Linear Control Systems Research Articles

Minimum-Time Control of a Linear System With Input Saturation: A Practical Approach

Abstract A simple algorithm is presented to find near-minimum-time control inputs for a controllable continuous-time linear time-invariant system with any number of states and inputs. The well-known digital deadbeat control algorithm is modified to satisfy input constraints, and the design issue is the selection of sampling period to minimize the time required to reach the desired final state. During each sampling period, the input required to place all the poles at the origin of z-plane (deadbeat control) via state feedback is computed. If the infinity norm of the deadbeat input exceeds the input constraint, the input is found by placing the poles as close to the origin as possible. A sufficient condition for the system stability is developed. Numerical examples illustrate that this algorithm can lead to true time-optimal control or near time-optimal control.

Control of the Motion of an Inverted Spherical Pendulum on a Moving Base. Hybrid Impact Approach

A new hybrid method for the construction of control actions of a linear control system with constant coefficients is considered in this paper. It is assumed in this paper that a part of the discussed system meets some conditions. Some states of the main system are considered to be control actions for a subsystem for which and LQR stabilizer is acquired. Then, those control actions of the subsystem are used to construct the control actions for the main system. In the problem of controlling the motion of a complex linear system of an inverted spherical pendulum on a moving base, a new approach to the construction of control actions (hybrid action method) was used. It is assumed that a component of the complex system under discussion satisfies certain conditions. The inertial forces at the center of mass of the base of the composite system are considered to be the controlling influences on the inversion of the pendulum, for which the LQR stabilizer was purchased. The determined internal control actions on the inverted pendulum are then used to construct external control actions on the base of the composite system. In the end, a numerical analysis was carried out.

Dynamic inversion and optimal tracking control on the ball-plate system based on a linearized nonholonomic multibody model

This paper addresses the optimal control of the ball-plate system, a well-known nonholonomic system in the context of nonprehensile manipulation, using a multibody dynamics approach. The trajectory tracking control of a steady-state circular motion of the ball on the plate, for any radius and potentially off-centric with respect to the plate’s pivoting point, is achieved by designing a Linear-Quadratic Regulator. A spatial multibody model of the ball-plate system is considered. A key contribution is the analytical computation of the circular steady motion of the ball by dynamic inversion, including the control actions to achieve this reference solution. This enables the analytical computation of the linearized equations along this reference motion, resulting in a periodic linear time-varying (LTV) system, and the application of linear controllability criteria for LTV systems. A controllable linear system, involving the Cartesian coordinates of the contact point and the yaw angle of the sphere, is obtained using a convenient coordinate partition in the linearization. Compared to existing results on the same problem, closed-loop stability about the desired trajectory is achieved for any radius of the circular trajectory.

Static Output Energy-to-Peak Control for Semi-Markov Jump Linear Systems With Phase-Type Distributions and Hidden Modes

Static Output Energy-to-Peak Control for Semi-Markov Jump Linear Systems With Phase-Type Distributions and Hidden Modes

Adaptive Linear Quadratic Control for Stochastic Discrete-Time Linear Systems With Unmeasurable Multiplicative and Additive Noises

Adaptive Linear Quadratic Control for Stochastic Discrete-Time Linear Systems With Unmeasurable Multiplicative and Additive Noises

Hierarchical Stackelberg Control for a Two-Stroke Linear System in Population Dynamics

We study a unique hierarchical control problem for a two-stoke linear system, adjoint to an age and space dependent population dynamics problem. Using Stackelberg's method, we introduce two levels of control: a boundary control to achieve optimal flow regulation and a distributed control for null controllability. Our approach employs Carleman inequalities to address non-homogeneous Dirichlet boundary conditions, leading to new insights in controlling invasive species populations. These results highlight the applicability of hierarchical controls in ecological systems, providing a robust framework for future studies in control theory and population dynamics.

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.

Covariance Control for Uncrewed Aircraft Systems Under Correlated Uncertainty

Uncrewed aircraft systems and advanced air mobility are associated with uncertain conditions, such as wind prediction errors, and they may deviate from dedicated airspaces referred to as corridors. However, further studies on these aspects are required. This study focuses on the covariance control for a stochastic discrete-time linear system under correlated uncertainties. A covariance control problem with chance constraints can be formulated as a convex-programming problem. Numerical simulations of uncrewed aircraft systems path planning can be used to compare two control laws: covariance control laws that consider correlated and noncorrelated noise. Numerical simulations were conducted to verify that the covariance control law considering correlated noise generates a trajectory that appropriately satisfies the constraints. Consequently, the effect of considering correlated noise in the covariance control law was clarified. In addition, covariance control has been demonstrated to generate appropriate trajectories for various state constraints. Moreover, numerical simulations indicated that covariance control facilitated the control of uncertainty, and the effectiveness of the covariance control was evaluated. The insights gained from the results of this study can inform enhanced uncertainty management for improving reliability and safety in real-world applications of uncrewed aircraft systems and advanced air mobility.

Novel Droop-Based Techniques for Dynamic Performance Improvement in a Linear Active Disturbance Rejection Controlled-Dual Active Bridge for Fast Battery Charging of Electric Vehicles

Electric vehicles (EVs) are rapidly replacing fossil-fuel-powered vehicles, creating a need for a fast-charging infrastructure that is crucial for their widespread adoption. This research addresses this challenge by improving the control of dual active bridge converters, a popular choice for high-power EV charging stations. A critical issue in EV battery charging is the smooth transition between charging stages (constant current and constant voltage) which can disrupt converter performance. This work proposes a novel feedforward control method using a combination of droop-based techniques combined with a sophisticated linear active disturbance rejection control system applied to a single-phase shift-modulated dual active bridge. This combination ensures a seamless transition between charging stages and enhances the robustness of the system against fluctuations in both input voltage and load. Numerical simulations using MATLAB/Simulink R2024a demonstrated that this approach not only enables smooth charging but also reduces the peak input converter current, allowing for the use of lower-rated components in the converter design. This translates to potentially lower costs for building these essential charging stations and faster adoption of EVs.

Dispersiveness, higher stability and controllability of linear control systems on the Heisenberg group

Dispersiveness, higher stability and controllability of linear control systems on the Heisenberg group

Finite-Timeless Local Observability for Linear Control Systems on Lie Groups

In this work, we introduce finite-timeless observability for linear control systems on Lie groups. Observability problem is to distinguish the elements of the state space by looking the images of the solutions passing through those elements in the output space. For this aim, we define almost indistinguishability and show that it is an equivalence relation. We consider linear control systems on Lie groups, where the output function is a Lie group homomorphism and study properties of the image of almost indistinguishable points from the neutral element. Thus, we relate local observability of this kind of systems to its finite-timeless local observability.

Controllability and minimum energy control problem of two-dimensional continuous-time fractional linear systems

The effectiveness of this paper lies in a new formulation and solution of the minimum energy control problem for the fractional two-dimensional systems described by the Fornasini–Marchesini first models. The problem of controllability and minimum energy control of a general fractional two-dimensional Fornasini–Marchesini model is newly considered by the authors in this work. Necessary and sufficient controllability conditions are then established using the Gramian matrix. To carry out this study, a new test was described and used in order to give the existence conditions of the solution to the minimum energy control problem and the procedures for calculating an input minimising the given performance index. The usefulness and the precision of the proposed procedure is demonstrated by a numerical examples, Darboux equation and metal rolling process model as a practical application.

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Reinforcement Learning for Finite-Horizon H∞ Tracking Control of Unknown Discrete Linear Time-Varying System

Reinforcement Learning for Finite-Horizon H∞ Tracking Control of Unknown Discrete Linear Time-Varying System

A robust iterative learning control for linear system with variable initial state and trail length

AbstractTo address the variable initial state and trail length this paper first presents a robust PD‐type open‐closed‐loop iterative learning control (ILC) law for a multiple‐input‐multiple‐output (MIMO) linear discrete‐time system. It is demonstrated that the convergence condition is dependent on the PD‐type feed‐forward learning gains, while an appropriate feedback learning gain can improve the ILC convergence performance. As a special case of PD‐type open‐closed‐loop ILC law, P‐type and D‐type open‐closed‐loop ILC laws are deduced. The three developed ILC laws ensure that as the number of iterations approaches infinity, the expectation of ILC tracking error will be constrained within a limited range, where the boundary is proportional to the initial state variation. Through a numerical simulation, the effectiveness of the proposed ILC laws is illustrated.

Controllability and Optimal Fast Operation of Nonlinear Systems

A new method for solving the problem of controllability and optimal transient behavior of nonlinear systems subject to boundary conditions and constraints on control values was proposed. Unlike existing methods, this new approach is based on constructing a general solution of the integral equation for a linear controlled system, followed by transforming the original problem into a special initial optimal control problem. We propose a new method for studying the global asymptotic stability of dynamical systems with a cylindrical phase space with a countable equilibrium position based on a non-singular transformation of the equation of motion and estimation of improper integrals along the solution of the system. Conditions for global asymptotic stability were obtained without involving any periodic Lyapunov function, as well as the frequency theorem. The effectiveness of the proposed method is shown with an example.

A linear dissipativity approach to incremental input-to-state stability for a class of positive Lur'e systems

Incremental stability properties are considered for certain systems of forced, nonlinear differential equations with a particular positivity structure. An incremental stability estimate is derived for pairs of input/state/output trajectories of the Lur'e systems under consideration, from which a number of consequences are obtained, including the incremental exponential input-to-state stability property and certain input–output stability concepts with linear gain. Our results show that an incremental version of the real Aizerman conjecture is true for positive Lur'e systems when an incremental gain condition is imposed on the nonlinear term, as we describe. Our argumentation is underpinned by linear dissipativity theory – a property of positive linear control systems.

On arbitrary matrix coefficient assignment for the characteristic matrix polynomial of block matrix linear control systems

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

Optimal quantized feedback control for linear quandratic Gaussian systems with input delay

AbstractThis paper is concerned with the optimal quantized feedback linear quadratic Gaussian (LQG) control problem for a discrete‐time stochastic system with input delay as well as the measurements to be quantized before transmitted to the controller. In this scenario, the system is presented with several choices of quantizers, along with the cost of using each quantizer. The objective is to jointly select the quantizers and synthesize the controller to maintain an optimal balance between control performance and quantization cost. It is shown that this problem can be decoupled into two optimization problems when the innovation signal is quantized instead of state: one for optimal controller synthesis and the other for optimal quantizer selection. More specifically, a necessary and sufficient condition is derived for the optimal control problem based on Pontryagin's maximum principle. On the other hand, the optimal quantizer selection policy is established by dealing with a certain Markov decision process (MDP).

Linear Active Disturbance Rejection Control System for the Travel Speed of an Electric Reel Sprinkling Irrigation Machine

The uniformity of the travel speed of electric reel sprinkling irrigation machines is a key factor affecting irrigation quality. However, conventional PID control is susceptible to sudden disturbances under complex farmland conditions, leading to reduced speed uniformity. To enhance the robustness of the control system, it is necessary to investigate new disturbance rejection control algorithms and their effects. Therefore, a kinematic model of the reel sprinkling irrigation machine and a brushless DC (BLDC) motor model were established, and a linear active disturbance rejection control (LADRC) strategy based on improved particle swarm optimization (IPSO) was proposed. The simulation results show that under variable speed conditions, the system exhibits no overshoot, with an adjustment time of 0.064 s; under variable load conditions, the speed vibration amplitude is less than 0.3%. The field test results indicate that at travel speeds of 10 m/h and 30 m/h, the maximum absolute deviation rate under IPSO-LADRC control is reduced by 27.07% and 13.98%, respectively, compared to PID control. The control strategy based on IPSO-LADRC effectively improves the control accuracy and robustness under complex farmland conditions, providing a reference for enhancing the control performance of other electric agricultural machinery.