Linear Quadratic Tracking Research Articles

Improved Control for Industrial Systems Over Model Uncertainty: A Receding Horizon Expanded State Space Control Approach

In this paper, an improved linear quadratic tracking control strategy using a new receding horizon expanded state space (ESS) model is proposed. Unlike traditional state space model, the new ESS model first facilitates the combination of the system state variables and tracking errors, then a subsequent new receding horizon improved linear quadratic tracking control (RHLQTC) is designed with more degrees of freedoms for the adjustment of control parameters, which yields improved control performance compared with traditional RHLQTC. Finally, simulations on a typical reverse process are done in comparison with traditional RHLQTC in terms of both servo and regulatory performance, where results show that the proposed controller provides improved performance under both situations.

A Modified Functional Observer-Based EID Estimator for Unknown Continuous-Time Singular Systems

This paper presents the design of a linear quadratic analog tracker (LQAT) based on the observer–Kalman-filter identification (OKID) method and the design of a modified functional observer-based equivalent input disturbance (EID) estimator for unknown square–non-square singular analog systems with unknown input and output disturbances. First, an equivalent mathematical model of the singular analog system is presented to simulate the time response of continuous-time linear singular analog systems to arbitrary inputs via the model conversion method. Then, for the unknown singular analog system, it constructs a linear quadratic analog tracker with state feedback and feed-forward gains based on the off-line OKID method. Furthermore, it extends the design methodology of the EID estimator for strictly proper regular systems with unknown matched–mismatched input and output disturbances to proper regular systems. It is important to mention that the newly developed modified functional observer for proper systems is used to estimate the unknown EID of singular analog systems and that the constraints on the dimensions of unknown disturbances can be eliminated by using the newly proposed EID estimation method. The contributions of this paper can be listed as follows: (1) based on both the OKID method and the discrete-to-continuous model conversion, the simulation of the time responses of the continuous-time linear singular models (which are not feasible using existing MATLAB toolboxes) become feasible; (2) for effective control of the unknown singular analog system, an off-line OKID method is proposed to design an LQAT with state feedback and feed-forward gains; and (3) based on the newly developed modified functional observer for the reduced-order proper regular system, the original EID estimator in the literature is newly extended to estimate the EID from the unknown strictly proper singular analog system, without the original dimensional constraints of the disturbances. It is important to mention that the disturbances of interest can be unknown matched–mismatched input and output disturbances.

Trajectory tracking control of a VTOL unmanned aerial vehicle using offset-free tracking MPC

Designing a stable and robust flight control system for an Unmanned Aerial Vehicle (UAV) is an arduous task. This paper addresses the trajectory tracking control problem of a Ducted Fan UAV (DFUAV) using offset-free Model Predictive Control (MPC) technique in the presence of various uncertainties and external disturbances. The designed strategy aims to ensure adequate flight robustness and stability while overcoming the effects of time delays, parametric uncertainties, and disturbances. The six degrees of freedom DFUAV model is divided into three flight modes based on its airspeed, namely the hover, transition, and cruise mode. The Dryden wind turbulence is applied to the DFUAV in the linear and angular velocity component. Moreover, different uncertainties such as parametric, time delays in state and input, are introduced in translational and rotational components. From the previous work, the Linear Quadratic Tracker with Integrator (LQTI) is used for comparison to corroborate the performance of the designed controller. Simulations are computed to investigate the control performance for the aforementioned modes and different flight phases including the autonomous flight to validate the performance of the designed strategy. Finally, discussions are provided to demonstrate the effectiveness of the given methodology.

Adaptive dynamic programming for model‐free tracking of trajectories with time‐varying parameters

SummaryRecently proposed adaptive dynamic programming (ADP) tracking controllers assume that the reference trajectory follows time‐invariant exo‐system dynamics—an assumption that does not hold for many applications. In order to overcome this limitation, we propose a new Q‐function that explicitly incorporates a parametrized approximation of the reference trajectory. This allows learning to track a general class of trajectories by means of ADP. Once our Q‐function has been learned, the associated controller handles time‐varying reference trajectories without the need for further training and independent of exo‐system dynamics. After proposing this general model‐free off‐policy tracking method, we provide an analysis of the important special case of linear quadratic tracking. An example demonstrates that our new method successfully learns the optimal tracking controller and outperforms existing approaches in terms of tracking error and cost.

Linear-Quadratic Tracking Control of a Commercial Vehicle Air Brake System

This article proposes to utilize linear-quadratic tracking (LQT) control to reduce the air brake system response time and vehicle stopping distance, and hence, to significantly improve the system performance of a commercial vehicle air brake system equipped with electro-pneumatic proportional valve actuators. The nonlinear dynamic model of the air brake system, consisting of a control actuator (a proportional valve) and braking actuator (the brake chamber), is developed and linearized using the q-Markov COVariance Equivalent Realization (q-Markov Cover) method in our early work. Based on the linearized dynamic model, an infinite horizon LQT controller is designed along with Kalman state estimation at each linearized operational condition. To apply the LQT control law over a wide operational range to track the target pressure, the designed controller was interpolated between the neighboring controllers to have a control law cover the entire operational range. To validate this control law, the control scheme is implemented into a dSPACE unit and validated through bench tests under different supply and reference pressures. The LQT control performance is also compared with the PID (proportional-integral-derivative) one. The bench test results confirm the effectiveness of the proposed control scheme.

Distributed Linear Quadratic Tracking Control for Leader-Follower Multi-Agent Systems: A Suboptimality Approach

In this paper, we extend the results from Jiao et al. (2019) on distributed linear quadratic control for leaderless multi-agent systems to the case of distributed linear quadratic tracking control for leader-follower multi-agent systems. Given one autonomous leader and a number of homogeneous followers, we introduce an associated global quadratic cost functional. We assume that the leader shares its state information with at least one of the followers and the communication between the followers is represented by a connected simple undirected graph. Our objective is to design distributed control laws such that the controlled network reaches tracking consensus and, moreover, the associated cost is smaller than a given tolerance for all initial states bounded in norm by a given radius. We establish a centralized design method for computing such suboptimal control laws, involving the solution of a single Riccati inequality of dimension equal to the dimension of the local agent dynamics, and the smallest and the largest eigenvalue of a given positive definite matrix involving the underlying graph. The proposed design method is illustrated by a simulation example.

AN OPTIMAL FEEDBACK LINEARIZATION APPROACH FOR SOFT LANDING OF ELECTROMECHANICAL VALVE ACTUATOR IN CAMLESS INTERNAL COMBUSTION ENGINES

The fuel efficiency, maximum torque and power, and pollutant emissions of internal combustion engines have improved rather than conventional crankshaft system. These improvements are the result of replacing the crankshaft with a camless actuation technology. Soft landing, tracking the desired trajectory, valve seating in a low velocity and having a small time for valve opening and closing are some of the main conflicting objectives in camless engines. Due to the nonlinear model of the system, feedback linearization control is presented for dealing with the system challenges in this paper. Moreover, to optimize the system response, the concept of feedback linearization control is incorporated into the linear quadratic tracker and makes the proposed structure named FLQT. The simulation results indicate the optimal feedback linearization controller has a good performance in the realization of desired goals of the system.

Design of Robust Trackers and Unknown Nonlinear Perturbation Estimators for a Class of Nonlinear Systems: HTRDNA Algorithm for Tracker Optimization

A robust linear quadratic analog tracker (LQAT) consisting of proportional-integral-derivative (PID) controller, sliding mode control (SMC), and perturbation estimator is proposed for a class of nonlinear systems with unknown nonlinear perturbation and direct feed-through term. Since the derivative type (D-type) controller is very sensitive to the state varying, a new D-type controller design algorithm is developed to avoid an unreasonable large value of the controller gain. Moreover, the boundary of D-type controller is discussed. To cope with the unknown perturbation effect, SMC is utilized. Based on the fast response of SMC controlled systems, the proposed perturbation estimator can estimate unknown nonlinear perturbation and improve the tracking performance. Furthermore, in order to tune the PID controller gains in the designed tracker, the nonlinear perturbation is eliminated by the SMC-based perturbation estimator first, then a hybrid Taguchi real coded DNA (HTRDNA) algorithm is newly proposed for the PID controller optimization. Compared with traditional DNA, a new HTRDNA is developed to improve the convergence performance and effectiveness. Numerical simulations are given to demonstrate the performance of the proposed method.

Yaw Angle Tracking Control Design of Underactuated AUV By Using State Dependent Riccati Equations (SDRE)-LQT

In realizing yaw angle control tracking on AUV, the use of the State Dependent Riccati Equations method based on Linear Quadratic Tracking (SDRE-LQT) is realized. This algorithm calculates changes in yaw angle tracking problems through calculation of parameter changes from online AUV with Algebraic Riccati Equations.So that the control signal given to the plant can follow the changing conditions of the plant itself.

Adaptive optimal output feedback tracking control for unknown discrete‐time linear systems using a combined reinforcement Q‐learning and internal model method

This study addresses the novel output feedback-based reinforcement Q-learning algorithms for optimal linear quadratic tracking problem of unknown discrete-time systems. An augmented system composed of the original controlled system and the reference trajectory dynamic is first constructed. Then, learning algorithms including on-policy and off-policy approaches are both developed to solve the optimal tracking control problem with unknown augmented system dynamics. In both the optimal tracking control policies, a two-stage framework is proposed composed of two controllers, where the internal model controller is used to collect some data for the next process, and then the output feedback Q-learning scheme is able to learn the optimal tracking controller online by using the past input, output, and reference trajectory data of the augmented system. Finally, simulation results are provided to verify the effectiveness of the proposed scheme.

Economic Adaptive Cruise Control for a Power Split Hybrid Electric Vehicle

An extended adaptive cruise control (ACC), called economic adaptive cruise control, is proposed to improve the fuel economy for power split hybrid electric vehicles (HEV) by optimizing the vehicle route, speed, and powertrain control simultaneously. The economic route and speed for a given origin-destination pair with the given expected trip time are optimized by a proposed vehicle macroscopic motion planning method, where the powertrain is optimized by a proposed global power distribution (GPD) strategy. The HEV powertrain controller, consisting of feedforward and feedback control schemes, is developed for real-time cruise control, where the former is based on the GPD strategy and the latter is based on the receding horizon linear quadratic tracking method. In addition, a mode-switch local optimization method is used to modify the reference speed for passing, traffic jam and light crossing. A co-simulation model, combining the SUMO traffic model and Simulink hybrid powertrain model, is developed and used for validating the proposed EACC strategy. The co-simulation results indicate that the proposed EACC is able to decrease the fuel consumption by more than 30% comparing with the power follower strategy adopting the fastest route. Note that even with the same powertrain controller, the economic route and speed stratgey is able to improve the fuel economy by 14.21%, comparing with the fastest route without optimization.

Adaptive LQT Valve Timing Control for an Electro-Hydraulic Variable Valve Actuator

This paper presents an optimal tracking control of engine valve timing for an electro-hydraulic variable valve actuator utilizing the adaptively estimated supply pressure that is the main source for valve timing variation. The system supply pressure is online estimated through a model-guided adaptive estimator with one-step prediction capability. The estimation model consists of a static guidance model, an in-event valve model, and a one-step prediction model. Using the estimation model, the adaptive estimator generates a nominal estimation with accelerated convergence, estimates the pressure residual error using in-event dynamics, and provides one-step forward pressure prediction along with the resulting pressure fluctuation. The linear quadratic tracking (LQT) control is used to track the desired valve timing based on an event-by-event valve model, where the adaptively estimated pressure variation is used as the feedforward control in the LQT control to improve the tracking accuracy and a Kalman filter is used to estimate the system states for the LQT state feedback control. The parameter estimation and control algorithms are validated through both the simulation and test bench studies by comparing different estimation and control schemes. Both the steady-state and transient operational conditions are investigated, and the effectiveness of the proposed adaptive LQT valve timing control is demonstrated.

A modified functional observer-based EID estimator for unknown sampled-data singular systems

This paper presents both an observer/Kalman filter identification (OKID) method-based linear quadratic digital tracker (LQDT) and a new functional observer-based discrete equivalent input disturbance (EID) estimator, for unknown square/non-square singular sampled-data systems with unknown input and output disturbances. Initially, it is shown that even if the solution of the impulsive mode-free singular linear quadratic tracker problem exists, under some quite general conditions the tracker-based generalised algebraic Riccati equation (GARE) might have no solution. To overcome this issue, an innovative methodology that involves an introduced reduced-order equivalent proper regular model (EPRM) of the singular system is presented, to make the tracker-based GARE solvable for a wide class of servo control problems for a singular system. Furthermore, a new functional observer-based design methodology for the discrete EID estimator of the discrete EPRM is proposed in this paper, for a proper sampled-data system with unknown matched/mismatched input and output disturbances. Finally, a discrete-time EPRM of the unknown singular sampled-data system is constructed using the off-line OKID method and the robust prediction-based state-estimate optimal LQDT, associated with the (plug-in) discrete EID estimator, is developed for the singular sampled-data system.

Optimal output anti‐synchronisation of cooperative‐competitive multi‐agent systems via distributed observer

This study considers an optimal output anti-synchronisation problem for linear multi-agent systems with non-identical dynamics and antagonistic interactions. It is assumed that the agents are non-introspective, i.e. they cannot get their own states or outputs, and can only get a linear combination of relative outputs in the neighbourhood. First, a state transformation is made, which aims at reducing the dimension of the model underlying the output anti-synchronisation errors by removing redundant modes that have no effect on the output anti-synchronisation errors. Then, optimal state feedback controllers are proposed for agents to realise the output anti-synchronisation by constructing discounted performance functions and by solving the discounted algebraic Riccati equation. Moreover, by using an iterative technique, an offline algorithm is provided to solve the linear quadratic tracking problem. In addition, a distributed node-based adaptive observer is proposed to produce an estimate of the state with only relative output information of neighbouring agents by utilising a high-gain approach. The convergence of the proposed adaptive observer is analysed with the help of a Lyapunov function method. Finally, numerical simulations are carried out to demonstrate the effectiveness of the proposed approach.

Design of controllers for omnidirectional robot based on the ystem identification technique for trajectory tracking

Omnidirectional wheeled mobile robots are gaining a lot of importance owing to its widespread applications in healthcare, industries and various other service sectors. Most of these applications demand the mobile base to track the desired trajectory with the desired speed, optimal time and energy consumption. This work focusses on the trajectory tracking of the four-wheeled mobile robot with omnidirectional wheels. Most of the existing works make use of the models developed using first principles but in this work, system modeling is carried out by the system identification technique using the data obtained from the fabricated mobile base. Trajectory tracking is then carried out using PID and Linear Quadratic Tracking (LQT) based controllers. The simulation results are compared for tracking errors and energy consumption and the results obtained in real time are presented.

An ADDHP-based Q-learning algorithm for optimal tracking control of linear discrete-time systems with unknown dynamics

This paper investigates a novel Q-learning algorithm based on action dependent dual heuristic programming (ADDHP) to solve the infinite-time domain linear quadratic tracker (LQT) for unknown linear discrete-time systems. The proposed method is conducted based on only system data without requiring the knowledge of the system matrices. After the reference system is determined, an augmented system composed of the original system and the reference system is constructed, and it is proved that the value function of LQT is quadratic concerning the state of the augmented system. Using the quadratic value function, the augmented algebraic Riccati equation (ARE) is derived to solve the LQT. Due to the difficulty of directly solving the augmented ARE, a Q-learning algorithm based on ADDHP structure is used to solve this problem. With unknown system matrices, a model neural network is developed to reconstruct system dynamics incorporating stability analysis. The estimated system matrices are employed to the proposed algorithm to calculate the optimal control by policy iteration. Moreover, the convergence of the algorithm is proved. Two simulation examples are used to validate the performance of the method, where all results have demonstrated the effectiveness of the proposed ADDHP-based Q-learning method without a priori knowledge of system matrices for LQT.

Trajectory optimization for mobile robots using model predictive control

The goal of this article is trajectory generation for biped robots based on Model Predictive Control (MPC) and the receding-horizon principle. Specifically, we want to minimize the error between the desired CoM and ZMP trajectory and the actual one and the cancellation of the shock gradient of the CoM and ZMP movements. Model predictive control (MPC) consist in a finite horizon optimal control scheme which uses a prediction model to predict vehicle response and future states, thus minimizing the current error and optimizing the future trajectory within the prediction horizon. The proposed algorithm will provide a trajectory of control inputs which will optimize the system states utilizing a quadratic form cost function similar to standard linear quadratic tracking. Specific to finite horizon control, the cost is summed over the finite prediction horizon of time length, rather than over an infinite time horizon. Many techniques have been proposed, developed, and applied to solve this constrained optimization problem for the mobile robots. With our aproach we try to investigate how is the MPC framework is applicable to trajectory generation for point-to-point problems with a fixed final time and to find a set of assumptions and methods that allow for real-time solutions.

A New Synthetic Minmax Optimization Design of ${H_{\infty} }$ LQ Tracking Control for Industrial Processes Under Partial Actuator Failure

To cope with the control problem of industrial processes with partial actuator failure and disturbance, an improved synthetic minmax optimization design-based ${H_{\infty} }$ linear quadratic (LQ) tracking control strategy is presented in this paper. A new state space model that integrates the dynamics of the process states and the tracking error is first formulated as the basic dynamic process representation. By introducing the new state space model formulation, a new minmax optimization of the ${H_{\infty} }$ norm of the system performance is presented for LQ tracking control design, where more degrees of freedom are offered through the simultaneous adjusting of the tracking dynamics and the state regulation in the cost function, resulting in the enhanced control performance over traditional ${H_{\infty} }$ LQ tracking control. Meanwhile, the nominal closed-loop system stability and robustness under uncertainty are discussed and the monotonicity conditions that are sufficient for both nominal and robust system stability are derived. The case studies on an injection molding process and a nonlinear batch reactor under partial actuator failures and disturbance show the effectiveness of the proposed strategy.

Profile Tracking for an Electro-Hydraulic Variable Valve Actuator Using Receding Horizon LQT

The camless valve system is able to provide flexible engine-valve profiles (timing, duration, lift, etc.) to optimize the performance of internal combustion engines. To provide a precise valve profile of an electro-hydraulic variable valve actuator and achieve the desired engine performance, an optimal tracking controller for the valve-rising duration, a key valve profile parameter, is presented in this paper. An event-by-event nonlinear model, connecting the system supply pressure dynamics to the valve-rising duration, is developed and linearized along the desired valve-rising trajectory. Based on the trajectory linearization, a receding horizon linear–quadratic tracking (LQT) controller is designed along with a Kalman optimal state estimation. The equilibrium control resulted from the model linearization is used as the LQT feedforward control. The control performance is compared with that of baseline controllers through both simulation study and bench tests. The transient and steady-state validation results confirm the effectiveness of proposed control scheme.

Modelling, design and analysis of three controllers based on LQR formulation for a non-linear hydraulic uniaxial seismic shake table

This study presents the modelling, design and analysis of three controllers applied to the non-linear model of a hydraulic uniaxial seismic shake table. Firstly, the system’s non-linear model is constructed based on the dynamic and mathematical analysis of the hydraulic actuator and the servo valve. Then, three control systems based on the LQR formulation are designed for the acceleration tracking: Linear quadratic integral (LQI), linear quadratic tracking (LQT) and a variation of LQT. Lastly, simulations are carried out using the non-linear model as the plant and the results showed that the variation of the LQT control exhibited the best acceleration tracking performance.