Linear Time-varying Research Articles

Finite horizon analysis of autolanded aircraft in final approach under crosswind

The paper presents a worst-case touchdown performance analysis of auto-landed aircraft under complex wind disturbance. It takes advantage of the fact that the approaching aircraft effectively follows a predefined trajectory provided by the instrument landing system. Thus, the aircraft dynamics’ linearization along the approach trajectory results in a finite horizon linear time-varying (LTV) representation. This naturally allows to include altitude triggered control law changes and changes in the flight dynamics, e.g., due to ground effect, in the analysis. To cover a broad range of environmental and aircraft parameter combinations in the worst-case analysis, a time-varying trajectory uncertainty description is introduced. The uncertainty’s input/output behavior is covered by integral quadratic constraints. Thus, recent advances on the worst-case gain analysis of finite horizon LTV systems can be used. The corresponding analysis condition is based on a parameterized Riccati differential equation’s solvability, which leads to a readily solvable nonlinear optimization problem. Applying the robust LTV framework, worst-cases for common touchdown criteria, such as vertical touchdown velocity, are calculated. These worst-cases cover the influence of complex wind fields and a large aircraft and environmental parameter set. The results are evaluated against corresponding Monte Carlo simulation on the original high fidelity, industry-sized nonlinear aircraft model.

Observability analysis for a cooperative range-based navigation system that uses a rotating single beacon

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. First, the observability analysis is carried out without taking into account the ocean current that affects the vehicle. To that end, an augmented state-space realization is obtained in order to convert the nonlinear system into a linear time varying (LTV) system. Then, the observability analysis is performed considering constraints on the initial conditions due to the state transformation. Based on these results, a second observability analysis is carried out taking into account ocean current in the model. We show that, by knowing the initial beacon’s position, the system is observable when the vehicle is moving in a trim trajectory with and without ocean currents. The obtained results include information about the initial conditions of the beacon and the ocean currents in the observability analysis. 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.

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.

Time‐dependent Luenberger‐type interval observer design for uncertain time‐varying systems

AbstractThis article investigates the state interval observation problem for a class of continuous and discrete‐time linear time‐varying (LTV) systems with bounded uncertainties separately. First, by utilizing both the LTV transformation and the solutions to a type of time‐varying fully‐actuated Sylvester equations, a time‐dependent standard interval observer (IO) is established in a parametric form with some design degrees of freedom. Second, a new time‐dependent structure of Luenberger‐type IOs is proposed to relax the limitation of the LTV transformation‐based design method. Further, its sufficient existence conditions are given in virtue of positive system theory. Meanwhile, a parametric design technique of such an IO for uncertain LTV systems is also raised under rank conditions. Third, the obtained results are extended to the discrete‐time systems. Finally, the correctness and benefits of the proposed methods are verified using some numerical examples.

Four MPC implementations compared on the Quadruple Tank Process Benchmark: pros and cons of neural MPC*

This study aims to aid understanding of Model Predictive Control (MPC) alternatives through comparing most interesting MPC implementations. This comparison will be performed intrinsically and illustrated using the four-tank benchmark, widely studied by academics taking care of industrial perspectives. Although MPC provides advanced control solutions for a wide class of dynamical systems, challenges arise in managing the compromise between accuracy, computational cost and resilience, depending on the type of model used. In this study, linear, linear time-varying and non-linear MPCs are compared to MPC that uses a neural network based predictive model identified from data. The tuning and implementation methods considered are discussed, and accurate simulation results provided and analyzed. Precisely, the performance of each method (linear, linear time-varying, non-linear MPC) are compared to the neural MPC. Pros and cons of neural MPC are highlighted.

Contributions to Output Controllability for Linear Time Varying Systems

The purpose of this letter is to provide some contributions to one notion of Output controllability for Linear Time Varying (LTV) systems which is the Complete State to Output Controllability (CSOC), notion introduced in the 60s by Sarachik and Kranc. More precisely, we consider LTV systems with direct transmission of the input to the output and establish criteria to ensure the CSOC in finite time of these systems. We also give, under the assumption of CSOC in finite time, an explicit expression of a continuous control built by means of a Gramian matrix.

Probabilistic Robustness Analysis of Uncertain LTV Systems in Linear Fractional Representation

This letter quantifies the effect of random model uncertainty on finite horizon linear time-varying (LTV) systems. Mean and standard deviation field are approximated with high accuracy and efficiency by a Hilbert space technique called polynomial chaos expansion (PCE). The deterministic expansion coefficients of the generalized Fourier series are determined via orthogonal projection, also known as Galerkin projection. We propose the projection of uncertain systems in linear fractional representation (LFR), which can have computational benefits. The technique is benchmarked on a two-link robotic manipulator.

Adaptive observer design for time-varying systems with relaxed excitation conditions*

In this paper we are interested in the problem of adaptive state observation of linear time-varying (LTV) systems where the system and the input matrices depend on unknown time-varying parameters which described unknown LTI dynamics with unknown initial conditions. Our main contribution is to propose a globally convergent state observer that requires only a weak excitation assumption on the system.

Path Planning on Large Curvature Roads Using Driver-Vehicle-Road System Based on the Kinematic Vehicle Model

Path planning is a critical part for improving the driving safety and driver comfort of autonomous vehicles (AVs), especially in complex maneuvering conditions. In addition, different drivers have different preferences for AVs, thus, how to provide personalized trajectories for different drivers is a vital issue for AVs. The collision-free path planning problem in conditions with large road curvatures is investigated in this paper, with the consideration of environmental safety constraints, drivers’ comfort, vehicle actuator constraints, etc. Firstly, a Driver-Vehicle-Road (DVR) system is established based on the combination of the kinematic vehicle model and the two-point visual preview driver model, such that the driver's individual handling characteristics can be considered in the controller. The kinematic vehicle model is modified to have the similar understeering characteristics with those of the nonlinear full car models, and then the proposed DVR system can satisfy different groups of drivers and cars. Secondly, for environmental constraints, a new artificial potential field (APF) method is proposed, which can form a banana-shaped 3-D dangerous imaginary mountain and a lane boundary cliff suitable for arbitrary curvature roads to generate a collision-free evasive path. Finally, the Linear-Time-Varying (LTV) model predictive control (MPC) method is adopted to design the path planner. The CarSim-Simulink joint simulation illustrates that with the proposed planner, the host vehicle is capable of avoiding obstacles with a safer and more comfortable maneuver on large curvature roads. And the proposed path planner can provide individually safe trajectories for different drivers with good maneuverability.

Impact-Angle and Terminal-Maneuvering-Acceleration Constrained Guidance against Maneuvering Target

A new, highly constrained guidance law is proposed against a maneuvering target while satisfying both impact angle and terminal acceleration constraints. Here, the impact angle constraint is addressed by solving an optimal guidance problem in which the target’s maneuvering acceleration is time-varying. To deal with the terminal acceleration constraint, the closed-form solutions of the new guidance are needed. Thus, a novel engagement system based on the guidance considering the target maneuvers is put forward by choosing two angles associated with the relative velocity vector and line of sight (LOS) as the state variables, and then the system is linearized using small angle assumptions, which yields a special linear time-varying (LTV) system that can be solved analytically by the spectral-decomposition-based method. For the general case where the closing speed, which is the speed of approach of the missile and target, is allowed to change with time arbitrarily, the solutions obtained are semi-analytical. In particular, when the closing speed changes linearly with time, the completely closed-form solutions are derived successfully. By analyzing the generalized solutions, the stability domain of the guidance coefficients is obtained, in which the maneuvering acceleration of the missile can converge to zero finally. Here, the key to investigating the stability domain is to find the limits of some complicated integral terms of the generalized solutions by skillfully using the squeeze theorem. The advantages of the proposed guidance are demonstrated by conducting trajectory simulations.

Optimal trajectory design accounting for the stabilization of linear time-varying error dynamics

This study is dedicated to the development of a direct optimal control-based algorithm for trajectory optimization problems that accounts for the closed-loop stability of the trajectory tracking error dynamics already during the optimization. Consequently, the trajectory is designed such that the Linear Time-Varying (LTV) dynamic system, describing the controller’s error dynamics, is stable, while additionally the desired optimality criterion is optimized and all enforced constraints on the trajectory are fulfilled. This is achieved by means of a Lyapunov stability analysis of the LTV dynamics within the optimization problem using a time-dependent, quadratic Lyapunov function candidate. Special care is taken with regard to ensuring the correct definiteness of the ensuing matrices within the Lyapunov stability analysis, specifically considering a numerically stable formulation of these in the numerical optimization. The developed algorithm is applied to a trajectory design problem for which the LTV system is part of the path-following error dynamics, which is required to be stable. The main benefit of the proposed scheme in this context is that the designed trajectory trades-off the required stability and robustness properties of the LTV dynamics with the optimality of the trajectory already at the design phase and thus, does not produce unstable optimal trajectories the system must follow in the real application.

Discrete-time distributed Kalman filter design for networks of interconnected systems with linear time-varying dynamics

This paper addresses the problem of designing distributed state estimation solutions for a network of interconnected systems modelled by linear time-varying (LTV) dynamics in a discrete-time framework. The problem is formulated as a classical optimal estimation problem, for the global system, subject to a given sparsity constraint on the filter gain, which reflects the distributed nature of the network. Two methods are presented, both of them able to compute a sequence of well-performing stabilising gains. Moreover, both methods are validated by resorting to simulations of: (i) a randomly generated synthetic LTV system; and (ii) a large-scale nonlinear network of interconnected tanks. One of the proposed methods relies on a computationally efficient solution, thus it is computed very rapidly. The other achieves better performance, but it is computationally more expensive and requires that a window of the future dynamics of the system is known. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that stabilise the estimation error dynamics. Both algorithms are scalable, being adequate for implementation in large-scale networks.

Time‐varying robustness analysis of launch vehicles under thrust perturbations

AbstractThis article presents a robustness analysis for launch vehicles during atmospheric ascent. It is based on recent advances in the worst‐case analysis of uncertain, finite horizon linear time‐varying (LTV) systems. Integral quadratic constraints are used to represent the uncertainties leading to a worst‐case gain condition based on a parameterized Riccati differential equation (RDE). This framework readily covers certain parametric uncertainties, for example, aerodynamic uncertainties. However, the effects of thrust perturbations are inherently harder to address. They directly result in mass and balance uncertainties, and a deviation from the planned trajectory. The latter amounts to a shift from the nominal/reference trajectory that has to be accounted for. Since thrust is proportional to the mass change of the launch vehicle, the thrust uncertainty is modeled as an external disturbance acting on the system. In order to cover the deviation from the reference trajectory, a dynamic uncertainty model is added in the analysis. A worst‐case aerodynamic loads and lateral drift analysis under wind disturbances is performed. The LTV results are validated against Monte Carlo simulations of the space launcher's nonlinear model.

Energy‐efficient cabin climate control of electric vehicles using linear time‐varying model predictive control

AbstractA cabin climate control system, often referred to as a heating, ventilation, and air conditioning (HVAC) system, is one of the largest auxiliary loads of an electric vehicle (EV), and the real‐time optimal control of HVAC brings a significant energy‐saving potential. In this article, a linear‐time‐varying (LTV) model predictive control (MPC)‐based approach is presented for energy‐efficient cabin climate control of EVs. A modification is made to the cost function in the considered MPC problem to simplify the Hessian matrix in utilizing quadratic programming for real‐time computation. A rigorous parametric study is conducted to determine optimal weighting factors that work robustly under various operating conditions. Then, the performance of the proposed LTV‐MPC controller is compared against a rule‐based (RB) controller and a nonlinear economic MPC (NEMPC) benchmark. Compared with the RB controller benchmark, the LTV‐MPC reaches the target cabin temperature at least 69 s faster with 3.2% to 15% less HVAC system energy consumption, and the averaged cabin temperature difference is 0.7°C at most. Compared with the NEMPC, the LTV‐MPC controller can achieve comparable performance in temperature regulation and energy consumption with fast computation time: the maximum differences in temperature and energy consumption are 0.4°C and 2.6%, respectively, and the computational time is reduced 72.4% on average with the LTV‐MPC.

On Symmetric Solutions to Linear Matrix Time-Varying Differential Equations

In this paper, we discuss when the solution to the initial value problem for a linear matrix time-varying differential equation is symmetric on a given interval. By symmetry, we mean that the solution does not change when transposed. Throughout the paper, we assume that the equation has coefficients of finite order of smoothness. We demonstrate that, in order to verify whether the solution to the initial value problem is symmetric on a given interval, it can be useful to construct two matrix sequences associated to the equation. Using these sequences, we prove a sufficient condition for the solution symmetry on a given interval. Assuming that the initial value problem for a linear matrix time-varying differential equation satisfies this condition, we derive a formula for a symmetric solution to this problem.

Finite‐time output regulation by bounded linear time‐varying controls with applications to the satellite formation flying

AbstractThis article studies the finite‐time output regulation problem for linear time‐invariant continuous‐time systems. By using the solution to a parametric Lyapunov equation (PLE) and regulator equations, three bounded linear time‐varying (LTV) state controllers composed of the LTV feedback gain and the LTV feedforward gain are designed, such that (prescribed) finite‐time output regulation is solved. As a further result, a linear LTV observer‐based controller is also designed. The most significant advantages of this article are that the system under consideration is more general and the output regulation problem is achieved within a user‐chosen regulation time. Finally, the developed LTV state controllers are utilized to the design of the satellite formation flying control system and simulation results verify the effectiveness of the proposed approaches.

Necessary and Sufficient Conditions for Asymptotic Decoupling of Stable Modes in LTV Systems

For a linear time varying (LTV) system containing stable and unstable modes, this article gives the necessary and sufficient conditions for asymptotic decoupling of stable modes. Establishment of the conditions relies on newly developed techniques for analyzing matrix exponential and state transition matrix. The research exploits fundamental properties of LTV systems and advances the knowledge base of linear control theory. The results can find a direct application in the autonomous synchronization problem for heterogeneous multiagent systems. The research methodology is based on rigorous theoretical proofs and numerical verification.

Adaptive boundary sine cosine optimizer with population reduction for robustness analysis of finite time horizon systems

This paper proposes an adaptive boundary sine cosine optimizer with population reduction. It is designed specifically to calculate an upper bound on the worst-case gain of a known finite horizon Linear Time Varying (LTV) system and a perturbation. The input/output behavior of the perturbation is described by a time domain Integral Quadratic Constraint (IQC). The analysis condition is formulated as a parametric Riccati differential equation which depends on the IQC representation. A nonlinear optimization problem is posed that minimizes the upper bound on the worst-case gain over the IQC parameterization. For industrial size applications like a space launcher, the number of decision variables can grow arbitrarily large depending on the number of considered perturbations as well as the type and representation of the IQC. This is aggravated by nonlinear constraints on the decision variables imposed by the Riccati differential equation making it challenging to solve. Several established Meta-Heuristics (MHs) along with the proposed algorithm are applied to an industry size worst case analysis of a space launcher during its atmospheric ascend. Their respective performances are evaluated to emphasize the advantages of the developed optimizer. This work builds the foundation of applying MHs to IQC based robustness analysis of finite horizon LTV systems.

Discrete‐time decentralized linear quadratic control for linear time‐varying systems

AbstractThis article addresses the problem of designing a decentralized control solution for a network of agents modeled by linear time‐varying (LTV) dynamics, in a discrete‐time framework. A general scheme is proposed, in which the problem is formulated as a classical linear quadratic regulator problem, for the global system, subject to a given sparsity constraint on the gain, which reflects the decentralized nature of the network. A method able to compute a sequence of well‐performing stabilizing regulator gains is presented and validated resorting to simulations of two randomly generated LTV systems, one stable and the other unstable. Moreover, a tracking solution is developed, building on the solution to the regulator problem. Both methods rely on a closed‐form solution, thus they can be computed very rapidly. Similarly to the centralized solution, both the presented methods require that a window of the future system dynamics is known. Both methods are validated resorting to simulations of: (i) a nonlinear network of four interconnected tanks; and (ii) a large‐scale nonlinear network of interconnected tanks. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well‐performing gains that track the desired output. Finally, both algorithms are scalable, being adequate for implementation in large‐scale networks.

Double-Conversion, Noise-Cancelling Receivers Using Modulated LNTAs and Double-Layer Passive Mixers for Concurrent Signal Reception With Tuned RF Interface

A double-conversion, noise-cancelling receiver architecture is presented that consists of a double-layer mixer-first branch and two quadrature-modulated LNTA branches. The two layers of passive mixing in the mixer-first branch up-convert the low-pass, baseband impedance and create concurrent, narrowband impedance matching at (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LO</sub> ±F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IF</sub> ), as well as concurrently receive signals around these two RF carriers while rejecting spurious responses without using any IF filters. To improve the noise performance, quadrature-modulated LNTA branches are incorporated to allow frequency-translational, noise cancellation for better receiver sensitivity. Double conversion is achieved in the LNTA branches by periodically varying the LNTA transconductance and current-mode, passive mixing. A generalized, linear time-varying (LTV) analysis of the receiver is presented and verified with behavioral-model simulation results.