Linear Time-varying Research Articles

Optimal Selection of Basis Functions for Minimum-Effort Tracking Control of Nonminimum Phase Systems Using Filtered Basis Functions

Abstract Accurate tracking of nonminimum phase (NMP) systems is well known to require large amounts of control effort. It is, therefore, of practical value to minimize the effort needed to achieve a desired level of tracking accuracy for NMP systems. There is growing interest in the use of the filtered basis functions (FBF) approach for tracking the control of linear NMP systems because of distinct performance advantages it has over other methods. The FBF approach expresses the control input as a linear combination of user-defined basis functions. The basis functions are forward filtered through the dynamics of the plant, and the coefficients are selected such that the tracking error is minimized. There is a wide variety of basis functions that can be used with the FBF approach, but there has been no work to date on how to select the best set of basis functions. Toward selecting the best basis functions, the Frobenius norm of the lifted system representation (LSR) of dynamics is proposed as an excellent metric for evaluating the performance of linear time varying (LTV) discrete-time tracking controllers, like FBF, independent of the desired trajectory to be tracked. Using the metric, an optimal set of basis functions that minimize the control effort without sacrificing tracking accuracy is proposed. The optimal set of basis functions is shown in simulations and experiments to significantly reduce control effort while maintaining or improving tracking accuracy compared to popular basis functions, like B-splines.

Distributed Consensus of Heterogeneous Linear Time-Varying Systems on UAVs–USVs Coordination

This brief addresses the distributed consensus of heterogeneous linear time-varying (LTV) systems composed of unmanned aerial vehicles (UAVs) and unmanned surface vehicles (USVs), where the UAVs and USVs are modeled as integrator agents and Euler–Lagrange (EL) agents, respectively. Given the connected communication topology, sufficient conditions of the designed distributed consensus protocol are derived to achieve the consensus. The parameter impact of EL agents on control performance to coordinate the consensus of UAVs–USVs are analyzed, whose effectiveness has been illustrated through numerical simulations as well.

On the finite-time boundedness of linear systems

This paper deals with the finite-time boundedness (FTB) of linear time-varying (LTV) systems; in this context, the novel contribution of our work goes in several directions. The first result is a sufficient condition for FTB when the plant input is generated by an exo-system; it is worth noting that constant, norm bounded signals, which most part of the previous literature has considered so far, belong to such class of inputs. To this regard, it is possible to compare our result with the existing literature, which shows that our approach yields less conservative conditions. The second result provided in the paper, is a sufficient condition for FTB when the input belongs to L2, the class of square integrable signals, which have never been considered so far. In both cases, the proposed results require the solution of a feasibility problem constrained by differential linear matrix inequalities (DLMIs). A third contribution of the paper proves that, from the system-theoretic point of view, the obtained FTB conditions can be interpreted in terms of the norm of a suitable input–output operator. As a consequence of this fact, we are able to find alternative and equivalent conditions for FTB, both for the exogenous and the L2 cases, which are based on the solution of differential Lyapunov equations (DLEs), which are much more efficient from the computational point of view. The DLMI-based condition, however, is fundamental to solve the design problem, which is also investigated in the paper. Some examples show that the proposed technique is less conservative with respect to the previous literature, when exogenous inputs are considered, and its effectiveness when L2 signals are considered. Moreover, one example is devoted to show that the analysis approach, based on the solution of DLE, is much more efficient from the computational point of view with respect to the one based on DLMIs; finally the last example investigates the application of the state feedback control technique.

Balanced Model Reduction for Linear Time-Varying Symmetric Systems

The goal of this paper is to develop balancing theory for the linear time-varying (LTV) symmetric systems. To this end, first, we extend the concept of symmetry in terms of the dual system. Then, we define the cross Gramian for the LTV systems. For LTV symmetric systems, we establish a connection among the controllability, observability, and cross Gramians. In particular, if one of these three Gramians is obtained, the other two Gramians can be constructed. Based on this fact, we show that the symmetry structure is preserved under balanced truncation if the Hankel singular values are pointwise distinct.

A periodic linear–quadratic controller for suppressing rotor-blade vibration

This paper presents an active control strategy, based on a time-varying linear–quadratic optimal control problem, to attenuate the tip vibration of a two-dimensional coupled rotor-blade system whose dynamics is periodic. First, a periodic full-state feedback controller based on the linear–quadratic regulator (LQR) problem is designed. If all the states are not available for feedback, then an optimal periodic time-varying estimator, using the Kalman–Bucy filter, is computed. Both the Kalman filter gain and the LQR gain are obtained as the solution of a periodic Riccati differential equation (PRDE). Together, these gains provide the observer-based linear–quadratic–Gaussian (LQG) controller. An algorithm to solve the PRDE is also presented. Both controller designs ensure closed-loop stability and performance for the linear time-varying rotor-blade equation of motion. Numerical simulations show that the LQR and the LQG controllers are able to significantly attenuate the rotor-blade tip vibration.

Three-dimensional trajectory tracking of a hybrid autonomous underwater vehicle in the presence of underwater current

In this work, a six degrees-of-freedom (DOF) nonlinear kinematic and dynamic model of a Hybrid Autonomous Underwater Vehicle (H-AUV) is derived for the two modes of locomotion, propelled and gliding modes. A comprehensive linearization algorithm is developed to include both modes of locomotion. Starting with a three-dimensional time-parameterized curve of position history, Frenet-Serret frames are used to specify the unknown nominal states, which in turn are used to obtain the nominal inputs. A linear time-variant (LTV) state-space model is obtained, and a linear quadratic regulator (LQR) is designed and applied to the nonlinear model. The performance of the devised controller is assessed via a metric that computes the error between the actual and the desired position. Simulation results show that the LQR provides accurate tracking performance, even in the presence of underwater currents with bounded flow velocity. Moreover, the controller autonomously switches modes between propulsion and gliding to ensure minimal trajectory tracking error and energy consumption.

Robust fault detection and isolation based on zonotopic unknown input observers for discrete-time descriptor systems

In this paper, we propose a robust fault detection and isolation (FDI) strategy based on zonotopic unknown input observers (UIOs) for discrete-time descriptor linear time-varying (LTV) systems subject to uncertainties and additive actuator faults. System uncertainties including state disturbances and measurement noise are unknown but bounded by predefined zonotopes. The uncertain state estimations and constructed residuals for robust FDI are propagated in a sequence of zonotopes. Based on a defined performance criterion, the fault detection (FD) observer gain is designed to be robust against uncertainties and meanwhile sensitive to faults. The explicit computational method for the FD observer gain is derived. In addition to include fault isolation, a bank of zonotopic UIOs are employed. Finally, we apply the proposed method into two case studies to show its effectiveness.

Deterministic learning from sampling data

In this paper, based on the deterministic learning mechanism, we present an alternative systematic scheme for dynamics identification from sampling data sequences. The proposed scheme belongs to a dynamical machine learning framework based on Lyapunov stability theory rather than optimization estimation approaches. Given a sampling data sequence collected from an unknown deterministic nonlinear dynamical system, we show that the inherent dynamics of the sampling data sequence can be locally accurately captured and represented as a converged constant neural network. This accurate identification result can be derived from the exponential stability of a class of linear time-varying (LTV) error systems usually appearing in adaptive control and identification fields. However, there are few existing results concerning the exponential stability problem of the discrete-time LTV case. To this end, by using the small gain theorem of input-to-state stability (ISS), we provide a rigorous proof for establishing the exponential stability of the LTV system under the persistent excitation (PE) condition. Based on the exponential stability results, the exponential convergence of neural weights to their optimal values can be analytically guaranteed, which leads to the achievement of the accurate dynamics identification. The significance of the paper is that we preliminarily explore the essential problem of machine learning of dynamical systems by innovatively transforming tools and methodologies from dynamic system stability and control theory. Numerical simulation results are provided to validate the effectiveness of the proposed scheme with comparison to previously mentioned approaches.

Finite-horizon multi-objective generalized [formula omitted] control with transients

This paper presents a variational approach to computing the finite-horizon gain of a linear time-varying (LTV) system which characterizes the worst-case peak value of a multiple output measured by the generalized L∞ norm in response to uncertain initial states and the external disturbance with a bounded energy. This induced operator norm is named the generalized H2 norm with transients and is determined in terms of the solution to some Lyapunov differential matrix equation and inequality. By using discretization, a semi-definite program is derived to compute the optimal control minimizing the generalized H2 norm with transients for a given output. It is shown that Pareto optimal controls minimizing the generalized H2 norms with transients for several outputs turn out to be the generalized H2 controls with transients with respect to a single multiple artificial output consisting of the parameterized outputs. As a byproduct, necessary and sufficient conditions in terms of the generalized H2 norm with transients are provided for a LTV system to be finite-time bounded in a modified formulation. The efficiency of the approach proposed is demonstrated on some problems of optimal protection from shock and vibration.

Adaptive model predictive control for linear time varying MIMO systems

A robust, adaptive Model Predictive Control (MPC) approach for asymptotically stable, constrained linear time-varying (LTV) systems with multiple inputs and outputs is proposed. The approach consists of two-steps, carried out on-line with a receding horizon strategy. In the first one, a real-time Set Membership identification algorithm exploits the measured input–output data and the available prior knowledge to build and refine a set of admissible models of the plant (Feasible Parameter Set, FPS). This set is guaranteed to contain also the true system dynamics under the considered working assumptions. In the second step, a robust finite-horizon optimal control problem is formulated and solved. The variation of system dynamics is taken into account by inflating the FPS over the prediction horizon, according to worst-case bounds, assumed a priori, on the parameters’ rate of change. The resulting optimal control sequence guarantees that the outputs of all possible plants inside the FPS satisfy the operational constraints, also considering all possible future parameter changes. The main theoretical properties of the proposed approach are demonstrated and the method is showcased in numerical simulations, highlighting the fundamental improvement over previous approaches not designed for LTV systems.

Observation and stabilization of LTV systems with time-varying measurement delay

This work deals with observer-based output-feedback stabilization of linear time-varying (LTV) systems with time-varying measurement delay. The delay is modeled by a first-order hyperbolic partial differential equation (PDE), whose boundary condition evolves according to an ordinary differential equation (ODE), namely, the plant. The overall system is thus represented by an ODE–PDE cascade, which is also the structure of the proposed observer. The exponential stability of the estimation error is established for arbitrarily large time-varying delays. An observer-based controller is also introduced and the closed-loop stability is proved. The separation principle holds and the design of both the observer and controller gains can be done as if there was no delay. Some simulations illustrate the feasibility of this approach.

A Parametric Method of Linear Functional Observers for Linear Time-varying Systems

This paper considers the design of a Luenberger observer to estimate the linear multiple states functional for linear time-varying (LTV) systems. Based on the solutions to a type of full actuated homogeneous generalized Sylvester matrix equations and the conditions for the existence of observers for LTV systems, general parametric solutions to Luenberger functional observers are established. With the proposed approach, the functional observers can be achieved at desired convergence rate of the observation error, and also without any particular transformation for LTV systems. Finally, a numerical example shows the feasibility and effectiveness of the proposed parametric design method for the functional observers in LTV systems.

Lyapunov conditions for finite-time stability of time-varying time-delay systems

In this paper, we develop the Lyapunov–Razumikhin method to finite-time stability (FTS) and finite-time contractive stability (FTCS) of time-delay systems. Several Lyapunov-based sufficient conditions for establishing these FTS properties are obtained. Then the theoretical results are applied to FTS and FTCS for a class of linear time-varying (LTV) time-delay system. The efficiency of the proposed criteria is illustrated by three numerical examples, where a stabilizing memoryless controller for FTCS of a second-order LTV system with time delay is proposed.

An easy design for interval observers

ABSTRACT This paper deals with the problem of interval estimation. The proposed approach is based on the use of successive derivatives in continuous time (or forward measurements in discrete time) in an algebraic (or local) interval observer. The main advantage of such an observer is that it relaxes the Metzler assumption widely used in the literature of cooperative interval observers. Moreover, no prior knowledge on the initial condition is needed, and no stability nor convergence issue do appear as the estimation is local. After introducing such an observer for Linear Time Variant (LTV) systems with bounded disturbances, some extensions are proposed for Linear Parameter Varying (LPV) systems as well as for functional systems and systems affected by Unknown Input (UI). Some examples illustrate the theoretical contributions.

Output Feedback Tracking Control of Flat Systems via Exact Feedforward Linearization and LPV Techniques

In this article, an output feedback control scheme is proposed for trajectory tracking of differentially flat systems. After applying the exact feedforward linearization, the tracking error dynamics is linearized as a linear time-varying (LTV) system, where only the state matrix is time-varying while the input and output matrices are time-invariant. Using linear parameter-varying (LPV) techniques based on polytopes, for the LTV system the controller matrices, which are set to be affine on time-varying parameters, are calculated by solving a set of linear matrix inequalities (LMIs). An example for trajectory tracking of a two-wheeled mobile robot is given to show the effectiveness of the proposed method.

Finite Horizon Worst Case Analysis of Launch Vehicles

Abstract This paper presents a robust linear time varying (LTV) analysis of a space launcher. It evaluates the worst case aerodynamic loads during the atmospheric flight phase under wind turbulence. The results of the LTV analysis are evaluated against a Monte Carlo simulation of the respective industry-sized nonlinear launcher model over a set of allowable uncertainties and disturbances. The recent extension of the strict/ finite horizon bounded real lemma (BRL) for integral quadratic constraints is applied, covering the robustness analysis of finite horizon linear time varying (LTV) systems. Here, an equivalent formulation of the strict BRL in the form of a Riccati differential equality (RDE) is used. The RDEs direct dependence on the IQC multiplier leads to a nonlinear optimization problem, which can be solved efficiently. Using this, it is possible to explicitly respect the time varying dynamics of the space launcher in the worst case analysis. Furthermore, the incorporation of IQCs opens the analysis to a broader range of analyzable perturbations.

Cooperative range-based navigation using a beacon with circular motion installed on board the support platform

This work addresses the observability analysis for a cooperative range-based navigation system that uses a beacon with circular motion installed on board the support platform to find the location of an underwater vehicle. A nonlinear model is first obtained in order to describe the motion of the vehicle and the beacon, that is attached to the tip of a 1 DoF rotating arm (deployed from a stationary support platform or vessel). A state-space realization is obtained and augmented in order to convert the nonlinear system into a Linear Time Varying (LTV) system. Then, the observability analysis is performed for the new system considering its equivalence since the state trajectories of the LTV system have one-to-one correspondence to the state trajectories of the nonlinear system. We show that, when the vehicle moves in circles or in straight lines, it is possible to ensure observability by rotating the arm that has the beacon attached. This result is important from a theoretical point of view, and represents a new way of implementing range-based navigation systems, without relying on deviation of the vehicle’s mission or the need of another vehicle to ensure observability.

Regularized Adaptive Observer to Address Deficient Excitation

Adaptive observers are recursive algorithms for joint estimation of both state variables and unknown parameters. Usually some persistent excitation (PE) condition is required for the convergence of adaptive observers. However, in practice, it may happen that the PE condition is not satisfied, because the available sensor signals do not contain sufficient information for the considered recursive estimation problem, which is ill-posed. To remedy the lack of PE condition, inspired by typical methods for solving ill-posed inverse problems, this paper proposes a regularized adaptive observer for general linear time varying (LTV) systems. Two regularization terms are introduced in both state and parameter estimation recursions, in order to preserve the state-parameter decoupling transformation involved in the design of the adaptive observer. Like in typical ill-posed inverse problems, regularization implies an estimation bias, which can be reduced by using prior knowledge about the unknown parameters.

Interval State Estimator Design Using the Observability Matrix for Multiple Input Multiple Output Linear Time-Varying Discrete-Time Systems

In this paper, a novel numerical scheme to set-membership interval state estimator design is proposed for the multiple-input-multiple-output (MIMO) linear time-varying (LTV) discrete-time systems using systems observability matrix and its past input/output values. The proposed method is more simple and efficient. First, an interval state estimator is designed that will generate a tight interval vector for the real state vector in a guaranteed way by employing interval analysis and consistency techniques for the single-input-single-output (SISO) systems. The proposed interval state estimator technique is then extended easily to the MIMO systems. Secondly, the estimation errors dynamics bounds are computed a-priori to measurements for the unknown but bounded uncertainties. Finally, the convergence of the width of the interval state vector towards a known value in finite time is provided to prove the boundedness of the interval state vector and estimation error that further quantify the accuracy of the developed technique. The performance and comparison with already existing techniques are highlighted through numerical examples.

MV Benchmark for Networked Control Systems with Random Communication Delays

Abstract This work considers the control performance assessment (CPA) of networked control systems (NCSs) with random communication delays. Two kinds of communication delays are considered simultaneously: sensor-to-controller communication delay and controller-to-actuator communication delay A closed-form solution of the linear time-variant (LTV) minimum variance (MV) control law is derived for this class of systems. Correspondingly the MV benchmark value is calculated based on the achieved output with minimum variance. These results can be used for evaluating the potential improvement of control performance for NCSs with random communication delays. Further, the proposed results are illustrated based on a numerical example.