Closed-loop Eigenvalues Research Articles

Decentralized Control of Complex Systems: Lyapunov Function Approach

In this contribution, we generalize existing methods for decentralized control design, providing a unified methodological framework that applies to linear continuous and discrete-time complex systems, as well as certain classes of nonlinear complex systems. Our approach leverages the direct connection between the stability properties of the overall complex system and those of its individual subsystems. By conducting the entire controller design process at the subsystem level, we circumvent the need for explicit interconnection values. Through numerical examples, we demonstrate that the proposed method ensures asymptotic stability of the full complex system within a specified region, and also guarantees stability of the isolated subsystems. In particular, we obtain quantifiable stability margins (e.g., γc bounds) and closed-loop eigenvalue placements that verify the effectiveness of the design. These results highlight not only the method’s theoretical robustness, but also its practical significance in simplifying the design process, reducing computational overheads, and enhancing scalability for large and interconnected engineering systems.

Design of LQ regulators with prescribed degree of stability for dual-rate systems based on reinforcement learning

This paper considers dual-rate systems, where the output is measured at a relatively slow rate while the control signal is adjusted at a faster rate. The output sampling time is an integer multiple of the input sampling time. The paper examines dual-rate inferential control systems, which consist of a fast model, a slow model, and a switch. Missing output samples are estimated using the fast single-rate model. The single-rate control algorithm is then implemented at the fast-sampling rate. The fast-sampling discrete-time model is derived from the plant’s continuous-time model using the first-order hold (FOH) element. A discrete LQ regulator is proposed for this plant model, with a prescribed degree of stability (all closed-loop eigenvalues are within the range 0 < λ < 1 in magnitude). The matrix gain is calculated offline, and an online method for calculating the regulator gain is provided. The regulator gain is calculated using policy iteration, specifically Hewer’s algorithm. Finally, it is demonstrated that the presented inferential control system remains effective in the presence of multiplicative unmodeled dynamics. The main contributions of the paper are: (i) Designing the LQ regulator with a prescribed degree of stability using reinforcement learning (RL) (generalized policy iteration); and (ii) Considering the robust stability of the inferential control system in the presence of multiplicative unmodeled dynamics using the lifting technique.

A robust regional eigenvalue assignment problem using rank-one control for undamped gyroscopic systems

<abstract><p>Considering the advantages of economic benefit and cost reduction by using rank-one control, we investigated the problem of robust regional eigenvalue assignment using rank-one control for undamped gyroscopic systems. Based on the orthogonality relation, we presented a method for solving partial eigenvalue assignment problems to reassign partial undesired eigenvalues accurately. Since it is difficult to achieve robust control by assigning desired eigenvalues to precise positions with rank-one control, we assigned eigenvalues within specified regions to provide the necessary freedom. According to the sensitivity analysis theories, we derived the sensitivity of closed-loop eigenvalues to parameter perturbations to measure robustness and proposed a numerical algorithm for solving robust regional eigenvalue assignment problems so that the closed-loop eigenvalues were insensitive to parameter perturbations. Numerical experiments demonstrated the effectiveness of our method.</p></abstract>

On the State-Feedback Controller Design for Polynomial Linear Parameter-Varying Systems with Pole Placement within Linear Matrix Inequality Regions

The present paper addresses linear parameter-varying systems with high-order time-varying parameter dependency known as polynomial LPV systems and their controller design. Throughout this work, a procedure ensuring a state-feedback controller from a parameterized linear matrix inequality (PLMI) solution is presented. As the main contribution of this paper, the controller is designed by considering the time-varying parameter rate as a tuning parameter with a continuous control gain in such a way that the closed-loop eigenvalues lie in a complex plane subset, with high-order time-varying parameters defining the system dynamics. Simulation results are presented, aiming to show the effectiveness of the proposed controller design.

Nonlinear Damping Output-Feedback Voltage Control for DC/DC Converters via Model-Independent Voltage Derivative Observer

This paper proposes an output-feedback solution for DC/DC converter applications that considers model-plant mismatches and load variations as constraints in industrial applications. The following two major contributions of this study are particularly advantageous: (a) nonlinear damping terms to inject the critical damping performance into the closed-loop system with the cooperation of the feedback gain structure depending on the damping coefficients, and (b) estimation of the output voltage derivative by the model-independent observer with only one simple closed-loop eigenvalue and current feedback removal. The experimental study highlights the practical merit of the proposed solution by demonstrating the critical damping performance that suppresses the peak current level using a 3-kW bidirectional DC/DC converter.

Distributed Boundary Control of the Heat Equation on a Disk

We consider a Linear Quadratic Regulator (LQR) for the heat equation on the unit disk using distributed boundary control. This leads to a Riccati PDE for the kernel of optimal cost. In a simple case we are able to explicitly solve this PDE using Fourier series and compute the closed loop eigenvalues and eigenfunctions.

Minimal-norm state-feedback globally non-overshooting/ undershooting tracking control of multivariable systems

The non-overshooting/undershooting (NOUS) tracking control objective in the output response of the closed-loop system has several practical applications in “positioning” control problems. Recently, a state-feedback gain matrix based on Moore’s eigenstructure assignment technique to achieve the above control objective with a desired convergence rate in the output error from any initial condition is designed, referred to as globally monotonic tracking control, which is a convex optimization problem. From a practical viewpoint, a feedback gain matrix of a minimal norm is often required to install low-cost actuators in the overall closed-loop system. However, there is always a trade-off between the control objectives of achieving a NOUS tracking response with the desired convergence rate and a minimal norm of the feedback gain matrix. Besides, the latter control requirement renders the globally NOUS control optimization problem nonconvex. In this paper, a convex optimization-based iterative algorithm is developed to synthesize a minimal-norm state-feedback globally NOUS tracking controller. An upper bound on the achievable settling time in all output responses for the closed-loop eigenvalues lying in a prespecified region of the complex plane is also provided. The effectiveness of the proposed algorithm is demonstrated through a numerical example, followed by experimental validation on a multitank system.

Design of Platooning Controllers That Achieve Collision Avoidance by External Positivity

This paper solves the problem of designing local adaptive cruise controllers that guarantee collision avoidance of the vehicles in a platoon. Whereas previous studies have solved this problem for specific vehicle models, the present paper shows a general solution for a wider class of linear vehicle models. In particular, the problem of finding a controller that renders a vehicle externally positive to guarantee string stability and collision avoidance is studied in detail. The result is an algorithm that finds a set of closed-loop eigenvalues that correspond to an externally positive feedback loop. The developed methods are applied to an example and the effectiveness is verified in a laboratory experiment with multiple robots driving in a platoon.

Discrete State-Feedback Control Design with D-Stability and Genetic Algorithm for LED Driver Using a Buck Converter

In this paper, a discrete state-feedback controller is applied to control the current in power LEDs by using a buck converter. Sufficient conditions for the existence of the controller are given in terms of linear matrix inequalities (LMIs) with D-stability theory. In order to consider the specific closed-loop dynamics of the closed-loop LED driver, the genetic algorithm (GA) technique is applied, as a complement of the D-stability theory. Therefore, the proposed GA technique is used to search offline the optimal closed-loop eigenvalues to have a desired response of the closed-loop system, subject to a proposed cost function equation. The main contribution of this work is the design and the experimental validation of both the state-feedback controller and the GA to achieve the desired closed-loop dynamics of the LED-driver system. Finally, simulations and experimental results are done in order to illustrate the effectiveness of the proposed methodology.

Eigenvalue sensitivity minimisation for robust pole placement by the receptance method

The problem of robust pole placement in active structural vibration control by the method of receptance is considered in this paper. Expressions are derived for the eigenvalue sensitivities to parametric perturbations, which are subsequently minimised to improve performance robustness of the control of a dynamical system. The described approach has application to a vibrating system where variations are present due to manufacturing and material tolerances, damages and environment variabilities. The closed-loop eigenvalue sensitivities are expressed as a linear function of the velocity and displacement feedback gains, allowing their minimisation with carefully calculated feedback gains. The proposed algorithm involves curve fitting perturbed frequency response functions, FRFs, using the rational fraction polynomial method and implementation of a polynomial fit to the individual estimated rational fraction coefficients. This allows the eigenvalue sensitivity to be obtained entirely from structural FRFs, which is consistent with the receptance method. This avoids the need to evaluate the M,C,K matrices which are typically obtained through finite element modelling, that produces modelling uncertainty. It is also demonstrated that the sensitivity minimisation technique can work in conjunction with the pole placement and partial pole placement technique using the receptance method. To illustrate the working of the proposed algorithm, the controller is first implemented numerically and then experimentally.

Learning Controllers for Performance Through LMI Regions

In an open-loop experiment, an input sequence is applied to an unknown linear time-invariant system (in continuous or discrete time) affected also by an unknown-but-bounded disturbance sequence (with an energy or instantaneous bound); the corresponding state sequence is measured. The goal is to design directly from the input and state sequences a controller that enforces a certain performance specification on the transient behaviour of the unknown system. The performance specification is expressed through a subset of the complex plane where closed-loop eigenvalues need to belong, a so called LMI region. For this control design problem, we provide here convex programs to enforce the performance specification from data in the form of linear matrix inequalities (LMI). For generic LMI regions, these are sufficient conditions to assign the eigenvalues within the LMI region for all possible dynamics consistent with data, and become necessary and sufficient conditions for special LMI regions. In this way, we extend classical model-based conditions from a seminal work in the literature to the setting of data-driven control from noisy data. Through two numerical examples, we investigate how these data-based conditions compare with each other.

Automated Controller Design for the PMSM Using Dynamic Mode Decomposition

This work presents a method to automatically generate a high performance controller for the permanent magnet synchronous motor (PMSM). The method consists of two components, a nominal system identification and a harmonic component identification. Both identification methods are based on dynamic mode decomposition (DMD). The nominal system identification is used to assign the feedback gaines by matching the desired closed-loop eigenvalues and eigenvectors. The harmonic system identification is used to generate vectors that are multiplied by a delay embedding of the current to predict harmonic components at the next time-step. The method is applied to two experimental test setups, one interior permanent magnet (IPM) and one surface mount permanent magnet (SPM) motor. It is shown that the automatically generated feedback controller is able to achieve a more precise transient response than the traditional rule-based PI controller. It is also shown the harmonic compensation method is able to reduce total demand distortion (TDD) on phase currents better than a traditional adaptive filter approach without the need for gain tuning. This work shows a novel approach to using DMD for the complete system identification of the PMSM, and lays the foundation for using DMD with delay embeddings to analyze and manipulate harmonic signals in a predictive way.

Robust coupling control using pole region assignment

In this paper, we develop a constructive method for designing robust linear coupling controllers with respect to the closed-loop eigenvalues and eigenstructure. The design method is based on a multi-model approach for parameter-varying systems, which is used to calculate constraints for a complete state feedback. The associated controller structure assures the fulfillment of given algebraic coupling conditions and is finally parameterized in a pole region assignment procedure. Necessary and sufficient conditions for the existence of a robust coupling controller are determined and an approximate method is developed for the case in which these conditions are not fulfilled. The developed method is applied to an example system.

Formula omitted]-optimal vibration control using receptance-based regional eigenvalue assignment

This paper proposes a method of H2-optimal vibration control using regional eigenvalue assignment based only on measured receptances, typically from a modal test. The sum of the squared weighted H2-norm of closed-loop receptance matrix and the transfer matrix from the external input to the control input is minimized by exploiting the flexibility from assigning the closed-loop eigenvalues to prescribed stable regions with single- or multiple-input control. It is found that regional eigenvalue assignment and the optimization of a free parameter matrix (in the multiple-input case) provide extra flexibility for improving the closed-loop performance over and above the selection of weighting matrices, as in standard H2 control. Therefore, the internal stability, satisfactory response, and minimum control energy are readily guaranteed by the proposed method, whereas the standard approach is dependent upon the arbitrarily chosen weighting matrices. The method inherits the advantages of both optimal control and eigenvalue assignment by the receptance method. The working of the method and its advantages over the standard H2 control are demonstrated numerically.

Modified neural network algorithm based robust design of AVR system using the Kharitonov theorem

This paper proposes a new control law for designing an optimal proportional-integral-derivative (PID) controller for flexible automatic voltage regulator (AVR) system using modified neural network algorithm (MNNA). First, the exploration capability of neural network algorithm (NNA) is enhanced by addition of a learning factor, α in MNNA. Then, to evaluate the performance of MNNA in terms of its exploration and exploitation capabilities, extensive statistical analysis has been carried out on 23 benchmark functions consisting of unimodal, multimodal and fixed dimension multimodal functions against 12 state of the art algorithms. The results are encouraging and marks the superiority of MNNA against NNA as well as 11 other state of the art techniques. It is followed by application of Kharitonov theorem as a design tool to derive the Interval AVR system. Next, NNA and MNNA have been used for tuning of PID controller parameters in such a way that maximum value of the closed loop eigen values of K-extreme polynomials is minimized. Further to show the robustness of the proposed methods, the results are taken on set point tracking, noise suppression, load disturbance rejection, and minimum controller effort utilized by these controllers and compared against seven state of the art techniques namely PSO, GA, ABC, MOEO, NSGA-II, FSA and variants of constrained GA. The results demonstrate the high performance of the PID controller optimized using MNNA for control of I-AVR and guarantees stability for a wider range of system parameter uncertainty.

Discrete-time backstepping-state feedback approach for current control of LCL grid-tied converters

Voltage source converters connected to the grid through an LCL filter effectively attenuate the high frequency harmonics generated by the PWM modulation. However, due to the intrinsic resonant peak of this filter and due to the discretization process, the controller design is commonly a challenge and it may not guarantee robustness against abnormalities of the grid. In order to overcome these problems, this paper proposes a new discrete-time current control scheme applied to grid-tied converters based on a backstepping algorithm and state feedback controller in a multi-loop approach. The tracking of converter-side current for a given reference, which is generated by the outer loop, is performed by the backstepping algorithm. In addition, the controller gain design methodology is presented taking into account an extended model, that includes the delay resulted from the discrete-time implementation. An analysis of the gain choice effect on the system closed-loop eigenvalues is demonstrated, which allows the designer to select an appropriate set of controller gains. As outer loop, a state feedback method combined with a state-space resonant controller is used for the grid-side current tracking. In this structure, the harmonic compensation is easily preformed, in addition, the feedback gains are chosen using an optimum algorithm. The robustness of the multi-loop backstepping based control structure is improved in comparison with a full state feedback controller. Simulation and experimental results are obtained to demonstrate the effectiveness of the proposed control scheme under different test conditions, as reference changes, operating during electric faults and grid impedance variation.

Embedded model control for underactuated systems: An application to Furuta pendulum

The main goal of the paper is to test the Embedded Model Control (EMC) design and implementation on a typical underactuated apparatus, like the Furuta pendulum, by comparing experimental results with a Linear Quadratic Regulator (LQR). EMC can be considered as a disturbance rejection control strategy, since the state predictor is extended to explicitly include disturbance dynamics, in charge of predicting the uncertainty to be rejected by control law. Essential in EMC design is the separation between controllable and not controllable dynamics, a task which allows us to find the controllable channel of underactuated systems from the low-dimensional command to the whole system degrees of freedom (DF). Pursuing this objective, a rather generic method is shown, which is applicable to other underactuated systems. The result is a very simple controllable dynamics from the single pendulum command to pendulum DF arranged in a single series of controllable integrators. The neglected feedback channels, including the unstable gravity feedback, are treated as unknown thus posing a challenge to disturbance prediction and closed loop stability. Typical in EMC, closed loop eigenvalues are chosen to guarantee stability, a pre-requisite to performance. Experimental results point out effectiveness and advantage, with respect to LQR, of design and implementation under adverse conditions, due to a disturbance pulse, in which command saturates.

Robust partial quadratic eigenvalue assignment for the damped vibroacoustic system

In this paper, we consider the robust partial quadratic eigenvalue assignment problem for the damped vibroacoustic system by active feedback control. Based on the special structures of the mass and stiffness matrices of the damped vibroacoustic system, we provide a new orthogonality relation and give solution to partial quadratic eigenvalue assignment problem which only needs a few unwanted eigenvalues of the open-loop pencil and associated eigenvectors. With the QR decomposition of the control matrix, we propose a new method to characterize the solution to the quadratic eigenvalue assignment problem, which does not need to solve the Sylvester equation. Finally, we propose two different gradient-based methods for the robust and minimum norm partial quadratic eigenvalue assignment problems, in which the closed-loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously. Numerical experiments demonstrate the effectiveness of our methods.

Controllability of voltage- and calcium-driven cardiac alternans in a map model.

Certain cardiac arrhythmias are preceded by electrical alternans, a state characterized by beat-to-beat alternation in cellular action potential duration. Cardiac alternans may arise from different mechanisms including instabilities in voltage or intracellular calcium cycling. Although a number of techniques have been proposed to suppress alternans, these methods have mainly been tested using models that do not support calcium-driven alternans. Therefore, it is important to understand how control methods may perform when alternans is driven by instabilities in calcium cycling. In this study, we applied controllability analysis to a discrete map of alternans dynamics in a cardiac cell. We compared two different controllability measures to determine to what extent different control strategies could suppress alternans and tested these predictions using three feedback controllers. We found a modal controllability measure, unlike the minimum singular value of the controllability matrix, consistently indicated the control strategies requiring the least control effort and yielding the smallest closed-loop eigenvalue. In addition, action potential duration was identified as the most effective variable through which control can be applied, regardless of alternans mechanism, although sarcoplasmic reticulum calcium load was also useful for the calcium-driven alternans cases.

Quasi‐LPV PI control of TRMS subject to actuator saturation

This study is concerned on the design of a quasi-linear parameter varying (qLPV) proportional–integral (PI) controller for twin rotor multiple-input multiple-output systems (TRMS). The non-linear model is represented as a qLPV polytopic plant with an affine dependence on a non-linear parametric function of the pitch angle. This representation retains the exact model as opposed to the conventional linearisation around an operating point. Due to the availability of the pitch angle measurement, the non-linear parameter can be obtained in real-time and the controller is designed using qLPV technique. To deal with limited control input for such systems, the proposed controller design also considers the actuator saturation that yields controller with practical gains without any additional gain bound criterion. Further, the transient tracking performance is also considered in the design by using closed-loop eigenvalues assignment in desired damping regions. The control synthesis problem is formulated in the form of linear matrix inequalities for L 2 gain based performance criterion. The designed controller is validated on a two-degree of freedom helicopter experimental setup. Finally, to demonstrate the effectiveness of the proposed design, a comparative analysis is done with the existing algorithms. Also, the efficacy of the decentralised controller vis-a-vis the centralised one is presented.