regulator in terms of the singular values of the matrices of the regulator state-space description. These margins are compared to their continuous-time counterparts and their implication on the regulator robustness is discussed.

Model predictive control (MPC) is a multi-objective control technique that can handle system constraints. However, the performance of an MPC controller highly relies on a proper prioritization weight for each objective, which highlights the need for a precise weight tuning technique. In this paper, we propose an analytical tuning technique by matching the MPC controller performance with the performance of a linear quadratic regulator (LQR) controller. The proposed methodology derives the transformation of a LQR weighting matrix with a fixed weighting factor using a discrete algebraic Riccati equation (DARE) and designs an MPC controller using the idea of a discrete time linear quadratic tracking problem (LQT) in the presence of constraints. The proposed methodology ensures optimal performance between unconstrained MPC and LQR controllers and provides a sub-optimal solution while the constraints are active during transient operations. The resulting MPC behaves as the discrete time LQR by selecting an appropriate weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. In this paper, the effectiveness of the proposed technique is investigated in the application of a novel vehicle collision avoidance system that is designed in the form of linear inequality constraints within MPC. The simulation results confirm the potency of the proposed MPC control technique in performing a safe, feasible and collision-free path while respecting the inputs, states and collision avoidance constraints.

This paper designs a discrete-time state-feedback output tracking control for the heart rate during treadmill exercise. Initially, the nonlinear model describing the relationship between the heart rate and the speed of a treadmill is discretized. Afterwards, a feedback linearization-based control law is proposed to achieve perfect output tracking. The control objective is to make the runner's heart rate follow a heart rate reference profile set by especialists as reference. The set-up of the problem in discrete time allows taking into consideration the effect of sampling during the controller design procedure instead of relegating it to the implementation stage. It will be shown that a linear state feedback controller is enough to make the nonlinear model's output track the reference profile regardless its possibly complex time variation. Since the full state is not available for measurement a reduced order state observer is incorporated into the discrete-time control law. Then, the continuous control command is generated by using a zero order hold (ZOH). The designed control system is tested on the original continuous-time nonlinear model by computer simulation to demonstrate the effectiveness of the proposed method to achieve the required objective.

In this paper, we consider a discrete-time stochastic control problem with uncertain initial and target states. We first discuss the connection between optimal transport and stochastic control problems of this form. Next, we formulate a linear-quadratic regulator problem where the initial and terminal states are distributed according to specified probability densities. A closed-form solution for the optimal transport map in the case of linear-time varying systems is derived, along with an algorithm for computing the optimal map. Two numerical examples pertaining to swarm deployment demonstrate the practical applicability of the model, and performance of the numerical method.

This paper presents a new method for the design of control laws for systems with on-off nonlinear actuators. The new methodology improves the design of control laws for pulse-modulated systems by allowing the use of continuous-time multi-inputlmultioutput design procedures. Discrete-time state feedback control gains are developed from the of continuous-time feedback gains, based on a geometric series approximation, to closely match the states of the closed-loop hybrid system to those of the original designed closed-loop continuous-time system at each sampling instant. A delay is then incorporated into a pulse-width modulator design to closely match the states of the modulated system to those of the discrete-time pulse-amplitude modulated system. The delay time is improved from previous work to provide a close match between the states for pulse durations of all sizes, in effect matching the states of the pulse-width modulated system to those of the closed-loop continuous-time system. The new control law design is applied to the problem of attitude control of the Space Shuttle Orbiter with the Hubble Space Telescope deployed on its remote manipulator. * Senior Member Technical Staff. Member A I M . t Professor, Electrical Engineering. Member AIAA. tt Aerospace Engineer. Member AIAA. Copyright O 1995 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Many paradigms are available for the synthesis of control laws for continuous-time systems, and though many methods also exist for developing discrete-time or digital controllers, it is often desirable to design a control law for the continuous-time plant. It is often more direct to determine the control specifications in the continuous-time or frequency domain. Also, direct digital control law design has difficulty in predetermining the hybrid control specifications and ignores inter sampling effects. Yet, these continuoustime control laws are generally implemented via discrete-time computer control, requiring that a discrete-time control, equivalent to the continuous-time control, be developed. The process of developing discrete-time control laws that provide closed-loop performance of the hybrid system equivalent to the performance of the designed closed-loop continuoustime system is known as redesign. Several techniques have been developed to digitally redesign continuous-time feedback controllers[l] ,[2] ,[3]. Unfortunately, the principal applied for the design of discrete-time control laws, i.e., that the amplitude of the control signal can take any finite value, which is why discrete-time control is often referred to as pulse-amplitude control (PAM), is not valid for all computer controlled systems. Systems that utilize onoff control devices, such as the reaction jet or electronic relay, can only provide a single finite value of control. To provide control with these types of actuator, either complex non-linear algorithms[4] must be developed or the control must be applied by modulating the pulse width, the pulse frequency or both the pulse width and frequency. These modulation techniques, in particular pulse-width modulation (PWM), allow the design of linear control laws to control non-linear systems. Although methods exist for the design of linear control laws for pulse width modulation systems[5], they still

In this paper, a novel 2-DOF high-speed/high-accuracy robotic system is proposed, which consists of two permanent linear DC servomotors and a planar parallel mechanism. Analysis of robotic dynamical model verifies that load variation is violent. Due to the direct-drive principle of a linear motor, the load disturbance can have a significant effect on position accuracy. For this purpose, a discrete-time disturbance observer, feedback controller and ZPECT feedforward controller is presented. The disturbance observer, which is applied to the plant velocity loop, provides robustness for design of both the feedback controller and the feedforward controller. The feedback controller is intended to make the command following properties fast, and the feedforward term realized as a Zero Phase Error Tracking Controller improves tracking. In the experiment, robotic max acceleration is equal to 20m/s2, max velocity is more than 1 m/s, and resolution is 0.5 μm. While the tracking errors are less than 50 μm, the steady-state errors are less than 1 μm, and steady time is less then 42 ms. Experiment results show that: high-performance motion system can be obtained with this discrete-time controller, even in large load variance motion system.

This paper presents a hierarchical control architecture for autonomous mobile robots (AMRs) that enhances obstacle avoidance and trajectory smoothness in industrial-like navigation tasks. A global A*-based planner generates a collision-free path on an occupancy grid map and triggers replanning when newly detected obstacles invalidate the current route; dynamic obstacles are handled by projecting their short horizon predicted occupancy as virtual walls on the grid. A timed elastic band (TEB) local planner refines the global path in the spatiotemporal domain to satisfy nonholonomic constraints, velocity and acceleration limits, and obstacle- avoidance requirements, producing a smooth reference trajectory with moderated curvature variations. For tracking, a four-wheel steering (4WS) 2-degrees of freedom (DOF) lateral dynamics model is controlled by a discrete-time linear quadratic regulator (LQR) feedback controller combined with an Ackermann-based feedforward term to proactively compensate the reference curvature and stabilize lateral and heading errors under steering constraints. Simulations in MATLAB and a high-fidelity multi-DOF vehicle dynamics simulation program compare five schemes under both static and dynamic obstacle scenarios: A* with Pure Pursuit (short/medium/long look-ahead), A* with LQR/feedforward, and the proposed A*-TEB-LQR/feedforward architecture. The proposed method successfully avoids obstacles and achieves the smallest average curvature and shortest arrival time among collision-free algorithms, while Pure Pursuit suffers from look-ahead sensitivity and A*-LQR/feedforward shows increased local curvature near obstacles.

It is very challenging to improve the dynamic response speed and stability of digital control system, due to the inherent defects of digital controller such as quantization effect and delay in sampling step. This brief proposes an adaptive logarithmic state feedback control (ALSFC) for three-phase grid-tied and off-grid inverters which can be specially applied in digital control system. In the state feedback control system, the feedback coefficient is designed as adaptive logarithmic function of the difference value between the actual state variable and the reference, so as to make the closed-loop system converge to new equilibrium point after about one switching cycle. Comparative experiments verify that the proposed control, carried out by digital controller, has the dynamic response characteristics of quasi one-switching-cycle with its response speed is significantly higher than that of PI control, and has good stability with lower THD.

This paper introduces an application of Multi-Objective Evolution Algorithm (MOEA) to design Q and R weighting matrices in Linear Quadratic regulators (LQR). Considering the difficulty of designing weighting matrices for a linear quadratic regulator, a multi-objective evolutionary algorithm based approach is proposed. The LQR weighting matrices, state feedback control rate and consequently the optimal controller are obtained by means of establishing the multi-objective optimization model of LQR weighting matrices and applying MOEA to it, which makes control system meet multiple performance indexes simultaneously. Controller of double inverted pendulum system is designed using the proposed approach. Simulation results show that it has shorter adjusting time and smaller amplitude value deviating from steady-state than a Non-dominated Sorting Genetic Algorithm LQR ( NSGA- LQR )weighting matrices design approach.

For the continuous infinite horizon time-invariant linear quadratic regulator problem (LQR), in the paper, the optimal state feedback controller based on the estimated state of the observer can be decoupled by the proposed approach which resulting one continuous time algebraic Riccati equation (CARE) for the controller design and one matrix equality equation (MEE) for the observer design. A coupling term related the CARE of the controller is found to be existed in the MEE of the observer. Unlike the separate principle to design the controller and observer separately without any coupling term, the design of the observer should consider the coupling term related to the CARE of the controller. The coupling problem between the controller and the observer usually exists in the linear matrix inequality (LMI) approach, and it is the main problem to be solved. The two-stage scheme is popular in the LMI approach, and the proposed method is similar to it, but adopting equality instead of inequality.

In this paper, an optimal tracking control approach based on adaptive dynamic programming (ADP) algorithm is proposed to solve the linear quadratic regulation (LQR) problems for unknown discrete-time systems in an online fashion. First, we convert the optimal tracking problem into designing infinite-horizon optimal regulator for the tracking error dynamics based on the system transformation. Then we expand the error state equation by the history data of control and state. The iterative ADP algorithm of policy iteration (PI) and value iteration (VI) are introduced to solve the value function of the controlled system. It is shown that the proposed ADP algorithm solves the LQR without requiring any knowledge of the system dynamics. The simulation results show the convergence and effectiveness of the proposed control scheme.

SUMMARY The development of control strategies for controllable passive dampers — ‘semiactive’ damping devices — is complicated by the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-loop system with any form of parametric feedback control. Control design for nonlinear systems is often achieved by designing a control for a linearized model since strategies for linear systems are straightforward. One such approach is clipped-optimal control in which the desired damper forces are determined from an optimal controller (e.g., linear quadratic regulator, linear quadratic Gaussian, H2 and H∞), which is designed assuming that the damping devices are fully linear actuators that can exert any forces (dissipative or nondissipative), and a secondary bang–bang controller commands the controllable damper to exert forces as close as possible to the desired forces. However, designs using any linearized model generally result in suboptimal (and sometimes very poor) performance because the linear actuator assumption differs from the actual implementation with a dissipative damping device. Thus, one must generally resort to a large-scale parameter study in which the nonlinear system is simulated many times to determine control strategies that are actually optimal for the nonlinear controlled closed-loop system. This paper demonstrates how an approach developed by the authors, which can reduce the model of a system with local modifications to a lower order set of Volterra integral equations in the modification forces, can significantly decrease the computational burden of a complex control design study for controllable dampers. The approach is first adapted to use a state feedback control with a bang–bang on–off secondary controller. Then, the structural and damping device models are detailed. The optimal semiactive control design is demonstrated via both a parameter study and an optimization algorithm. Numerical examples demonstrate that the approach is effective and, compared with a conventional simulation approach, can reduce computation time by two orders of magnitude. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a novel synchronous Q-learning method is proposed for solving discrete-time linear quadratic regulator (LQR) problems. To begin with, the Bellman equation corresponding to the optimal Q-function is reformulated into a consistency equation on the parameters of the optimal Q-function and the optimal controller. Then an actor-critic structure is introduced to learn the optimal Q-function and the optimal controller online in real time by using the state samples generated by the behavior policy. Particularly, the proposed synchronous Q-learning scheme simultaneously updates the Q-function approximation and the optimal controller approximation, rather than iterating between policy evaluation and policy improvement. The proposed control scheme is proved to be uniformly ultimately bounded (UUB) under appropriate learning rates, provided that certain persistence of excitation (PE) conditions are satisfied. Besides, the PE conditions can be easily met by injecting appropriate exploration noise into the behavior policy without causing any excitation noise bias. Finally, one simulation example is provided to verify the effectiveness of the proposed synchronous Q-learning method.

The Active Magnetic Bearing system (AMB) is a mechatronic device which is used to suspend rotating parts of a machine so that they rotate without contact to the stationary part of the machine. AMBs are highly nonlinear, non-minimum phase and inherently unstable. This paper has aimed to obtain a robust optimal state-feedback control system for the stabilization of the AMB System, with the help of Genetic algorithm (GA) as an optimization tool which will solve the tedious manual tuning of the weighting matrices in the design of linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG).The system’s mathematical model has been developed and also the properties of the uncontrolled system have been analyzed. Since AMB is a MIMO system, the interaction of the inputs with the outputs has been analyzed using relative gain array analysis and frequency domain analysis of the system transfer functions. Then, the optimal state feedback controllers have been developed. Here, LQR and LQG controllers are developed.Finally, Comparative analysis between the controllers and between the design methods was carried out. The proposed GA based design methodology has resulted good Performance. In addition, the GA based design has also resulted improvements in robustness of the control systems. As far as gain margin (GM) and phase margin (PM) are concerned GA has resulted increase of 8.0*10–4 db and 6.02*10–3° in GM and PM respectively for LQR. Whereas, in LQG GA has resulted an increase of 3.8*10–5 db and 2.54*10–4°. KeywordsActive magnetic bearingGenetic algorithmLinear quadratic GaussianKalman filter