Finite-horizon Linear Quadratic Regulator Research Articles

Hierarchical bi-level load frequency control for multi-area interconnected power systems

To overcome the shortcomings of proportional–integral (PI) control, a linear quadratic regulator (LQR) is proposed in this study, to determine the set points of the PI controllers in a two-level hierarchical manner. The application of the proposed scheme is studied on load frequency control (LFC) of a power system with multiple control areas (CAs). At the lower control level, a discrete PI controllers are used to regulate the frequency and tie-line powers of each of the CAs. To achieve an improved closed loop performance, the local PI controllers are optimally tuned using moth flame optimization (MFO) algorithm by minimizing the Integral Square Error (ISE) of the state errors with respect to local information and interactions with neighboring CAs. While at the upper control layer, an optimized finite-horizon LQR is applied as a guide to establish the optimal set-points of the lower level PI controllers. Since only few of the state variables can be accessed, Kalman filter (KF) is applied as an observer for the estimation of the remaining states. The efficacy of the proposed scheme is verified by implementing it on a three-CA system perturbed with a step and random net disturbances formed by combining the fluctuations in the output power of DFIG-based wind turbine generator and random load demand. From the simulation results, it is established that the proposed LQR-PI supervisory scheme has outperformed the conventional PI control scheme in optimality and stability.

Wireless networked control over lossy uplinks abstracted by finite-state Markov channels

Networked control systems using wireless links to convey information among sensors, controllers, and actuators greatly benefit from having an accurate estimate of the communication channel condition. To this end, the finite-state Markov channel abstraction allows for reliable channel state estimation. This paper develops a Markov jump linear system representation for wireless networked control with intermittent channel state observation, message losses, and generalized hold-input dropout compensation. Furthermore, it exploits the emerging structural properties of the system to solve the finite-horizon linear quadratic regulation problem efficiently.

Robust Optimal Output-Tracking Control of Constrained Mechanical Systems With Application to Autonomous Rovers

This article presents a robust optimal output-tracking control strategy for underactuated mechanical systems whose motion is restricted by mixed holonomic and nonholonomic constraints. Autonomous rovers/cars and unmanned underwater/aerial vehicles are a few examples of such systems that must often operate in harsh environments and under uncertain conditions. We present a comprehensive control analysis of this large class of nonlinear systems, including the existing studies on local reachability and static state feedback linearization. We also propose a local observability analysis of the feedback transformed input–output linearized systems. Based on the input–output linearization of the holonomically restricted nominal system, we develop a sliding mode control strategy that is robust against the projected effects of uncertainties and disturbances on the system’s output. Asymptotic stability of the output toward a bounded desired trajectory is proven using Lyapunov’s direct method, while the system’s internal stability (in the sense of boundedness) is investigated based on the notion of tracking-error zero dynamics. Time-dependent bounded matched uncertainties in the inertia parameters and disturbance forces arising in the unrestricted system are considered in this study. We propose an optimal sliding manifold according to the finite-horizon linear-quadratic regulator design problem with split boundary value conditions. The developed control strategy is implemented on a six-wheel autonomous Lunar rover in a simulation environment and its performance is compared with that of an optimal proportional–integral–derivative feedback, feedforward controller. The optimal sliding mode controller shows superior performance in trajectory tracking with acceptable control actions.

Model-free finite-horizon optimal tracking control of discrete-time linear systems

Conventionally, the finite-horizon linear quadratic tracking (FHLQT) problem relies on solving the time-varying Riccati equations and the time-varying non-causal difference equations as the system dynamics is known. In this paper, with unknown system dynamics being considered, a Q-function-based model-free method is developed to solve the FHLQT problem. First, an augmented system consisting of the controlled system and the desired trajectory system is formulated, and the FHLQT problem transforms to the finite-horizon linear quadratic regulator (FHLQR) problem with the augmented system. Then, a time-varying Q-function which depends explicitly on the control input is defined. With the defined time-varying Q-function, a model-free finite-horizon control method is developed to approximate the solutions of the time-varying Riccati equations of the transformed FHLQR problem. At last, simulation studies are carried out to verify the validity of the developed method.

Model Predictive Control of a Tandem-Rotor Helicopter With a Nonuniformly Spaced Prediction Horizon

This paper considers model predictive control of a tandem-rotor helicopter. The error is formulated using the matrix Lie group SE2(3). A reference trajectory to a target is calculated using a quartic guidance law, leveraging the differentially flat properties of the system, and refined using a finite-horizon linear quadratic regulator. The nonlinear system is linearized about the reference trajectory enabling the formulation of a quadratic program with control input, attitude keep-in zone, and attitude error constraints. A non-uniformly spaced prediction horizon is leveraged to capture the multi-timescale dynamics while keeping the problem size tractable. Monte-Carlo simulations demonstrate robustness of the proposed control structure to initial conditions, model uncertainty, and environmental disturbances.

Preview-based leader-following consensus control of distributed multi-agent systems

The preview-based leader-following consensus control of distributed multi-agent systems (MAS) with a directed acyclic graph is investigated in this paper. The external disturbance considered in this paper is the step function that can be previewed, and the information of the leader can be obtained in advance. The leader-following consensus control of MAS is solved by using the state augmentation technique and the preview control method that transforms the consensus control problem into the finite horizon linear quadratic regulator (LQR) problem and the infinite horizon LQR problem. In the finite horizon, the minimum principle is utilized to design the optimal controller. In the infinite horizon, the standard infinite horizon LQR is applied to devise the corresponding controller. Assuming the original systems are stabilisable and detectable, the distributed consensus controller guarantees the asymptotic consensus of the preview MAS. The simulation illustrates the effectiveness of the distributed leader-following consensus controller designed in this paper.

Optimal actuator switching synthesis of observer-based event-triggered state feedback control for distributed parameter systems

The study aims to explore the optimal actuator switching scheme of observer-based event-triggered state feedback control for distributed parameter systems. The performance of distributed parameter systems is improved through the observer-based event-triggered control, in which the state feedback is updated only when a triggered event happens. In such an event-triggered mechanism, the event-based closed-loop system and minimum time interval between consecutive events are bounded. Based on finite horizon linear quadratic regulator (LQR) optimal control, the optimal switching algorithm is proposed based on the event-triggered mechanism during an unfixed time interval. Finally, the proposed scheme is verified through a simulation case.

Data-Based Receding Horizon Control of Linear Network Systems

We propose a distributed data-based predictive control scheme to stabilize a network system described by linear dynamics. Agents cooperate to predict the future system evolution without knowledge of the dynamics, relying instead on learning a data-based representation from a single sample trajectory. We employ this representation to reformulate the finite-horizon Linear Quadratic Regulator problem as a network optimization with separable objective functions and locally expressible constraints. We show that the controller resulting from approximately solving this problem using a distributed optimization algorithm in a receding horizon manner is stabilizing. We validate our results through numerical simulations.

LQR Design under Stability Constraints

The solution of classic discrete-time, finite-horizon linear quadratic regulator (LQR) problem is well known in literature. By casting the solution to be a static state-feedback, we propose a new method that trades off low LQR objective value with closed-loop stability.

Solution for the Continuous-Time Infinite-Horizon Linear Quadratic Regulator Subject to Scalar State Constraints

This letter provides a solution for the continuous-time linear quadratic regulator (LQR) subject to a scalar state constraint. Using a dichotomy transformation, novel properties for the finite-horizon LQR are derived; the unknown boundary conditions are explicitly expressed as a function of the horizon length, the initial state, and the final state or cost of the final state. Practical relevance of these novel properties are demonstrated with an algorithm to compute the continuous-time LQR subject to a scalar state constraint. The proposed algorithm uses the analytical conditions for optimality, without a priori discretization, to find only those sampling time instances that mark the start and end of a constrained interval. Each subinterval consists of a finite-horizon LQR, hence, a solution can be efficiently computed and the computational complexity does not grow with the horizon length. In fact, an infinite horizon can be handled. The algorithm is demonstrated with a simulation example.

Data-driven Linear Quadratic Regulation via Semidefinite Programming

Abstract This paper studies the finite-horizon linear quadratic regulation problem where the dynamics of the system are assumed to be unknown and the state is accessible. Information on the system is given by a finite set of input-state data, where the input injected in the system is persistently exciting of a sufficiently high order. Using data, the optimal control law is then obtained as the solution of a suitable semidefinite program. The effectiveness of the approach is illustrated via numerical examples.

Nonlinear Model Predictive Control of Reentry Vehicles Based on Takagi-Sugeno Fuzzy Models

In this paper, we apply a discrete-time Takagi-Sugeno Fuzzy Model (TSFM) based model predictive controller (MPC) to a Martian aerocapture vehicle following an arbitrary trajectory. We compare two baseline controllers: a continuous-time TSFM based parallel distributed controller (PDC) and a finite-horizon linear quadratic regulator (LQR). We evaluate the change in velocity (ΔV) required to bring the orbit of the controlled exit conditions to the orbit of the reference trajectory exit conditions over a range of initial condition errors and perturbations to atmospheric density. The LQR controller was least robust but performed best in a smaller range of perturbations. The PDC controller was most robust but performed the worst. The MPC based controllers demonstrate a balance of robustness and performance.

Robust Control of the Sit-to-Stand Movement for a Powered Lower Limb Orthosis

The sit-to-stand movement is a key feature for wide adoption of powered lower limb orthoses for patients with complete paraplegia. In this paper we study the control of the ascending phase of the sit-to-stand movement for a minimally actuated powered lower limb orthosis at the hips. First, we generate a pool of finite horizon Linear Quadratic Regulator feedback gains, designed under the assumption that we can control not only the torque at the hips but also the loads at the shoulders that in reality are applied by the user. Next we conduct reachability analysis to define a performance metric measuring the robustness of each controller against parameter uncertainty, and choose the best controller from the pool with respect to this metric. Then, we replace the presumed shoulder control with an Iterative Learning Control algorithm as a substitute for human experiments. Indeed this algorithm obtains torque and forces at the shoulders that result in successful simulations of the sit-to-stand movement, regardless of parameter uncertainty and factors deliberately introduced to hinder learning. Thus it is reasonable to expect that the superior cognitive skills of real users will enable them to cooperate with the hip torque controller through training.

Modeling and Control of Underwater Mine Explorer Robot UX-1

This paper presents the design and experimental assessment of the control system for the UX-1 robot, a novel spherical underwater vehicle for flooded mine tunnel exploration. Propulsion and maneuvering are based on an innovative manifold system. First, the overall design concepts of the robot are presented. Then, a theoretical six degree-of-freedom (DOF) dynamic model of the system is derived. Based on the dynamic model, two control systems have been developed and tested, one based on the principle of nonlinear state feedback linearization and another based on a finite horizon linear quadratic regulator (LQR). A series of experimental tests have been carried out in a controlled environment to experimentally identify the complex parameters of the dynamic model. Furthermore, the two proposed controllers have been tested in underwater path tracking experiments designed to simulate navigation in mine tunnel environments. The experimental results demonstrated the effectiveness of both the proposed controllers and showed that the state feedback linearization controller outperforms the finite horizon LQR controller in terms of robustness and response time, while the LQR appears to be superior in terms of fall time.

Fast Trajectory Tracking of a Flexure-Based, Multiaxis Nanopositioner With 50-mm Travel

The precise, high-speed control of nanopositioning stages is critical for many microscale additive manufacturing systems, such as microscale selective laser sintering (μ-SLS), where high throughput is needed. In μ-SLS, the positioning stage requires achieving 10 Hz steps with sub-100 nm accuracy and a travel range of 50 mm. The open-loop resolution and settling time of the flexure stage presented in the study are found to be 63 nm and 1.27 s, respectively. This settling time is too large for its application in high-throughput μ-SLS. To improve the tracking performance of the stage for fast varying signals up to 10 Hz and achieve better resolution, this paper presents the design of a finite horizon linear quadratic regulator controller for the XY stage. Owing to the good damping properties of the controller, the resolution is improved to 8 nm and input signals with 10-Hz frequency are effectively tracked. With the addition of a resonant shifting control, the tracking bandwidth is improved to 23 Hz. The paper presents a unique combination of low cost, large travel, sub-10 nm resolution and high-bandwidth XY positioning system, which is not to be found in either off-the-shelf nanopositioning stages or in research laboratories.

Finite-horizon LQR controller for partially-observed Boolean dynamical systems

This paper proposes an approach for finite-horizon control of partially-observed Boolean dynamical systems (POBDS) with uncertain continuous control input and infinite observation space. To cope with the partial observability of states, the proposed method first maps the POBDS to an unnormalized belief space. The nonlinear dynamics in this continuous belief space are linearized over a nominal trajectory. Then, the optimal feedback controller is derived, based on the well-known linear quadratic regulator (LQR), to push the system to follow the nominal trajectory. This nominal trajectory is computed in a planning stage before starting execution, and updated efficiently during execution, whenever the system is found to deviate from the nominal trajectory. We prove that, under mild regularization conditions, the proposed controller approaches the cost of the nominal trajectory as the linearization error approaches zero. The performance of the proposed controller is demonstrated by numerical experiments with a Melanoma gene regulatory network observed through noisy gene expression measurements.

Hands-off Control for Discrete-time Linear Systems subject to Polytopic Uncertainties

Abstract This paper develops approaches to the hands-off control problem subject to performance constraints for discrete-time linear systems. The approaches minimize the l1-norm of the control input to acquire the hands-off property, while satisfying the performance constraints that are given in terms of the quadratic cost of states and inputs with respect to the optimal solution to the finite-horizon linear quadratic regulator problem. We consider three kinds of the input and state matrices for the system; 1) known, 2) uncertain but contained in a known discrete set, and 3) uncertain but contained in a known polytopic uncertainty set. For the first two cases, we show that each problem is formulated as an l1 optimization that is expressed as a second-order cone programming. We also show that the last case leads to a second-order cone programming after relaxations. A numerical example is included to illustrate the validity of the proposed approach.

Hamiltonian Decomposition for Model Predictive Control

We present the application of an eigenvalue decomposition for the solution of the optimal control problem in model predictive control (MPC). This approach can be used as an alternative to the Riccati recursion that occurs within interior-point solvers, which are used to compute the solution of the optimization problem in constrained MPC. We first demonstrate that a known method applied to the finite-horizon linear quadratic regulator (LQR) for free-final state can be extended to the LQ tracking problem. Likewise, the receding-horizon idea is implemented on these two optimal control problems. In addition, the implications for online implementation are discussed.

Dual-loop iterative optimal control for the finite horizon LQR problem with unknown dynamics

Achieving optimal performance over a finite-time horizon has gained a lot of attention in many engineering applications. Among them, the Finite Horizon Linear Quadratic Regulator (FHLQR) formulation for continuous-time linear time-varying systems has been well studied, with an optimal solution characterised by the Differential Riccati Equation (DRE). The solution of the DRE requires that the exact system dynamics are known. However, this assumption may not always hold, as the plant model might not be completely known or may change over time due to wear and tear. This paper proposes a dual-loop iterative algorithm to find the optimal solutions of the FHLQR formulation for continuous-time LTV systems. The inner loop utilises input trajectories based on an estimate of the optimal control gain with the addition of some excitation noise, and produces measured state trajectories. The outer loop improves the estimate of the optimal control gain utilising these measured state trajectories. It is shown in this work that with appropriate selection of the discretisation parameter T and the set of excitation signals, the proposed dual-loop iterative algorithm can converge to an arbitrarily small neighbourhood of the optimal solution. A simulation example demonstrates the effectiveness of the proposed method.

Optimal control data scheduling with limited controller-plant communication

This paper considers optimal control data scheduling for finite-horizon linear quadratic regulation (LQR) control of scalar systems with limited controller-plant communication. Both the single-system and multiple-system scenarios are studied. For the first scenario, we derive the necessary and sufficient condition for a comparison function to be positive. Using this condition, the optimality of an explicit schedule is extended from unstable systems in the existing work to general systems. For the second scenario, we are able to construct explicit optimal scheduling policies for three particular classes of problems. Numerical examples are provided to illustrate the proposed results.