Time-invariant Control Research Articles

Data-driven stabilization of an oscillating flow with linear time-invariant controllers

This paper presents advances towards the data-based control of periodic oscillator flows, from their fully developed regime to their equilibrium stabilized in closed loop, with linear time-invariant (LTI) controllers. The proposed approach directly builds upon the iterative method of Leclercq et al. (J. Fluid Mech., vol. 868, 2019, pp. 26–65) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments. First, we use input–output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with linear quadratic Gaussian synthesis, which is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method for the classic flow past a cylinder at Reynolds number $Re=100$ . Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.

Controller switching with the anti-bump adapter

Switching on a new or retuned controller while keeping the control loop engaged tends to generate transient “bumps” in the commands and plant outputs. For a linear time-invariant (LTI) controller in state–space form, these bumps can be mitigated or even suppressed by resetting the controller state to an appropriate non-zero value adapted to the pre-switching trajectories of the feedback loop signals. This paper proposes a simple way to compute this adapted controller state by minimizing the difference between the commands actually applied immediately before switching and the virtual command trajectory which the new controller would have generated if it had been activated in parallel. This adapted state can be recursively computed as the output of an LTI system in standard state–space form, the anti-bump adapter. Computing the anti-bump adapter involves only standard matrix algebra and control concepts (e.g., observability and controllability matrices/Gramians). Calculations are very simple and can be translated into compact computer code. This bump suppression procedure is applicable to any discrete-time or continuous-time LTI SISO or MIMO controller, without any assumption on or knowledge of either the previously active controller or the plant. Illustrative applications to continuous-time and discrete-time switching problems and a Matlab® code for computing the anti-bump adapter are presented.

Robust adaptive deep brain stimulation control of in-silico non-stationary Parkinsonian neural oscillatory dynamics

Objective. Closed-loop deep brain stimulation (DBS) is a promising therapy for Parkinson’s disease (PD) that works by adjusting DBS patterns in real time from the guidance of feedback neural activity. Current closed-loop DBS mainly uses threshold-crossing on-off controllers or linear time-invariant (LTI) controllers to regulate the basal ganglia (BG) Parkinsonian beta band oscillation power. However, the critical cortex-BG-thalamus network dynamics underlying PD are nonlinear, non-stationary, and noisy, hindering accurate and robust control of Parkinsonian neural oscillatory dynamics. Approach. Here, we develop a new robust adaptive closed-loop DBS method for regulating the Parkinsonian beta oscillatory dynamics of the cortex-BG-thalamus network. We first build an adaptive state-space model to quantify the dynamic, nonlinear, and non-stationary neural activity. We then construct an adaptive estimator to track the nonlinearity and non-stationarity in real time. We next design a robust controller to automatically determine the DBS frequency based on the estimated Parkinsonian neural state while reducing the system’s sensitivity to high-frequency noise. We adopt and tune a biophysical cortex-BG-thalamus network model as an in-silico simulation testbed to generate nonlinear and non-stationary Parkinsonian neural dynamics for evaluating DBS methods. Main results. We find that under different nonlinear and non-stationary neural dynamics, our robust adaptive DBS method achieved accurate regulation of the BG Parkinsonian beta band oscillation power with small control error, bias, and deviation. Moreover, the accurate regulation generalizes across different therapeutic targets and consistently outperforms current on-off and LTI DBS methods. Significance. These results have implications for future designs of closed-loop DBS systems to treat PD.

Stability analysis of hybrid integrator-gain systems: A frequency-domain approach

The hybrid integrator-gain system (HIGS) has been introduced recently with the aim to overcome fundamental limitations of linear time-invariant (LTI) control systems. To support the analysis and design of HIGS-based controllers, in this paper a novel frequency-domain condition for stability analysis of the feedback interconnection of an LTI system and HIGS is presented. Compared to existing frequency-domain stability conditions such as the one extending the circle-criterion, the condition presented in this paper exploits explicit knowledge regarding HIGS’ switching strategy, thereby potentially providing a significantly less conservative condition. In particular, the novel condition in this paper guarantees the existence of a quadratic Lyapunov function that does not need to be positive definite within the full state space. The proposed condition can be verified graphically in a manner that is reminiscent of the classical Popov plot, as will be illustrated in an experimental case-study.

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

Tracking and Vibration Control with a Parallel Structure Controller Based on a Flexible Ball Screw Drive System

In this paper, we present a parallel structure controller for flexible ball screw drive systems with dynamic variations mainly caused by variations in table position and workpiece mass. The controller consists of two parts: a linear quadratic regulator (LQR) controller with the aim of tracking reference trajectories with high response and accuracy and an interpolated gain-scheduled controller used to restrain system vibration. To damp out the varied resonant modes, the controller is obtained by a set of μ-synthesis linear time-invariant (LTI) controllers interpolated via Youla parameterization. Comparison experiments are conducted to confirm the performance of the proposed controller with a ball screw drive experimental setup. We demonstrate that the parallel structure controller achieves high performance in tracking, vibration suppression and disturbance rejection.

Tracking the Uneven Outcomes of COVID-19 on Racial and Ethnic Groups: Implications for Health Policy.

The socioeconomic shocks of the first COVID-19 pandemic wave disproportionately affected vulnerable groups. But did that trend continue to hold during the Delta and Omicron waves? Leveraging data from the Johns Hopkins Coronavirus Resource Center, this paper examines whether demographic inequalities persisted across the waves of COVID-19 infections. The current study utilizes fixed effects regressions to isolate the marginal relationships between socioeconomic factors with case counts and death counts. Factors include levels of urbanization, age, gender, racial distribution, educational attainment, and household income, along with time- and state-specific COVID-19 restrictions and other time invariant controls captured via fixed effects controls. County-level health outcomes in large metropolitan areas show that despite higher incidence rates in suburban and exurban counties, urban counties still had disproportionately poor outcomes in the latter COVID-19 waves. Policy makers should consider health disparities when developing long-term public health regulatory policies to help shield low-income households from the adverse effects of COVID-19 and future pandemics.

Frequency-Domain Data-Driven Position-Dependent Controller Synthesis for Cartesian Robots

Cartesian robots have position-dependent dynamics that must be taken into account for high-performance applications. Traditional methods design linear time-invariant (LTI) controllers that are robustly stable with respect to position variations but result in reduced performance. Advanced methods require linear parameter varying (LPV) models and LPV controller design methods that are not well-established in the industry. On the other hand, the classical model-based gain-scheduled technique involves parametric identification, high-performance controller design for each position, interpolation of the controller parameters, and real-time controller validation, making it time-consuming and costly. Our approach uses frequency response at different operating points to design an LPV controller using a convex optimization algorithm based on second-order cone programming. The approach is applied to an industrial three-axis Cartesian robot, showing significant improvements over state-of-the-art control design strategies. Data acquisition and controller design can be performed automatically, reducing significantly the engineering costs for controller synthesis.

Data-driven analysis for disturbance amplification in car-following behavior of automated vehicles

This paper presents a data-driven framework to quantitatively analyze the disturbance amplification behavior of automated vehicles in car-following (CF). The data-driven framework can be applied to unknown CF controllers based on the concept of empirical frequency response function (FRF). Specifically, a well-known signal processing method, Welch's method, together with a short time Fourier transformation is developed to extract the empirical transfer functions from vehicle trajectories. The method is first developed assuming a generic linear controller with time-invariant CF control features (e.g., control gains) and later extended to capture time-variant features. The proposed methods are evaluated for estimation consistencies via synthetic data-based simulations. The evaluation includes the performances of the linear approximation accuracy for a linear time-invariant controller, a nonlinear controller, and a linear time-variant controller. Results indicate that our framework can provide reasonably consistent results as theoretical ones in terms of disturbance amplification. Further it can perform better than a linear theoretical analysis of disturbance amplification, particularly when nonlinearity in CF behavior is present. The methods are applied to existing field data collected from vehicles with adaptive cruise control (ACC) on the market. Findings reveal that all tested vehicles tend to amplify disturbances, particularly in low frequency (< 0.5 Hz). Further, the results demonstrate that these ACC vehicles exhibit time-varying features in terms of disturbance amplification ratio depending on the leading vehicle trajectories.

A combined method for stability analysis of linear time invariant control systems based on Hermite‐Fujiwara matrix and Cholesky decomposition

AbstractIn this paper, we have developed an integrative method for checking the stability of linear time‐invariant (LTI) systems as well as nonlinear continuous‐time ones. In our method, we first apply the iterative Faddeev–Leverrier algorithm to obtain the characteristic polynomial of the LTI system. Subsequently, the associated Hermite‐Fujiwara matrix will be evaluated by a particularly efficient technique for the calculation of the Bézoutian matrices. The positive‐definiteness of the Hermite‐Fujiwara form, as the stability criterion, is next tested by performing the Cholesky decomposition. Our method is extended to assess the local stability of nonlinear continuous‐time systems with the help of the Jacobian matrix concept. The proposed method is demonstrated to approximately be 2.2 times faster than the classical Hurwitz algorithm in average, at least for matrices with dimensions less than 40, according to a performed central processing unit (CPU) time analysis. For the sake of illustration, four numerical examples are given, including dynamical models for a real‐world hydrolysis reactor and a well‐mixed bioreactor.

Topological equivalence of linear time-varying control systems

Abstract In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $\dot{x}\left ( t\right )=A(t)x(t)+B(t)u(t)$ defined on an interval $I \subset \mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.

Learning-Based Synthesis of Robust Linear Time-Invariant Controllers

Recent advances in learning for control allow to synthesize controllers from learned system dynamics and maintain robust stability guarantees. However, no approach is well-suited for training robustly-stabilizing linear time-invariant (LTI) controllers using arbitrary learned models of the dynamics. This article introduces a method to do so. It uses a robust control framework to derive robust stability criteria. It also uses simulated policy rollouts to obtain gradients on the controller parameters, which serve to improve the closed-loop performance. By formulating the stability criteria as penalties with computable gradients, they can be used to guide the controller parameters toward robust stability during gradient descent. The approach is flexible as it does not restrict the type of learned model for the simulated rollouts. The robust control framework ensures that the controller is already robustly stabilizing when first implemented on the actual system and no data is yet collected. It also ensures that the system stays stable in the event of a shift in dynamics, given the system behavior remains within assumed uncertainty bounds. We demonstrate the approach by synthesizing a controller for simulated autonomous lane-change maneuvers. This work thus presents a flexible approach to learning robustly stabilizing LTI controllers that takes advantage of modern machine learning techniques.

Dynamic Controller That Operates Over Homomorphically Encrypted Data for Infinite Time Horizon

In this article, we present a dynamic feedback controller that computes the next state and the control signal over encrypted data using homomorphic properties of cryptosystems, whose performance is equivalent to the linear dynamic controllers over real-valued data. Assuming that the input as well as the output of the plant is encrypted and transmitted back to the controller, it is shown that the state matrix of any linear time-invariant controller can be always converted to a matrix of integer components. This allows the dynamic feedback controller to operate for infinite time horizon without decryption or reset of its internal state. For implementation in practice, we illustrate the use of a cryptosystem that is based on the Learning With Errors problem, which allows both multiplication and addition over encrypted data. It is also shown that the effect of injected random numbers during encryption for security can be maintained within a small bound by way of the closed-loop stability.

Satellite Dynamics Toolbox Library: a tool to model multi-body space systems for robust control synthesis and analysis

The level of maturity reached by robust control theory techniques nowadays contributes to a considerable minimization of the development time of an end-to-end control design of a spacecraft system. The advantage offered by this framework is twofold: all system uncertainties can be included from the very beginning of the design process; the validation and Verification (V&V) process is improved by fast detection of worst-case configurations that could escape to a classical sample-based Monte Carlo simulation campaign. Before proceeding to the control synthesis and analysis, a proper uncertain plant model has to be available in order to push these techniques to their limits of performance. In this spirit, the Satellite Dynamics Toolbox Library (SDTlib) offers many features to model a spacecraft system in a multi-body fashion on SIMULINK. Parametric models can be easily built in a Linear Fractional Transformation (LFT) form by including uncertainties and varying parameters with minimal number of repetitions. Uncertain Linear Time Invariant (LTI) and uncertain Linear Parameter-Varying (LPV) controllers can then be synthesized and analyzed in a straightforward way. The authors present in this article a tutorial, that can be downloaded at https://nextcloud.isae. fr/index.php/s/XDfRfHntejHTmmp, to show how to deal with an end-to-end robust design of a spacecraft mission and to provide to researchers a benchmark to test their own algorithms.

Frequency-domain stability conditions for split-path nonlinear systems

This paper considers the class of control systems containing so-called split-path nonlinear (SPAN) filters, which are designed to overcome some of the well-known fundamental limitations in linear time-invariant (LTI) control. In this work, we are interested in developing tools for the stability analysis of such systems using frequency-domain techniques. Hereto, we explicitly show the equivalence between a set of linear matrix inequalities (LMIs) with S-procedure terms, guaranteeing stability of the closed-loop (SPAN) system, and a frequency-domain condition. We also provide a systematic procedure for verifying the frequency-domain condition in a graphical manner. The results are illustrated through a nummerical case study.

Linear Parameter Varying Controller Design For Satellite Attitude Control*

This paper presents a systematic linear parameter varying (LPV) control approach for the 3-axis attitude control of an Earth-observation satellite in a sun-synchronous orbit. The dynamics of the satellite depend on the orientation of the solar array which completes a full rotation every orbit, thus it is used as a scheduling parameter in the design. The satellite has two additional flexible appendages; these are 2 scatterometers. The control objective is to precisely track a given reference attitude using reaction wheels, while rejecting external torque disturbances and sensor noise. The design follows a mixed-sensitivity approach, applying a recently introduced weighting scheme. It allows traceable and effective controller tuning by using a low number of physically interpretable weights. The controller is synthesised by solving the induced L2-norm of the closed-loop interconnection of the controller and weighted plant. Scheduling with the solar array orientation leads to an LPV notching behaviour in the controller that effectively mitigates the effects of the array's most prominent flexible modes. This behaviour enables increased performance, when compared to a linear time invariant controller, while maintaining robustness. The pointing performance of the synthesised controller over the complete satellite lifecycle is verified using the European Space Agency's standards for spacecraft attitude control.

A Small-Gain Approach to Incremental Input-to-State Stability Analysis of Hybrid Integrator-Gain Systems

Incremental input-to-state stability plays an important role in the analysis of nonlinear systems, as it opens up the possibility for accurate performance characterizations beyond classical approaches. In this paper, we are interested in deriving conditions for incremental stability of a specific class of discontinuous dynamical systems containing a so-called hybrid integrator. Recently, it was shown that hybrid integrators have the potential for overcoming fundamental performance limitations of linear time-invariant control, thereby making them interesting for use in, e.g., high-precision motion control applications. The main contribution of this paper is to show that these hybrid integrators have incremental input-to-state stability properties, and that, under an incremental small-gain condition, the feedback interconnection of a hybrid integrator and a linear time-invariant plant is incrementally input-to-state stable.

Contraction analysis of Volterra integral equations via realization theory and frequency-domain methods

&lt;p style='text-indent:20px;'&gt;Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the contraction analysis of a class of nonlinear Volterra integral equations in some Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in some weighted Sobolev space which has the same stability properties as the Volterra equation.&lt;/p&gt;

Implementing prescribed-time convergent control: sampling and robustness

According to recent results, convergence in a prespecified or prescribed finite time can be achieved under extreme model uncertainty if control is applied continuously over time. This paper shows that this extreme amount of uncertainty cannot be tolerated under sampling, not even if sampling could become infinitely frequent as the deadline is approached, unless the sampling strategy were designed according to the growth of the control action. Robustness under model uncertainty is analyzed and the amount of uncertainty that can be tolerated under sampling is quantified in order to formulate the least restrictive prescribed-time control problem that is practically implementable. Some solutions to this problem are given for a scalar system. Moreover, either under a-priori knowledge of bounds for initial conditions, or if the strategy can be selected after the first measurement becomes available, it is shown that the real, practically achievable objectives can also be reached with linear time-invariant control and uniform sampling. These derivations serve to yield insight into the real advantages that implementation of prescribed-time controllers may have.