Unknown Linear Time-invariant System Research Articles

Robust data-driven predictive control for unknown linear time-invariant systems

This paper presents a new robust data-driven predictive control scheme for unknown linear time-invariant systems by using input-state-output or input–output data based on whether the state is measurable. To remove the need for the persistently exciting (PE) condition of a sufficiently high order on pre-collected data, a set containing all systems capable of generating such data is constructed. Then, at each time step, an upper bound on a given objective function is derived for all systems in the set, and a feedback controller is designed to minimize this bound. The optimal control gain at each time step is determined by solving a set of linear matrix inequalities. We prove that if the synthesis problem is feasible at the initial time step, it remains feasible for all future time steps. Unlike current data-driven predictive control schemes based on behavioral system theory, our approach requires less stringent conditions for the pre-collected data, facilitating easier implementation. The effectiveness of our proposed methods is demonstrated through application to an unknown and unstable batch reactor.

Combining Prior Knowledge and Data for Robust Controller Design

We present a framework for systematically combining data of an unknown linear time-invariant system with prior knowledge on the system matrices or on the uncertainty for robust controller design. Our approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data. The design procedures rely on a combination of multipliers inferred via prior knowledge and learnt from measured data, where for the latter a novel and unifying disturbance description is employed. While large parts of the paper focus on linear systems and input-state measurements, we also provide extensions to robust output-feedback design based on noisy input-output data and against nonlinear uncertainties. We illustrate through numerical examples that our approach provides a flexible framework for simultaneously leveraging prior knowledge and data, thereby reducing conservatism and improving performance significantly if compared to black-box approaches to data-driven control.

Linear Data-Driven Economic MPC with Generalized Terminal Constraint⋆

In this paper, we propose a data-driven economic model predictive control (EMPC) scheme with generalized terminal constraint to control an unknown linear time-invariant system. Our scheme is based on the Fundamental Lemma to predict future system trajectories using a persistently exciting input-output trajectory. The control objective is to minimize an economic cost objective. By employing a generalized terminal constraint with artificial equilibrium, the scheme does not require prior knowledge of the optimal equilibrium. We prove that the asymptotic average performance of the closed-loop system can be made arbitrarily close to that of the optimal equilibrium. Moreover, we extend our results to the case of an unknown linear stage cost function, where the Fundamental Lemma is used to predict the stage cost directly. The effectiveness of the proposed scheme is shown by a numerical example.

Fast identification and stabilization of unknown linear systems

In the present work, a simple algorithm for stabilizing an unknown linear time-invariant system is proposed, assuming only that this system is stabilizable. The suggested algorithm is based on first performing a partial identification of the system and then stabilizing the controllable subsystem. It should be emphasized that our approach does not depend on any prior model knowledge and requires only a minimal number of samples.

A novel constraint‐tightening approach for robust data‐driven predictive control

In this paper, we present a data-driven model predictive control (MPC) scheme that is capable of stabilizing unknown linear time-invariant systems under the influence of process disturbances. To this end, Willems' lemma is used to predict the future behavior of the system. This allows the entire scheme to be set up using only a priori measured data and knowledge of an upper bound on the system order. First, we develop a state-feedback MPC scheme, based on input-state data, which guarantees closed-loop practical exponential stability and recursive feasibility as well as closed-loop constraint satisfaction. The scheme is extended by a suitable constraint tightening, which can also be constructed using only data. In order to control a priori unstable systems, the presented scheme contains a prestabilizing controller and an associated input constraint tightening. We first present the proposed data-driven MPC scheme for the case of full state measurements, and also provide extensions for obtaining similar closed-loop guarantees in case of output feedback. The presented scheme is applied to a numerical example.

On the prescribed-time attractivity and frozen-time eigenvalues of linear time-varying systems

A system is called prescribed-time attractive if its solution converges at an arbitrary user-defined finite time. In this note, necessary and sufficient conditions are developed for the prescribed-time attractivity of linear time-varying (LTV) systems. It is proved that the frozen-time eigenvalues of a prescribed-time attractive LTV system have negative real parts when the time is sufficiently close to the convergence moment. This result shows that the ubiquitous singularity problem of prescribed-time attractive LTV systems can be avoided without instability effects by switching to the corresponding frozen-time system at an appropriate time. Consequently, it is proved that the time-varying state-feedback gain of a prescribed-time controller, designed for an unknown linear time-invariant system, approaches the set of stabilizing constant state-feedback gains.

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.

Online Convex Optimization for Data-Driven Control of Dynamical Systems

We propose an algorithm based on online convex optimization for controlling discrete-time linear dynamical systems. The algorithm is data-driven, i.e., does not require a model of the system, and is able to handle a priori unknown and time-varying cost functions. To this end, we make use of a single persistently exciting input-output sequence of the system and results from behavioral systems theory which enable it to handle unknown linear time-invariant systems. Moreover, we consider noisy output feedback instead of full state measurements and allow general economic cost functions. Our analysis of the closed loop reveals that the algorithm is able to achieve sublinear regret, where the measurement noise only adds an additional constant term to the regret upper bound. In order to do so, we derive a data-driven characterization of the steady-state manifold of an unknown system. Moreover, our algorithm is able to asymptotically exactly estimate the measurement noise. The effectiveness and applicational aspects of the proposed method are illustrated by means of a detailed simulation example in thermal control.

Data-driven model predictive control: closed-loop guarantees and experimental results

Abstract We provide a comprehensive review and practical implementation of a recently developed model predictive control (MPC) framework for controlling unknown systems using only measured data and no explicit model knowledge. Our approach relies on an implicit system parametrization from behavioral systems theory based on one measured input-output trajectory. The presented MPC schemes guarantee closed-loop stability for unknown linear time-invariant (LTI) systems, even if the data are affected by noise. Further, we extend this MPC framework to control unknown nonlinear systems by continuously updating the data-driven system representation using new measurements. The simple and intuitive applicability of our approach is demonstrated with a nonlinear four-tank system in simulation and in an experiment.

Time-Domain Adaptive Higher Harmonic Control for Rejection of Sinusoidal Disturbances

Abstract This paper presents two new time-domain feedback controllers that reject sinusoidal disturbances with known frequencies acting on an asymptotically stable linear time-invariant (LTI) system. The first controller is time-domain higher harmonic control (TD-HHC), which is effective for uncertain LTI systems. The second controller is time-domain adaptive higher harmonic control (TD-AHHC), which is effective for completely unknown LTI systems. TD-HHC requires an estimate of the control-to-performance transfer function evaluated at the disturbance frequencies. In contrast, TD-AHHC does not require any information regarding the LTI system. We analyze the stability and closed-loop performance of TD-HHC and TD-AHHC. For both TD-HHC and TD-AHHC, we show that the controller asymptotically rejects the disturbance. We present numerical simulations comparing TD-HHC and TD-AHHC with frequency-domain higher harmonic control (FD-HHC), which is an existing frequency-domain controller for rejection of sinusoidal disturbances. We also present results from acoustic disturbance rejection experiments, which demonstrate the practical effectiveness of both TD-HHC and TD-AHHC.

Switched Fractional Order Model Reference Adaptive Control for Unknown Linear Time Invariant Systems

Abstract This paper presents a model reference adaptive controller for linear time invariant systems with unknown parameters. The adaptive laws used to estimate the controller parameters are based on fractional differential equations, whose orders are switched among a fractional value in the interval (0,1) and 1 at certain time instants. Boundedness of the signals in the resulting controlled system is proved using recent results. A simulation study is provided for a second order system to show how the proposed control strategy can improve the behavior and decrease the control energy used, compared to the classic model reference adaptive controllers and fractional order model reference adaptive controllers with non-switched adaptive laws.

Structured exploration in the finite horizon linear quadratic dual control problem

This paper presents a novel approach to synthesize dual controllers for unknown linear time-invariant systems with the tasks of optimizing a quadratic cost while reducing the uncertainty. To this end, a synthesis problem is defined where the feedback law has to simultaneously gain knowledge of the system and robustly optimize the cost. By framing the problem in a finite horizon setting, the trade-offs arising when the tasks include both identification and control are formally captured in the optimization problem. Results show that efficient exploration strategies are achieved when the structure of the problem is exploited.

Data-Driven Adaptive LQR for Completely Unknown LTI Systems

In this paper, a data-driven on-policy optimal control design is proposed for continuous-time linear time invariant (LTI) systems with completely unknown dynamics. An online system identifier and control gain parameter estimator, which use past and current data together with standard gradient descent update laws, facilitate the design of an adaptive optimal controller that guarantees parameter convergence without the need of persistence of excitation (PE). Unlike the classical approach of enforcing the restrictive PE condition on the regressor, the data-driven approach is verifiable online and establishes parameter convergence from information rich past stored data simultaneously with the current data. A state feedback controller is designed using a dynamic gain parameter which is shown to converge to the neighborhood of the optimal LQR gain. Semi-global uniformly ultimately bounded (UUB) stability of the overall system is established using Lyapunov-based analysis. Simulation results further validate the developed result.

Data-driven robust receding horizon fault estimation

This paper presents a data-driven receding horizon fault estimation method for additive actuator and sensor faults in unknown linear time-invariant systems, with enhanced robustness to stochastic identification errors. State-of-the-art methods construct fault estimators with identified state-space models or Markov parameters, without compensating for identification errors. Motivated by this limitation, we first propose a receding horizon fault estimator parameterized by predictor Markov parameters. This estimator provides (asymptotically) unbiased fault estimates as long as the subsystem from faults to outputs has no unstable transmission zeros. When the identified Markov parameters are used to construct the above fault estimator, stochastic identification errors appear as model uncertainty multiplied with unknown fault signals and online system inputs/outputs (I/O). Based on this fault estimation error analysis, we formulate a mixed-norm problem for the offline robust design that regards online I/O data as unknown. An alternative online mixed-norm problem is also proposed that can further reduce estimation errors at the cost of increased computational burden. Based on a geometrical interpretation of the two proposed mixed-norm problems, systematic methods to tune the user-defined parameters therein are given to achieve desired performance trade-offs. Simulation examples illustrate the benefits of our proposed methods compared to recent literature.

Adaptive sinusoidal disturbance cancellation for unknown LTI systems despite input delay

Solutions exist for the problems of canceling sinusoidal disturbances and compensating input delays. In this paper, two problems are considered simultaneously and an adaptive controller is designed to cancel unknown sinusoidal disturbances forcing an unknown linear time-invariant system in controllable canonical form despite input delay. The design is based on three steps, (1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters, (2) representation of the delay as a transport PDE, (3) design of the adaptive controller by using the backstepping boundary control technique for PDEs. It is proven that the equilibrium of the closed loop system is stable and the state of the considered system converges to zero as t→∞ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a second order system.

State derivative feedback for adaptive cancellation of unmatched disturbances in unknown strict-feedback LTI systems

Solutions exist for the problem of canceling sinusoidal disturbances by the measurement of the state or by the measurement of an output for linear and nonlinear systems. In this paper, an adaptive backstepping controller is designed to cancel sinusoidal disturbances forcing an unknown linear time-invariant system in controllable canonical form which is augmented by a linear input subsystem with unknown system parameters. The state-derivatives of the original subsystem and the state of the input subsystem are the only measurements that are used in the design of the controller. The design is based on four steps, (1) parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters, (2) design of an adaptive disturbance observer for both disturbance and its derivative, (3) design of an adaptive controller for the virtual control input, and (4) design of the final adaptive controller by using the backstepping procedure. It is proven that the equilibrium of the closed-loop adaptive system is stable and the state of the considered original subsystem converges to zero as t→∞ with perfect disturbance estimation. The effectiveness of the controller is illustrated with a simulation example of a third order system.

Adaptive spatiotemporal modelling and estimation of the event-related fMRI responses

Functional magnetic resonance imaging (fMRI) data analysis is a challenging problem due to the underlying physiological complexity of the brain and the scanning process. From engineering perspective, the fMRI data analysis can be viewed as a system modelling problem. In this paper, assuming the fMRI signal as the output of an unknown linear time-invariant system, a spatiotemporal adaptive filter is proposed to model the spatial activation patterns as well as the haemodynamic response (HDR) to the event-related stimulus. The well-known least mean square adaptive algorithm is used for estimating the coefficients of the spatiotemporal filter. The proposed method is shown to be equivalent to the canonical correlation analysis method. It is then extended to multiple event type scenarios to estimate the HDRs of each event type. Results from simulated as well as real fMRI data show that these adaptive modelling schemes can capture the variations of the HDR at different regions of the brain and hence enhance the estimation accuracy of the activation patterns.

Multiple models for adaptive control to improve the performance of minimum variance regulators

The design of a minimum variance regulator for systems operating in dynamical uncertain environments is considered. For minimum variance self-tuning control of an unknown linear time-invariant system, a stable control strategy is developed that improves the transient response by using multiple models to describe the different environments and switching between the relative controllers. The controller is determined at every time instant by the model which best approximates the plant. The performance of the controller is evaluated by a set of simulation tests.

Model reference adaptive control of the output stochastic distributions for unknown linear stochastic systems

This paper presents a model reference based adaptive control approach for the control of the output probability density function for unknown linear time-invariant stochastic systems. Different from most existing models used in stochastic control, it is assumed here that the measured control input directly affects the distribution of the system output in probability sense. As such, the purpose of control is to make the shape of the probability density function of the system output as close as possible to a prespecified one. Using the weighted integration of the measurable output probability density functions, two adaptive on-line updating rules are developed which guarantee the global stability for theclosed loop adaptive control system under certain conditions. Ithas been shown, when there is no external disturbance, that the so-formed closed loop system also realizes the perfect tracking (i.e., the probability density function of the system output approaches a class of given distributions asymptotically). A simulated example is included to illustrate the use of the developed control algorithm and desired results have been obtained.

Fourier series based nonminimum phase model for statistical signal processing

In this paper, a parametric Fourier series based model (FSBM) for or as an approximation to an arbitrary nonminimum-phase linear time-invariant (LTI) system is proposed for statistical signal processing applications where a model for LTI systems is needed. Based on the FSBM, a (minimum-phase) linear prediction error (LPE) filter for amplitude estimation of the unknown LTI system together with the Cramer-Rao (CR) bounds is presented. Then, an iterative algorithm for obtaining the optimum LPE filter with finite data is presented that is also an approximate maximum-likelihood algorithm when data are Gaussian. Then three iterative algorithms using higher order statistics (HOS) with finite non-Gaussian data are presented to estimate parameters of the FSBM followed by some simulation results as well as some experimental results with real speech data to support the efficacy of the proposed algorithms using the FSBM. Finally, we draw some conclusions.