Linear Time-invariant Control Research Articles

Linear Parameter Varying Models-Based Gain-Scheduling Control for Lane Keeping System With Parameter Reduction

In this paper, a robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> controller based on Linear Parameter Varying (LPV) model with scheduling variable reduction is applied for a lane-keeping system. On a curved road section, varying longitudinal speed and roll motion lead to multiple parameter variations in the lateral vehicle dynamics. To trade-off between the complexity of the multiple scheduling variables and the accuracy of the LPV model, an order reducing method by Autoencoder (AE) is performed to obtain a reduced model and diminishes the conservatism of LPV-based controller design. Further, to ensure the continuous convex set membership of the reduced scheduling variables in online test scenarios, the Lipschitz constant of the offline trained neural network (NN) is tightly estimated by solving a Semidefinite Program (SDP). The convex set of local Linear Time-Invariant (LTI) controllers is designed according to the estimated Lipschitz bound of the trained NN. The LPV-based robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> feedback controller is then designed by solving a set of Linear Matrix Inequalities (LMIs). Numerical simulations with full vehicle dynamics from CarSim are given to demonstrate the performance of the proposed system on various test roads. The continuous finding of the reduced scheduling variable membership is guaranteed in the online tests, and the effectiveness of the proposed method is confirmed with the mean value of lateral offset error reduced by about 50% compared with LTI-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{2}$ </tex-math></inline-formula> controllers.

Generalized framework for computing the [formula omitted]-induced norm of sampled-data systems

This paper develops a generalized framework for computing the L∞-induced norm of sampled-data systems, by which we mean those consisting of a continuous-time linear time-invariant (LTI) plant and a discrete-time LTI controller, and such systems have a linear periodically time-varying (LPTV) nature. In the authors’ preceding studies, the input/output relation of sampled-data systems was described in an operator-theoretic fashion, and the kernel and input functions of an input operator and the hold function of an output operator were approximated by piecewise lth order polynomials with l=0,1. This leads to an approximate computation of this norm, where the computation errors were shown to converge to 0 in the order of 1/Ml+1 as the approximation parameter M becomes larger. Along this line, we aim at improving the computation performance compared to the preceding studies by considering the freedom in the point around which the corresponding functions are expanded to a Taylor series. This leads to a generalization of the piecewise lth order approximations, and we show that taking the central point for the Taylor expansion leads to quantitatively improved accuracy than that of our preceding results that take an edge point.Finally, the effectiveness of the developed method is verified through a numerical example.

Stealthy attacks and attack-resilient interval observers

Industrial control systems have been frequent targets of cyber attacks during the last decade. Adversaries can hinder the safe operation of these systems by tampering with their sensors and actuators, while ensuring that the monitoring systems are not able to detect such attacks in time. This paper presents methods to design and overcome stealthy attacks on linear time-invariant control systems that estimate their state using an interval observer, in the presence of unknown but bounded noise and perturbations. We analyze scenarios in which a malicious agent compromises the sensors and/or the actuators of the system with additive attack signals to steer the state estimate outside of the bounds provided by the interval observer. We first show that maximally disruptive attack sequences that remain undetected by a linear monitor can be computed recursively via linear programming. We then design an attack-resilient interval observer for the system’s state, identifying sufficient conditions on the sensor data for such an observer to be realizable. We propose a computational method to determine optimal observer gains using semi-definite programming and compute bounds for the unknown attack signal as well. In numerical simulations, we illustrate and compare the ability of such interval observers to still provide accurate estimates when under attack.

Distinguishability criteria of conformable hybrid linear systems

Abstract We relate this article to the emerging idea of distinguishability of conformable linear hybrid time-invariant control systems. To obtain the necessary and sufficient conditions of α \alpha -distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techniques, a more simple criterion of α \alpha -distinguishability for the fractional linear system is developed.

Conversion From Unstructured LTI Controllers to Observer-Structured Ones for LPV Systems

This paper considers the conversion problem from unstructured linear time-invariant (LTI) controllers to observer-structured LTI controllers, whose structure is similar to but not exactly the same as the so-called “Luenberger observer–based controllers,” for linear parameter-varying (LPV) plant systems. In contrast to Luenberger observer–based controllers, observer-structured LTI controllers can be defined and constructed even if the plant systems are given as LPV systems. In the conversion problem, the state-space matrices of the observer-structured LTI controller are parameterized with those of the given unstructured LTI controller, one free matrix, and a state transformation matrix. We also show a method to obtain the optimal state transformation matrix with respect to the convergence of the discrepancy between the plant state and the observer-structured controller state for a stochastically defined non-zero initial plant state. Several toy examples are included to illustrate the effectiveness and the usefulness of observer-structured LTI controllers, and the utility of the proposed conversion parametrization.

Decentralized Implementation of a Class of Centralized LTI Controllers for Two-Channel Systems Using Periodic Control

Given a centralized linear time-invariant (LTI) controller, this article investigates if and when the same can be implemented in a decentralized fashion using continuous-time, high-frequency, periodic controllers, recently reported in literature, without compromising input–output performance in averaged sense. In this regard, it is shown that for a strongly connected two-channel system having each subsystem of equal sizes, there exists a relation to compute the decentralized periodic controller gains in terms of the gains of a given centralized LTI controller if the latter is minimum-phase, stable, and satisfies some relative degree conditions. A method to satisfy this relative degree constraint while designing the centralized LTI controller is developed. A suitable example is considered to show that the proposed decentralized periodic controller can mimic a centralized LTI controller in averaged sense. At the end, it is found that the proposed decentralized periodic controller, however, does not have enough number of periodic gains to obtain the above relation to compute its parameters for systems with more than two channels.

Delay-based stabilisation and strong stabilisation of LTI systems by nonsmooth constrained optimisation

Stabilisation and strong stabilisation (i.e. the problem of stabilising by a stable controller) of linear time-invariant (LTI) multi-input multi-output systems are considered. The considered systems may be finite-dimensional or time-delay systems, which may have multiple discrete time delays. The controllers to be designed may also be either finite-dimensional or time-delay LTI controllers. In the controller design phase, a constrained nonsmooth optimisation-based design method is employed. The effectiveness of the proposed method is illustrated through various benchmark examples which are solved by using the recently developed Matlab-based software called DrStabilization . The examples also show that, in many cases, time-delay controllers may have certain advantages over conventional finite-dimensional controllers.

Designing and validating a robust adaptive neuromodulation algorithm for closed-loop control of brain states

Objective. Neuromodulation systems that use closed-loop brain stimulation to control brain states can provide new therapies for brain disorders. To date, closed-loop brain stimulation has largely used linear time-invariant controllers. However, nonlinear time-varying brain network dynamics and external disturbances can appear during real-time stimulation, collectively leading to real-time model uncertainty. Real-time model uncertainty can degrade the performance or even cause instability of time-invariant controllers. Three problems need to be resolved to enable accurate and stable control under model uncertainty. First, an adaptive controller is needed to track the model uncertainty. Second, the adaptive controller additionally needs to be robust to noise and disturbances. Third, theoretical analyses of stability and robustness are needed as prerequisites for stable operation of the controller in practical applications. Approach. We develop a robust adaptive neuromodulation algorithm that solves the above three problems. First, we develop a state-space brain network model that explicitly includes nonlinear terms of real-time model uncertainty and design an adaptive controller to track and cancel the model uncertainty. Second, to improve the robustness of the adaptive controller, we design two linear filters to increase steady-state control accuracy and reduce sensitivity to high-frequency noise and disturbances. Third, we conduct theoretical analyses to prove the stability of the neuromodulation algorithm and establish a trade-off between stability and robustness, which we further use to optimize the algorithm design. Finally, we validate the algorithm using comprehensive Monte Carlo simulations that span a broad range of model nonlinearity, uncertainty, and complexity. Main results. The robust adaptive neuromodulation algorithm accurately tracks various types of target brain state trajectories, enables stable and robust control, and significantly outperforms state-of-the-art neuromodulation algorithms. Significance. Our algorithm has implications for future designs of precise, stable, and robust closed-loop brain stimulation systems to treat brain disorders and facilitate brain functions.

Intuitive LTI energy-maximising control for multi-degree of freedom wave energy converters: The PeWEC case

Energy-maximising wave energy conversion control strategies are commonly based upon direct optimal control theory, where the control problem is discretised and transcribed into a nonlinear programme, and a solution is found via numerical routines. Though appealing from an optimality viewpoint, the real-time application of such strategies to realistic (complex) wave energy systems, such as the PeWEC device, can become potentially challenging, due to its intrinsic multiple degree-of-freedom (DoF) nature. Furthermore, this pendulum-based system is not only multi-DoF in its nature, but also underactuated, i.e. only one mode, associated to the pendulum mechanism installed inside the wave-excited floating body, can be effectively actuated. We propose, in this paper, a set of four simple and intuitive energy-maximising controllers for the PeWEC system based, upon linear time-invariant (LTI) systems. We achieve this by deriving the so-called impedance-matching conditions for the PeWEC, and extending well-established LTI controllers, originally designed for fully actuated single-DoF systems, to this multi-DoF underactuated case. In particular, we explore, design, and synthesise both feedback, and feedforward configurations, making explicit emphasis in their main characteristics. Furthermore, we provide a performance assessment for each of the proposed controllers, showing their energy-maximising capabilities for the wave resource characterising the Mediterranean Sea.

Design of Observer-Structured H∞ Flight Controllers Converted from Unstructured Conventional H∞ Flight Controllers

We address the conversion problem from a priori designed unstructured Linear Time-Invariant (LTI) controllers to observer-structured LTI controllers, whose structure is similar to but is not exactly the same as the so-called “Luenberger observer-based controllers”, for Linear Parameter-Varying (LPV) or LTI Parameter-Dependent (LTIPD) plant systems. For this problem, we propose a method by designing appropriate state transformation matrices which minimize the errors between plant systems’ states and transformed controllers’ states for external inputs via parametrically multi-affine Linear Matrix Inequality (LMI) conditions. The proposed method embeds existing LTI controllers with the ability of observers while their control performance is unchanged. We introduce a numerical example and illustrate the usefulness of our method by embedding observer property to an a priori designed H∞ flight controller.

Robust cascaded horizontal-plane trajectory tracking for fixed-wing unmanned aerial vehicles

This paper investigates the problem of horizontal-plane trajectory tracking for fixed-wing unmanned aerial vehicles(UAVs) subjected to external disturbances and uncertainties including coupling and unmodeled dynamics. Under the assumption there exist ideal inner-loop controllers, the 12-state model is reduced to a 6-state translational motion model, which is described by a group of simplified nonlinear equations with equivalent disturbances via introducing general aerodynamic models. Then a new cascaded control structure consisting of an outer-loop controller for position control and inner-loop controllers for attitude and thrust control is proposed. Based on feedback linearization technology and signal compensation theory, the proposed controller applied for position control incorporates a nominal linear time-invariant controller and a robust compensator, the latter of which is introduced to restrain the effects of uncertainties and disturbances. The robust performance of the closed-loop system is proved. Actual experimental results conducted on a small fixed-wing aircraft demonstrate that the proposed control approach is effective.

Decentralized periodic control of interconnected systems with fixed modes

It is well known that decentralized linear time-invariant controllers fail to stabilize interconnected systems with unstable decentralized fixed modes. This paper employs a high-frequency (ω), continuous-time, linear periodic controller to solve this problem. In addition to handling a general class of strongly connected systems, the proposed controller ensures negligible ripples (O(1/ω) at most) in the plant output, unlike O(1) ripples with the existing ones. An H∞-loop-transfer-recovery based sequential method is used to design the controller in such a way that it achieves some centralized linear quadratic performance along with some desired loop properties in averaged sense. The results are illustrated using a suitable example.

An Application of the Z-Transform in Extracting the Density Regulation from the Periodic Output of an Age-Structured Population Model

The application of the Z-transform, a manipulation tool from the discrete signal processing (DSP) toolbox, on an ecological model was motivated by the mathematical similarities between an age-structured fish population model with a non linear density regulation and a linear time invariant (LTI) control system. Both models include a switching mechanism in regulating stock/signal throughput in accordance with a given density limitation/set value and both models can be expressed in terms of a negative feedback loop difference equations (Getz &amp; Haight,1989; °Astr¨om &amp; Murray, 2008). In the fish model, the switching mechanism is a density regulated stock-recruitment (SR) function which models the strategies implemented by the population in keeping the vulnerable egg-larvaejuvenile densities within an environmental limitation thereof (Subbey et al, 2014). A switching mechanism is also present in control engineering, for example, in the mechanism associated with cruise control in cars which keeps traveling speed close to a chosen set value midst varying weather and road conditions (Antsaklis and Gao, 2005). In both cases, the choosing of the control action and the tuning of its parameters requires careful consideration to avoid failures such as incorrectly timed switching actions in a control plant (see Kuphaldt (2019)) and errors in estimating total allowable catch (TAC) in the fishing industry (see Borlestean et al (2015), Skagen et al (2013) and Taboadai and R. Anadn (2016)). The Z-transform has proven itself useful in tuning LTI controlmodels for a desired control action (see Orfanidis, (2010) and Smith, (1999)) and it is on this account that its application was extended to the ecological model in pursuit of a more efficient way of estimating SR parameters to simulate an already existing output. It was however found that it could not be used for parameter tuning but rather for the extraction of the SR component hidden in the output together with components resulting from the age structure itself. Such an extraction can greatly assist in the mathematical identification of the SR, reducing the complexity of its choosing as there are many different types used in the fishing industry such as the classic Beverton-Holt model, the Ricker model and Shepherd model (Myers, 2001; Iles, 1994; Shepherd, 1982). It can also be used to monitor changes in the SR over time which can indicate the presence of strategy evolution (Apaloo et al, 2009; Br¨annstr¨om et al, 2013). In 1998 Schoombie and Getz investigated the latter by subjecting the Shepherd SR to strategy optimization with regards to a parameter associated with population interventions in regulating recruitment throughput and it is because of this versatility that the Shepherd SR is chosen for the intended extraction. In true control style, Simulink, a graphic environment for designing control simulations, is used to visualize the production of the output as well as the extraction of the SR from it. This paper showcases the versatility of the Z transform and the possibilities and unexpected finds when applied to similar systems designed to regulate signals or, in this case, recruitment densities.

Parameterized Trajectory Optimization and Tracking Control of High Altitude Parafoil Generation

Parafoil trajectory directly affects the power generation of a high-altitude wind power generation (HAWPG) device. Therefore, it is particularly important to optimize the parafoil trajectory and then to track it effectively. In this paper, the trajectory of the parafoil at high altitudes is optimized and tracked in a comprehensively parameterized manner. Both the complex dynamic characteristics of the parafoil and the dexterous demand of the high-altitude controller are considered. Firstly, the trajectory variables and control signals are parameterized as Lagrange polynomials in terms of the corresponding values at the selected nodes. Then, the Radau pseudospectral method (PSM) is employed to reformulate the original dynamic trajectory optimization problem into a static nonlinear programming (NLP) problem. By doing so, the parameterized optimal trajectory, which has the maximum net power generation, can be obtained. To attenuate the strong nonlinear, multivariable and coupling characteristics of the flexible parafoil, a bandwidth parameterized linear extended state observer (ESO) is used to estimate and reject these dynamics explicitly in a unified way. Finally, the simulation results demonstrate the effectiveness of the proposed parameterized trajectory optimization and control strategies. The main contribution of this study is that complicated nonlinear parafoil dynamics with a complex trajectory can be well regulated by a PID-type linear time-invariant controller, which is appealing for practitioners.

Design and implementation of decoupled periodic control scheme for a laboratory-based quadruple-tank process

It is well known that linear time-invariant controllers fail to provide desired robustness margins (e.g. gain margin, phase margin) for plants with non-minimum phase zeros. Attempts have been made in literature to alleviate this problem using high-frequency periodic controllers. But because of high frequency in nature, real-time implementation of these controllers is very challenging. In fact, no practical applications of such controllers for multivariable plants have been reported in literature till date. This article considers a laboratory-based, two-input–two-output, quadruple-tank process with a non-minimum phase zero for real-time implementation of the above periodic controller. To design the controller, first, a minimal pre-compensator is used to decouple the plant in open loop. Then the resulting single-input–single-output units are compensated using periodic controllers. It is shown through simulations and real-time experiments that owing to arbitrary loop-zero placement capability of periodic controllers, the above decoupled periodic control scheme provides much improved robustness against multi-channel output gain variations as compared to its linear time-invariant counterpart. It is also shown that in spite of this improved robustness, the nominal performances such as tracking and disturbance attenuation remain almost the same. A comparison with [Formula: see text]-linear time-invariant controllers is also carried out to show superiority of the proposed scheme.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This paper deals with the exponential stability analysis of decentralized, sampled-data, Linear Time Invariant (LTI) control systems with asynchronous sensors and actuators. We consider the case where each controller in the decentralized setting has its own sampling and actuation frequency, which translates to asynchrony between sensors and actuators. Additionally, asynchrony may be induced by delays between the sampling instants and actuation update instants as relevant in a networked context. The decentralized, asynchronous LTI system is represented as the feedback interconnection of a continuous-time LTI system operator and an operator that captures the effects of asynchrony induced by sampling and delay. By characterizing the properties of the operators using small-gain type Integral Quadratic Constraints (IQC), we provide criteria for exponential stability of the asynchronous, decentralized LTI state-space models. The approach provided in this paper considers two scenarios, namely the ‘large-delay’ case and the ‘small-delay’ case where the delays are larger and smaller than the sampling interval, respectively. The effectiveness of the proposed results is corroborated by a numerical example.

Application of Structured Robust Synthesis for Flexible Aircraft Flutter Suppression

This article presents a method for structured robust control design for systems with a mixture of parametric and dynamic uncertainty. The proposed method alternates between an analysis step and a synthesis step. Samples of the parametric uncertainty are computed during the analysis steps, thus yielding an array of uncertain systems containing only dynamic uncertainty. The controller is then synthesized on this array of uncertain models. This synthesis step itself involves an alternation between constructing a D-scale for each of the uncertain systems and tuning a single controller for the entire collection of scaled plants. The controller tuning is performed using structured control design techniques. The proposed method is utilized to design a flutter suppression controller for a flexible aircraft. The aircraft dynamics are described by both a high-fidelity and a reduced-order model. The design objectives for flutter suppression are to achieve robust stabilization in the presence of mixed uncertainty. The proposed structured design method yields a single, low-order, linear time-invariant (LTI) controller, which increases the flutter speed by 15%. Additional robustness analyses and high-fidelity simulations are provided to assess the controller performance.

Overcoming Performance Limitations of Linear Control with Hybrid Integrator-Gain Systems

Abstract It is well-known that the performance of linear time-invariant (LTI) feedback control is hampered by fundamental limitations. In this paper, it is shown that by using a so-called hybrid integrator-gain system (HIGS) in the controller, important fundamental LTI performance limitations can be overcome. In particular, in this paper, we show this for two well-known limitations, where overshoot in the step-response of the system has to be present for any stabilizing LTI controller. For each case, it is shown that by using HIGS-based control, one can avoid overshoot in the step-response of the system. Key design considerations for HIGS-based controllers as well as the stability of the resulting closed-loop interconnections are discussed.

Matrix eigenvalue spectrum assignment for linear control systems by static output feedback

For a linear time-invariant control system defined by a linear differential equation of the n-th order with a multidimensional state, input and output, we set and study the problem of arbitrary matrix eigenvalue spectrum assignment by linear static output feedback. We obtain controllability-like rank conditions that are necessary and sufficient for the problem of arbitrary matrix eigenvalue spectrum assignment and are sufficient for the problem of arbitrary eigenvalue spectrum assignment by linear static output feedback. It is proved that, in particular cases, when the system has block scalar matrix coefficients, these conditions can be relaxed. The obtained results generalize the known results for the corresponding problem by static state feedback and by static output feedback in the case of the one-dimensional equation. Examples are presented to illustrate the results.

On the observability of conformable linear time-invariant control systems

&lt;p style='text-indent:20px;'&gt;In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.&lt;/p&gt;