Optimal Closed-loop System Research Articles

Indefinite Robust Linear Quadratic Optimal Regulator for Discrete-Time Uncertain Singular Markov Jump Systems.

The robust LQ optimal regulator problem for discrete-time uncertain singular Markov jump systems (SMJSs) is solved by introducing a new quadratic cost function established by the penalty function method, which combines the penalty function and the weighting matrices. First, the indefinite robust optimal regulator problem for uncertain SMJSs is transformed into the robust optimal regulator problem with positive definite weighting matrices for uncertain Markov jump systems (MJSs). The transformed robust LQ problem is settled by the robust least-squares method, and the condition of the existence and analytic form of the robust optimal regulator are proposed. On the infinite horizon, the optimal state feedback is obtained, which can guarantee the regularity, causality, and stochastic stability of the corresponding optimal closed-loop system and eliminate the uncertain parameters of the closed-loop system. A numerical example and a practical example of DC motor are used to verify the validity of the conclusions.

A Quantitative Parametric Study on Output Time Delays for Autonomous Underwater Cleaning Operations

Offshore pipelines and structures require regular marine growth removal and inspection to ensure structural integrity. These operations are typically carried out by Remotely Operated Vehicles (ROVs) and demand reliable and accurate feedback signals for operating the ROVs efficiently under harsh offshore conditions. This study investigates and quantifies how sensor delays impact the expected control performance without the need for defining the control parameters. Input-output (IO) controllability analysis of the open-loop system is applied to find the lower bound of the H-infinity peaks of the unspecified optimal closed-loop systems. The performance analyses have shown that near-structure operations, such as pipeline inspection or cleaning, in which small error tolerances are required, have a small threshold for the time delays. The IO controllability analysis indicates that off-structure navigation allow substantial larger time delays. Especially heading is vulnerable to time delay; however, fast-responding sensors usually measure this motion. Lastly, a sensor comparison is presented where available sensors are evaluated for each ROV motion’s respective sensor-induced time delays. It is concluded that even though off-structure navigation have larger time delay tolerance the corresponding sensors also introduce substantially larger time delays.

Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case

The indefinite linear quadratic (ILQ) optimal control problem has many important applications in financial, economic systems, etc., which has been widely researched in stochastic systems and descriptor systems, etc. However, for ILQ problem of rectangular descriptor Markov jump systems (DMJSs), because there are impulses simultaneously in descriptor subsystems and at the switching time, it is theoretically difficult to research and there is no result yet. This paper discusses the ILQ optimal control problem for continuous-time linear rectangular DMJSs. Firstly, under some rank conditions and inequality conditions, the ILQ problem for rectangular DMJSs can be equivalently transformed into standard LQ problem for Markov jump systems (MJSs) by using elementary linear algebra method. Then based on the LQ theory of MJSs, the solvable sufficient condition of the ILQ problem for rectangular DMJSs and the non-negative optimal cost value are obtained. The optimal control can be synthesised as state feedback, and the resulting optimal closed-loop system has the stochastically stable solution. In addition, with some rank inequality assumptions, the differential subsystem of the resulting optimal closed-loop system can be ensured to have a unique solution. Finally, two numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.

Finite and infinite horizon indefinite linear quadratic optimal control for discrete-time singular Markov jump systems

This paper concerns the indefinite linear quadratic (LQ) optimal control problem for discrete-time singular Markov jump systems (MJSs) with finite and infinite horizon, where the weight matrices for state and control of cost function are all indefinite. Firstly, the indefinite LQ problem for singular MJSs is equivalently transformed into indefinite LQ problem for MJSs under a series of equivalent transformations. Then, the sufficient and necessary condition is proposed for the solvability of finite horizon case, the optimal control and optimal cost value are given, and the resulting optimal closed-loop system is regular, casual. Next, some sufficient and necessary conditions are obtained to ensure the transformed equivalent LQ problem for MJSs to be definite one, which can guarantee the generalized algebraic Riccati equation with Markov jump has a unique semi-positive definite solution. Meanwhile, the optimal control and nonnegative optimal cost value in infinite horizon are acquired, and the resulting optimal closed-loop system is stochastically admissible. Finally, three examples are presented to illustrate the theoretical results.

Linear Model Predictive Control for a Coupled CSTR and Axial Dispersion Tubular Reactor with Recycle

This manuscript addresses a novel output model predictive controller design for a representative model of continuous stirred-tank reactor (CSTR) and axial dispersion reactor with recycle. The underlying model takes the form of ODE-PDE in series and it is operated at an unstable point. The model predictive controller (MPC) design is explored to achieve optimal closed-loop system stabilization and to account for naturally present input and state constraints. The discrete representation of the system is obtained by application of the structure properties (stability, controllability and observability) preserving Cayley-Tustin discretization to the coupled system. The design of a discrete Luenberger observer is also considered to accomplish the output feedback MPC realization. Finally, the simulations demonstrate the performance of the controller, indicating proper stabilization and constraints satisfaction in the closed loop.

A Design Methodology of Digital Control System for MEMS Gyroscope Based on Multi-Objective Parameter Optimization.

This paper presents a novel multi-objective parameter optimization method based on the genetic algorithm (GA) and adaptive moment estimation (Adam) algorithm for the design of a closed-loop control system for the sense mode of a Microelectromechanical systems (MEMS) gyroscope. The proposed method can improve the immunity of the control system to fabrication tolerances and external noise. The design procedure starts by deriving a parameterized model of the closed-loop of the sense mode. The loop parameters are then optimized by the GA. Finally, the ensemble of optimized loop parameters is tested by Monte Carlo analysis to obtain a robust optimal solution. Simultaneously, the Adam-least mean square (LMS) demodulator, which is appropriate for the demodulation of very noisy signals, is also presented. Compared with the traditional method, the time consumption of the design process is reduced significantly. The digital control system is implemented by the print circuit board based on embedded Field Programmable Gate Array (FPGA). The experimental results show that the optimized control loop has achieved a better performance, the system bandwidth in open-loop and optimal closed-loop control system is about 23 Hz and 101 Hz, respectively. Compared to a non-optimized closed-loop system, the bias instability reduced from 0.0015°/s to 7.52 × 10−4°/s, the scale factor increased from 17.7 mV/(°/s) to 23 mV/(°/s) and the non-linearity of the scale factor reduced from 0.008452% to 0.006156%.

The minimum norm multi-input multi-output receptance method for partial pole placement

Abstract A closed-form analytical solution is developed for the first time that fully addresses the problem of choosing feedback gains that minimize the control effort required for partial pole placement in multi-input, multi-output systems. The norm of the feedback gain matrix is shown to take the form of an inverse Rayleigh quotient, such that the optimal closed-loop system eigenvectors are given as a function of the dominant (highest) eigenvectors of the matrix in the quotient. The feedback gains that deliver the required pole placement with minimum effort may then be determined using standard procedures. The original formulation by the receptance method proposed an arbitrary choice of the closed loop eigenvectors that assigned the poles exactly but was generally wasteful of control effort that might otherwise be conserved or put to good use in satisfying additional control objectives. The analytical solution is validated against a set of numerical examples.

Kyle--Back Equilibrium Models and Linear Conditional Mean-Field SDEs

In this paper we study the Kyle--Back strategic insider trading equilibrium model in which the insider has instantaneous information on an asset, assumed to follow an Ornstein--Uhlenback-type dynamics that allows possible influence by the market price. Such a model exhibits some further interplay between an insider's information and the market price, and it is the first time being put into a rigorous mathematical framework of the recently developed conditional mean-field stochastic differential equation (CMFSDE). With the help of the “reference probability measure" concept in filtering theory, we shall first prove a general well-posedness result for a class of linear CMFSDEs, which is new in the literature of both filtering theory and mean-field SDEs and will be the foundation for the underlying strategic equilibrium model. Assuming some further Gaussian structures of the model, we find a closed form of optimal intensity of trading strategy as well as the dynamic pricing rules, and we substantiate the well-posedness of the resulting optimal closed-loop system, whence the existence of Kyle--Back equilibrium.

On the generalized algebraic Riccati equations

Three hundred years have passed since Jacopo Francesco Riccati analyzed a quadratic differential equation that would have been of crucial importance in many fields of engineering and applied mathematics. Indeed, countless variations and generalizations of this equation have been considered as they proved to be the right mathematical tool to address important problems. This paper is focused on a generalized version of the matrix Riccati equation where the matrix that in the classical Riccati equation is inverted can be singular: we analyze the equation obtained by substituting the inverse operator with the Moore-Penrose pseudo-inverse. The equations obtained by this substitution are known as generalized Riccati equations. The relation between these equations — both in continuos-time and in discrete-time — and singular Linear Quadratic (LQ) optimal control problem are examined. A geometric characterization of the set of solutions of the generalized Riccati equation is illustrated. It is shown that in this general setting there are LQ optimal control problems for which the optimal closed-loop system is stable also in cases where the Riccati equation does not possess a stabilizing solution.

Optimal Human–Machine Teaming for a Sequential Inspection Operation

A novel mixed initiative optimal control system for intelligence, surveillance and reconnaissance (ISR) operations which entails human–machine teaming has been developed. The scenario entails a camera-equipped unmanned air vehicle sequentially overflying geolocated objects of interest, which need to be classified as either a true or false target by a human operator. The vehicle is allowed a prespecified number of revisits, such that an object can be looked at, a second time, under better viewing conditions. The overarching goal is to correctly classify the objects and minimize the false alarm (FA) and missed detection (MD) rates. We design a stochastic controller that computes if and when a revisit is necessary and also the optimal revisit state, i.e., viewing altitude and aspect angle. The concept of operation is such that the critical task of detection/pattern recognition is relegated to the human operator, whereas optimal decision making is entrusted to the machine. The stochastic dynamic programming-based decision algorithm is, however, informed about the performance of the human operator via an empirical human perception model. The model is experimentally obtained in the form of state-dependent confusion matrices. The optimal closed-loop ISR system is shown to experimentally achieve a FA rate of 5% and MD rate of 12%, which are significantly lower than the open-loop operator-only performance metrics. The performance improvements that were observed are relevant to a particular operator, and thus, the study suggests that the same improvements could conceivably be achieved with other test subjects.

Optimal regulation of a cable suspended robot equipped with cable interfering avoidance controller

Closed-loop regulation of a spatial cable suspended robot is performed in this paper subject to maximizing the Dynamic Load Carrying Capacity (DLCC) of the end-effector while the cable interference is avoided actively. Optimization is performed between two predefined boundaries and considering the cable interference constraint. This constraint is satisfied by designing a controller which prevents the cables’ collision. The overall formulation of the closed-loop optimal control based on Feedback linearization is derived in this paper for planning the optimal path with the highest load capacity. Then a complementary adaptive controller is designed and implemented to the main controller which is responsible for providing cable interfering avoidance. The efficiency of the designed controller for preventing the cables’ collision is shown by performing and analyzing some comparative simulations conducted on an under constrained cable robot with six cables and six DOFs. All results related to regulation, tracking and DLCC are compared between the simple optimal closed-loop system and the system which is equipped with the proposed cable interfering avoidance controller. It is proved that the planned path satisfies cable interference constraint while its DLCCs are optimized.

Quadratic partial eigenvalue assignment in large-scale stochastic dynamic systems for resilient and economic design

Failure of structural systems under dynamic loading can be prevented via active vibration control which shifts the damped natural frequencies of the systems away from the dominant range of a loading spectrum. The damped natural frequencies and the dynamic load typically show significant variations in practice. A computationally efficient methodology based on quadratic partial eigenvalue assignment technique and optimization under uncertainty has been formulated in the present work that will rigorously account for these variations and result in economic and resilient design of structures. A novel scheme based on hierarchical clustering and importance sampling is also developed in this work for accurate and efficient estimation of probability of failure to guarantee the desired resilience level of the designed system. Finally the most robust set of feedback matrices is selected from the set of probabilistically characterized optimal closed-loop system to implement the new methodology for design of active controlled structures. Numerical examples are presented to illustrate the proposed methodology.

Exact Symbol Error Rate and Diversity Analysis of Orthogonalized Spatial Multiplexing Systems with Optimal Precoding

Orthogonalized spatial multiplexing (OSM) systems with optimal precoding, denoted by POSM, was recently proposed which allows a single-symbol decodable maximum likelihood receiver in closed-loop multiple-input multiple-output (MIMO) systems. It was shown that the performance of the POSM is identical to that of the optimal closed-loop MIMO system in terms of the minimum distance. In this letter, we derive an exact symbol error rate (SER) expression of the POSM for 4-QAM and 16-QAM in Rayleigh fading channels. By using polar coordinates and the singular value distributions, the final expression of the SER can be evaluated with a single-integral form. Also, in the high SNR regime, we obtain a simple average SER expression of the POSM which shows that the POSM with two transmitting antennas achieves full diversity. Simulation results confirm the accuracy of our derived analysis.

On the Continuous Time-Varying JLQ Problem

This paper is concerned with the optimal control of time-varying, continuous-time linear systems both with parameters depending on time and the process being a finite-state Markovian one. The performance index to be minimized is the infinite-time quadratic cost functional. The solution of this time-varying jump linear quadratic control problem consists of the study of nonnegative definite global and bounded solution of coupled differential Riccati equation. Necessary and sufficient conditions for existence of such a solution are obtained in terms of optimizability and detectability. Moreover, the conditions for stability of the optimal closed-loop system are established.

Discrete-time optimal fuzzy controller design: global concept approach

Proposes a systematic and theoretically sound way to design a global optimal discrete-time fuzzy controller to control and stabilize a nonlinear discrete-time fuzzy system with finite or infinite horizon (time). A linear-like global system representation of a discrete-time fuzzy system is first proposed by viewing such a system in a global concept and unifying the individual matrices into synthetic matrices. Then, based on this kind of system representation, a discrete-time optimal fuzzy control law which can achieve a global minimum effect is developed theoretically. A nonlinear two-point boundary-value-problem (TPBVP) is derived as a necessary and sufficient condition for the nonlinear quadratic optimal control problem. To simplify the computation, a multi-stage decomposition of the optimization scheme is proposed, and then a segmental recursive Riccati-like equation is derived. Moreover, in the case of time-invariant fuzzy systems, we show that the optimal controller can be obtained by just solving discrete-time algebraic Riccati-like equations. Based on this, several fascinating characteristics of the resultant closed-loop fuzzy system can easily be elicited. The stability of the closed-loop fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal closed-loop fuzzy system can not only be guaranteed to be exponentially stable, but also stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant closed-loop fuzzy system possesses an infinite gain margin, i.e. its stability is guaranteed no matter how large the feedback gain becomes. An example is given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proven stability properties.

Optimal fuzzy controller design in continuous fuzzy system: global concept approach

We propose a design method for a global optimal fuzzy controller to control and stabilize a continuous fuzzy system with free- or fixed-end point under finite or infinite horizon (time). A linear-like global system representation of continuous fuzzy system is first proposed by viewing a continuous fuzzy system in global concept and unifying the individual matrices into synthetical matrices. Based on this, the optimal control law which can achieve global minimum effect is developed theoretically. The nonlinear segmental two-point boundary-value problem is derived for the finite-horizon problem and a forward Riccati-like differential equation for the infinite-horizon problem. The stability of the closed-loop fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal closed-loop fuzzy system cannot only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant closed-loop fuzzy system possesses an infinite gain margin.

On initial jumps for singular systems in connection with LQ optimal control

This paper is concerned with the connection between optimal state feedback gains and initial jumps for the linear quadratic optimal control problem involving singular systems. It is known that the optimal state feedback gain for such an optimal control problem is not unique. Thus, different optimal gains lead to different closed-loop systems. In this paper, it is pointed out that all such optimal closed-loop systems will generically have jumps and these initial jumps are always the same. More importantly, a set of conditions under which LQ optimal gains will yield no initial jumps in the closed-loop systems is derived.

Control of Linear Motors for Machine Tool Feed Drives: Experimental Investigation of Optimal Feedforward Tracking Control

This paper investigates the use of optimal l1 and H∞ model reference optimal feedforward control to enhance the tracking performance of a linear motor drive. Experimental work is presented which studies the effects of signal preview, tracking constraint, and reference model choice on tracking performance. Suboptimal l1 control where the closed-loop system has a zero on the unit circle due to integral action in the feedback controller is given special attention, and is seen to give near optimal performance for the system under study here. For the specific trajectory employed here, the best performing feedforward controllers were experimentally seen to reduce by more than half the maximum and rms tracking errors of the H∞ optimal feedback closed-loop systems.

Formula omitted]∞ Optimal control of SISO continuous-time systems

In this paper we study the problem of designing a controller that minimizes the weighted amplitude of the time response due to a given, fixed input signal for SISO continuous-time systems. The main result of the paper shows that this problem admits a minimizing solution in L ∞ and that the optimal closed-loop system has a special structure: a sum of delayed step functions, all having the same amplitude. Thus the optimal controller has a non-rational transfer function. Although in the general case finding this controller entails solving an infinite-dimensional linear-programming problem, we show that in some special cases the optimal solutions have closed-form expressions and can be found by solving a set of algebraic equations. Finally, we address the issue of selecting time and frequency domain weighting functions. This paper together with our paper ‘Rational L ∞-suboptimal controller for SISO continuous time systems’ ( IEEE Trans. Autom. Control, AC-41, 1358–1363 (1996)), which deals with the design of rational L ∞ suboptimal controllers for general systems, give a complete solution of the L ∞ control problem.

Minimax control under a bound on the partial covariance sequence of the disturbance

A discrete-time linear SISO plant that is acted on by a stochastic disturbance is considered. It is assumed that a restriction on the partial covariance sequence of the disturbance is given. The cost function is a maximum of the closed-loop system output variance over the class of disturbances. The problem is to find the optimal linear stabilizing regulator. The transfer function of the optimal closed-loop system is constructed in explicit form. It is shown that the minimax regulator is intermediate between the H ∞ and H 2 optimal regulators. As a result of the problem solution, generalizations of the Caratheodory-Fejer and Nevanlinna-Pick theorems are obtained.