Control Of Continuous-time Linear Systems Research Articles

State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems

In this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded σ-entropy, is studied. The σ-entropy norm defines a performance index of the system on the set of the aforementioned input signals. Two problems are considered. The first is a state-space σ-entropy analysis of linear systems, and the second is an optimal control design using the σ-entropy norm as an optimization objective. The state-space solution to the σ-entropy analysis problem is derived from the representation of the σ-entropy norm in the frequency domain. The formulae of the σ-entropy norm computation in the state space are presented in the form of coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations. Optimal control law is obtained using game theory and a saddle-point condition of optimality. The optimal state-feedback control, which minimizes the σ-entropy norm of the closed-loop system, is found from the solution of two algebraic Riccati equations, one Lyapunov equation, and the log determinant equation.

Fuzzy discretization and control for non-linear, multiple binary input systems

The control of continuous-time linear systems with binary inputs cannot benefit from existing control design techniques because they are based on continuous control actions. In particular, optimal control problems with binary inputs lead to combinatorial optimization problems, which are difficult to solve. In this article we provide an exact discretization model of the binary continuous-time system that results in a non-linear multiple input controlled system. The non-linear model is then converted into a fuzzy discrete Takagi-Sugeno model, thus allowing the use of optimal control techniques based on LMI design. The modelling of the non-linear model by a discrete Takagi-Sugeno model is a complex process but it can be automatically performed as shown in the article and the code of the application examples.

An Improved Design of Event-Triggered Feedback Controllers for Linear Systems Based on Fast and Slow Dynamics

In this article, an event-triggered technique is proposed to implement state feedback controllers for continuous-time linear systems. Toward this end, the closed-loop dynamic is decomposed into the fast and slow subsystems. Then, two individualized event-triggered mechanisms (ETMs) are designed to compute two sequences of updates. The first ETM determines when the fast subsystem should be sampled and its measured data should be sent to the controller. The second ETM decides the execution times for sampling all the system states, including both the fast and slow states. The proposed event-triggered technique leads to sampling policies in which the data of the slow states are sent to the controller with long interevent intervals whereas the fast dynamics are simultaneously allowed to be sampled with relatively shorter interevent intervals. In comparison with conventional methods with one ETM, the proposed event-triggered technique, equipped with an extra degree of freedom, leads to better results both in achieving smoother closed-loop responses and reducing the number of the transmitted packages from the system to the controller. Theoretical aspects of the proposed method including the boundedness of the system and its Zeno free property are investigated. Two simulation and experimental case studies are provided to show the effectiveness of the proposed event-triggered controller.

Event-triggered control for continuous-time linear systems with a delay in the input

We provide an event-triggered control technique for a family of linear time-varying continuous-time systems with a constant known pointwise delay in the input. We adopt a subpredictor based prediction technique, and we provide sufficient conditions that ensure that Zeno behavior does not occur. At each time, only delayed measurements are needed to implement the control. Also, the delay can be an arbitrarily large constant. We prove an input-to-state stability property for the closed-loop system, using the theory of cooperative systems. We apply our method to a gyroscopic control problem for a curve tracking dynamics arising in marine robotics.

Padé-Approximation-Based Preview Repetitive Control for Continuous-Time Linear Systems

This paper concerns a Padé-approximation-based preview repetitive control (PRC) scheme for continuous-time linear systems. Firstly, the state space representation of the repetitive controller is transformed into a nondelayed one by Padé approximation. Then, an augmented dynamic system is constructed by using the nominal state equation with the error system and the state equation of a repetitive controller. Next, by using optimal control theory, a Padé-approximation-based PRC law is obtained. It consists of state feedback, error integral compensation, output integral of repetitive controller, and preview compensation. Finally, the effectiveness of the method is verified by a numerical simulation.

Adaptive output-feedback optimal control for continuous-time linear systems based on adaptive dynamic programming approach

This paper investigates the adaptive output-feedback optimal control problem for a class of continuous-time (CT) linear systems with dynamic uncertainties. Data-based controller design algorithm with policy iteration form is developed through adaptive dynamic programming (ADP) technique. In the process of proposed algorithm, only measured input and output information of the system is used to learn optimal control gain, but the exactly knowledge of the system is not required. Different from the existing works, the adaptive controllers learned by the algorithm has certain robustness to the dynamic uncertainties. Three examples are given to test the effectiveness and feasibility of the proposed approach.

Parallel control for continuous-time linear systems: A case study

In this paper, a new parallel controller is developed for continuous-time linear systems. The main contribution of the method is to establish a new parallel control law, where both state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the parallel control and traditional feedback controls is presented. Considering the situations that the systems are controllable and incompletely controllable, the properties of the parallel control law are analyzed. The parallel controller design algorithms are given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are carried out to demonstrate the effectiveness and applicability of the present method.

Reinforcement Learning-Based Adaptive Optimal Exponential Tracking Control of Linear Systems With Unknown Dynamics

Reinforcement learning (RL) has been successfully employed as a powerful tool in designing adaptive optimal controllers. Recently, off-policy learning has emerged to design optimal controllers for systems with completely unknown dynamics. However, current approaches for optimal tracking control design either result in bounded tracking error, rather than zero tracking error, or require partial knowledge of the system dynamics. Moreover, they usually require to collect a large set of data to learn the optimal solution. To obviate these limitations, this paper applies a combination of off-policy learning and experience-replay for output regulation tracking control of continuous-time linear systems with completely unknown dynamics. To this end, the off-policy integral RL-based technique is first used to obtain the optimal control feedback gain, and to explicitly identify the involved system dynamics using the same data. Second, a data-efficient-based experience replay method is developed to compute the exosystem dynamics. Finally, the output regulator equations are solved using data measured online. It is shown that the proposed control method stabilizes the closed-loop tracking error dynamics, and gives an explicit exponential convergence rate for the output tracking error. Simulation results show the effectiveness of the proposed approach.

Control of Continuous-Time Linear Systems With Markov Jump Parameters in Reverse Time

This note introduces continuous-time linear systems with switching parameters driven by a Markov chain running in reverse time. Reversibility of the chain is not imposed, so the system cannot be recast as standard Markov jump linear systems (MJLS), demanding for a specific study. We develop the operator-based approach, which is elaborate in view of possible multiplicity of solutions for the second moment variable, as detailed in this note. We also study the optimal quadratic control problem based on Hamilton–Jacobi–Bellman equations and standard dynamic programming. The control solution is given in terms of a coupled Riccati equation that is harmonious with filtering of MJLS, which allows us to generalize the classic control-filtering duality for linear time varying systems.

Finite-time Extended State Observer Based Adaptive Optimal Control of Continuous-time Linear Systems

In this paper, a novel strategy iterative method based on finite-time extended state observer is proposed to find the online adaptive optimal controller for a class of linear continuous-time systems with completely unknown system dynamics and unmeasurable state variables. In this method, finite-time state observer estimates the system state variables by using the input and output information of the system and the policy iterative method uses the estimated state variables and system input information to iteratively solve the algebraic Riccati equation until the adaptive optimal controller is obtained. It is worth noting that the method proposed in this paper only uses the input and output information of the system, without requiring the apriori knowledge of the system matrices and the state variables. A numerical simulation example will be provided to demonstrate the effectiveness of the method.

Enhanced approach to cascade reconfiguration control design

Following the concept of fault tolerant control systems, the paper is concerned with the problem of reconfiguration to retain fault tolerance in control of linear continuous-time systems with system dynamics faults. The main idea is to use a reference model output to be followed when a fault occurs, while the nominal control loop structure is kept untouched and the controllers with nominal parameters remains a part of the reconfigured control loop scheme. The full state control principle is applied for nominal control strategy and the static output control principle is proposed for the compensation control law. Exploiting the D-stability circle region precept, new conditions for control laws parameter design are introduced and proven, as well as stability of the cascade-like reconfiguration structure is analysed in the paper. To illustrate oncoming properties, the proposed feasible procedure is compared with that which was obtained using the bounded real lemma principle. The results, offering the sufficient and necessary design conditions, are illustrated with a numerical example to note the effectiveness of the proposed approach and its applicability.

Enhanced approach to cascade reconfiguration control design

Following the concept of fault tolerant control systems, the paper is concerned with the problem of reconfiguration to retain fault tolerance in control of linear continuous-time systems with system dynamics faults. The main idea is to use a reference model output to be followed when a fault occurs, while the nominal control loop structure is kept untouched and the controllers with nominal parameters remains a part of the reconfigured control loop scheme. The full state control principle is applied for nominal control strategy and the static output control principle is proposed for the compensation control law. Exploiting the D-stability circle region precept, new conditions for control laws parameter design are introduced and proven, as well as stability of the cascade-like reconfiguration structure is analysed in the paper. To illustrate oncoming properties, the proposed feasible procedure is compared with that which was obtained using the bounded real lemma principle. The results, offering the sufficient and necessary design conditions, are illustrated with a numerical example to note the effectiveness of the proposed approach and its applicability.

Cooperative optimal preview tracking control of continuous-time multi-agent systems

ABSTRACTThis paper presents the cooperative preview control method for continuous-time multi-agent systems with a time-invariant directed communication topology. First, the cooperative tracking preview control problem is transformed into the optimal regulation problem of an augmented system. Next, by applying the results of the standard optimal preview control of continuous-time linear systems, a controller of the augmented system is obtained. Furthermore, the stabilisability and detectability of the augmented system are studied under the fixed digraph and the prescribed leader. Then, we derive the controller of multi-agent systems with error integral and preview action that can guarantee the achievement of cooperative optimal preview tracking. Finally, the effectiveness of the controller is shown by numerical simulations.

H∞ Control of Continuous-Time Linear Systems Using Dynamic and Static Output Controllers

The paper deals with the problem of the H ∞ control of continuous-time linear systems by proper dynamic output controllers of order equal to the plant model order and by static output controller. The design procedure is based on solution of the set of linear matrix inequalities and the matrix equality and ensures the closed-loop asymptotic stability using Lyapunov approach and the H ∞ performance, to guarantee the closed-loop system robustness to unknown disturbance. Numerical examples are given to illustrate the design procedures and the relevance of the method as well as to validate the performances of the proposed approach.

Online optimal tracking control of continuous-time linear systems with unknown dynamics by using adaptive dynamic programming

In this paper, a novel theoretic formulation based on adaptive dynamic programming (ADP) is developed to solve online the optimal tracking problem of the continuous-time linear system with unknown dynamics. First, the original system dynamics and the reference trajectory dynamics are transformed into an augmented system. Then, under the same performance index with the original system dynamics, an augmented algebraic Riccati equation is derived. Furthermore, the solutions for the optimal control problem of the augmented system are proven to be equal to the standard solutions for the optimal tracking problem of the original system dynamics. Moreover, a new online algorithm based on the ADP technique is presented to solve the optimal tracking problem of the linear system with unknown system dynamics. Finally, simulation results are given to verify the effectiveness of the theoretic results.

Decentralized state feedback control for interconnected systems with application to power systems

We consider the problem of constructing decentralized state feedback controllers for linear continuous-time systems. Different from existing approaches, where the topology of the controller is fixed a priori, the topology of the controller is part of the optimization problem. Structure optimization is done in terms of a minimization of the required feedback links and subject to a predefined bound on the tolerable loss of the achieved H∞-performance of the decentralized controller compared to an H∞-optimal centralized controller. We develop a computationally efficient formulation of the decentralized control problem by convex relaxations which makes it attractive for practical applications. The proposed design algorithm is applied to design sparse wide area control of a 3-area, 6-machine power system.

Insensitive dynamic output feedback control with mixed-H∞ norm sensitivity minimization

This paper will focus on designing insensitive output feedback controllers for linear continuous-time systems with mixed-H∞ norm sensitivity minimization. In order to reduce the sensitivity of the designed controllers, a type of coefficient sensitivity function with respect to additive controller coefficient variations is defined. What's more, the H∞ norms of the coefficient sensitivity functions are regarded as sensitivity indexes. Then, the considered problem in this paper is reduced to a multi-objective problem which is to be a non-convex optimization problem, where both the coefficient sensitivity constraint and the standard H∞ norm constraint are considered simultaneously. To solve this non-convex problem, a two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities (LMIs). It is worth to mention here that the non-fragile control method is adopted to obtain an effective initial solution of the two-step procedure. Finally, a numerical example is also given to demonstrate the effectiveness of the proposed design procedure.

Integral Q-learning and explorized policy iteration for adaptive optimal control of continuous-time linear systems

This paper proposes an integral Q-learning for continuous-time (CT) linear time-invariant (LTI) systems, which solves a linear quadratic regulation (LQR) problem in real time for a given system and a value function, without knowledge about the system dynamics A and B. Here, Q-learning is referred to as a family of reinforcement learning methods which find the optimal policy by interaction with an uncertain environment. In the evolution of the algorithm, we first develop an explorized policy iteration (PI) method which is able to deal with known exploration signals. Then, the integral Q-learning algorithm for CT LTI systems is derived based on this PI and the variants of Q-functions derived from the singular perturbation of the control input. The proposed Q-learning scheme evaluates the current value function and the improved control policy at the same time, and are proven stable and convergent to the LQ optimal solution, provided that the initial policy is stabilizing. For the proposed algorithms, practical online implementation methods are investigated in terms of persistency of excitation (PE) and explorations. Finally, simulation results are provided for the better comparison and verification of the performance.

Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics

This paper presents a novel policy iteration approach for finding online adaptive optimal controllers for continuous-time linear systems with completely unknown system dynamics. The proposed approach employs the approximate/adaptive dynamic programming technique to iteratively solve the algebraic Riccati equation using the online information of state and input, without requiring the a priori knowledge of the system matrices. In addition, all iterations can be conducted by using repeatedly the same state and input information on some fixed time intervals. A practical online algorithm is developed in this paper, and is applied to the controller design for a turbocharged diesel engine with exhaust gas recirculation. Finally, several aspects of future work are discussed.

Decentralized State Feedback Control for Interconnected Process Systems

We consider the problem of constructing decentralized state feedback controllers for linear continuous-time systems. Different from existing approaches, where the topology of the controller is fixed a-priori, the topology of the controller is part of the optimization problem. Structure optimization is done in terms of a minimization of the required feedback and subject to a predefined bound on the tolerable loss of the achieved H∞-performance of the decentralized controller compared to an H∞-optimal centralized controller. We develop a computationally efficient formulation of the decentralized control problem by convex relaxations which makes it attractive for practical applications.