Linear Time-invariant Control Research Articles

Matrix eigenvalue spectrum assignment for linear control systems by static output feedback

For a linear time-invariant control system defined by a linear differential equation of the n-th order with a multidimensional state, input and output, we set and study the problem of arbitrary matrix eigenvalue spectrum assignment by linear static output feedback. We obtain controllability-like rank conditions that are necessary and sufficient for the problem of arbitrary matrix eigenvalue spectrum assignment and are sufficient for the problem of arbitrary eigenvalue spectrum assignment by linear static output feedback. It is proved that, in particular cases, when the system has block scalar matrix coefficients, these conditions can be relaxed. The obtained results generalize the known results for the corresponding problem by static state feedback and by static output feedback in the case of the one-dimensional equation. Examples are presented to illustrate the results.

On the observability of conformable linear time-invariant control systems

<p style='text-indent:20px;'>In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.</p>

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Non-Fragile H ∞ Order Reduction of LTI Controllers

The eigenvalue perturbation theorem is used to propose a convex fragility criterion with application to control system design. The criterion can be considered as a non-normality measure of the controller state-space matrix. Non-normality of a matrix is defined as its distance to the nearest real normal matrix within a convex normal subspace. Based on the criterion, an $\mathcal {H}_{\infty }$ method for the order reduction of linear time-invariant (LTI) controllers is developed which leads to non-fragile reduced order controllers.

ON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS

A problem of description of algebraic invariants for a linear control system that determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.

Stealth identification strategy for closed loop system structure

Many identification strategies for closed loop system structure assume that an open loop system is closed by a feedback mechanism, which contains a known and linear controller. This assumption means these identification strategies are feasible under some prior knowledge of feedback controller. To relax this assumption for some complex systems with unknown controller or nonlinear controller, a new stealth identification strategy is proposed to tackle the identification problem for closed loop system with unknown controller or nonlinear controller. Stealth identification modifies closed loop system, so that the new prediction error and inverse covariance matrix are all independent of the unknown controller or nonlinear controller. This independence simplifies the problem of estimating parameter vector and designing optimal input. The Robbins-Monro algorithm is applied to identify the unknown parameter vector in closed loop system with unknown controller, and the nonlinear controller is replaced by its equivalent linear time invariant controller. Some consideration about the linear approximation to nonlinear system are studied, and two linear approximation forms are constructed to approximate the nonlinear system. The effectiveness of our proposed stealth identification strategy is demonstrated through simulation example.

A new Solution of the Fractional-order Singular Linear Time-Invariant System

A new solution, the distributional solution of the fractional-order singular linear time-invariant control system (FOSLTIS) is studied in the paper. On the basis of proposing the definition of the Caputo fractional derivatives of Dirac function δ(t), the Laplace transform was obtained, then was derived. With regard to FOSLTIS, the system was decomposed into fast subsystem and slow subsystem by restricted equivalent transformation. Using Laplace transformation and taking into account, distributional solution for the fast subsystem was obtained. Two examples illustrated the correctness of the distributional solution of the FOSLTIS.

An Energy-Maximising Linear Time Invariant Controller (LiTe-Con) for Wave Energy Devices

A Linear Time Invariant (LTI) energy-maximising control strategy for Wave Energy Converters (WECs) is proposed in this paper. Using the fundamental requirement of impedance-matching, the controller is tuned to maximise the energy obtained under polychromatic wave excitation. Given the LTI nature of the proposed controller, the design and implementation procedure is significantly simpler than well-established energy-maximising controllers, including state-of-the-art numerical optimisation routines, which are predominant in this field. Additionally, a LTI constraint handling mechanism is provided. The effectiveness of both the LTI control strategy and the constraint handling mechanism are assessed using regular and irregular waves in unconstrained and constrained cases. The resulting performance is compared to those obtained using existing WEC optimal control strategies. Finally, the benefits, in terms of power production, for both the controller and the constraint handling mechanism are explicitly highlighted by means of an application case.

Constant Time-Headway Spacing Policy with Limited Communication Range for Discrete Time Platoon Systems

Abstract We present a study of the scaling properties of the interconnection of n agents (e.g. vehicles) through an r-lookahead network. These networks are considered as a possible implementation for vehicle platooning, although we do not make any assumptions on what the agents represent, and we assume them to be linear time invariant (LTI) discrete time systems, locally controlled by an LTI controller. In particular, we show that the r-lookahead topology gives rise to dynamics which can be studied from the roots of polynomials with transfer functions as their coefficients. Through numerical simulations, we study aspects relating the use of lookahead measurements and their effect on the value of a time headway constant needed for the scalability property known as string stability.

Optimal Design of Multiple Tuned Mass Dampers to Reduce Vibrations of a Ram-Type Structure With Varying Dynamics via a Control Theoretic Framework

Abstract This paper investigates the use of a linear time-invariant (LTI) control framework to optimally design multiple tuned mass dampers (TMDs) that minimize unwanted vibrations caused by exogenous disturbance forces to a ram-type structure with varying dynamic characteristics. A key challenge for the development of the LTI control framework is the reformulation of the TMDs’ design parameters, which consist of linear and nonlinear parameters as static feedback gains. This paper proposes the use of extra cascade control inputs to reformulate the optimization problem into an LTI control framework for the simultaneous optimization of linear (i.e., stiffness and damping) and nonlinear (i.e., location) parameters. A rigid planar system with multiple attached TMDs is developed as a mathematical model. It is reconstituted as an LTI framework by connecting a control input for the location parameter with control inputs for the stiffness and damping parameters. The model is then optimized using multi-model H∞ synthesis. A commercial gantry-type machining center is used to validate the proposed approach. Results from the simulation and experiment show that the optimized multiple TMDs systematically designed by this approach improve the system's dynamic stiffness by up to 83% and increase the allowable maximum depth of cut from 1 mm to 1.5 mm.

Finite-time stabilization of linear systems by bounded linear time-varying feedback

Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.

Attitude Control in Ascent Phase of Missile Considering Actuator Non-Linearity and Wind Disturbance

During the ascent phase of a missile, a challenging problem occurs that blocks the construction of a high-precision attitude control scheme, which directly affects accurate modeling including disturbances: non-linearities of an actuator, rapidly time-varying parameters, un-modeled dynamics, etc. In order to improve the control performance, an active disturbance rejection control (ADRC) scheme, considering non-linear dynamics of the actuator and wind disturbance during the ascent phase, is proposed in this paper. An expand state observer (ESO) is planned to estimate and compensate the actuator’s non-linear dynamics, flight model uncertainties, and wind disturbance. Therefore, the complex non-linear time-varying control problem is simplified into a linear time-invariant control problem. The pitch attitude control system is controlled by the cascade method and ADRC controllers are designed for actuator close loop and attitude control loop, respectively. The simulation results show that ADRC has strong robustness under different dead-zones and external disturbances of the actuator. On the other hand, ADRC can effectively suppress the external atmospheric disturbance. Compared with the traditional gain-scheduling control scheme, the ADRC scheme can significantly reduce the overloading of the system and shows remarkable performance for tracking as well as wind resistance.

Analysis and Design of Continuous-Time Multivariable Periodic Controllers: An $\mathcal {H}_{\infty }$/LTR-Based Approach

This article proposes a new high-frequency continuous-time two degrees of freedom (DOF) periodic control scheme to compensate linear time-invariant (LTI), square/nonsquare multivariable plants. The proposed scheme is more effective when the plant is nonminimum phase, or has more inputs than outputs. It is proved that, in averaged sense, the design of periodic controller can be recast as designing a 2-DOF LTI controller for an equivalent LTI plant. The conditions for which this equivalent plant becomes minimum phase are also obtained. When this plant is minimum phase and has no more inputs than outputs, the synthesis of the above-mentioned LTI controller is carried out using an linear matrix inequality-based $\mathcal {H}_{\infty }$ /loop transfer recovery (LTR) approach to achieve linear quadratic regulator (LQR) loop behavior asymptotically. The design method being based on LTR, the reference-input-to-output transfer function of the averaged compensated system becomes same as that using LQR. The periodic controller is shown to exhibit negligible ripples in the plant output, but $O(1)$ ripples in control input. Suitable examples are considered to show the superiority of the proposed control.

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

Automated design of structured gain-scheduling oil pressure controllers by modern synthesis methods

This paper presents a new systematic approach for the design of a gain-scheduling-control system for the oil pressure system of internal combustion engines. Therefore, optimal linear time invariant controllers are designed at given operating points and their characteristics are interpolated by an analytical function. An optimization algorithm finds optimal parameters for a given linear controller structure with a fixed order and two degrees of freedom. To solve the optimization problem, constraints in the $$\mathcal{H}_{\infty}$$-norm are given for six different transfer functions of the closed loop. The same restrictions are applied for all operating points. For the gain-scheduling-approach, no classical superimposition of the output variables of the linear controllers is selected, as it is often the case with gain-scheduling-controllers. Instead, the poles and zeros of the gain-scheduling-controller are functionally calculated from the linear control loops depending on the scheduling variable. Therefore, the resulting gain-scheduling-controller shows the set behaviour of the linear controllers independently of the original operating points. A control loop that is independent of the nonlinearity of the controlled system is more efficient than conventional oil pressure control systems that use too high target oil pressures for robustness reasons. This results in increased efficiency of the oil pressure system, which can make a significant contribution to fuel savings.

Necessary and Sufficient Conditions for Optimal Stabilization of Quasi-Linear Stochastic Systems

A wide class of admissible control strategies that guarantee the mean-square stabilization of a stochastic system is considered. Necessary and sufficient conditions for the opti-mality of a linear time-invariant controller are established. The difference between the stated problem and the optimal control problem on an infinite time interval is demonstrated. The obtained optimality conditions are illustrated by the example of stabilization of an artificial Earth satellite in the neighborhood of a circular orbit.

АДАПТИВНОЕ ОЦЕНИВАНИЕ И УПРАВЛЕНИЕ ДЛЯ СЛОЖНЫХ СИСТЕМ:ПРОШЛЫЕ ДОСТИЖЕНИЯ И НЕРЕШЕННЫЕ ЗАДАЧИ

Engineering systems’ adaptability to the unknown and unexpectedly changing equip-ment condition and environmental data is provided artificially by equipping the main loop of control with the adaptor loop. The paper characterizes the current status of re-search and practical studies in the field. As shown for estimation and control systems, the adaptor’s task is to ensure the efficient and conforming solution of the following three problems. Problem 1 is the quickest change point detection in the acting model relative to the previous model. Problem 2 is the unbiased and adequately quick identification of the acting model. Problem 3 is the computationally efficient modifying the feedback in the main loop of control. The paper characterizes the current status of research and prac-tical studies in the field over many publications and some author’s developments. Among the latter, there is the active principle of adaptation for model identification in the class of linear time-invariant control systems.

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

AbstractThe cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PIλDμ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.

Simultaneous stabilization of discrete-time delay systems and bounds on delay margin

This paper studies the delay margin problem of linear time-invariant (LTI) systems. For discrete-time systems, this problem can also be posed as one of simultaneous stabilization of multiple unstable systems with different lengths of delay. Our contribution is threefold. First, for general LTI plants with a number of distinct unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on the delay margin. We show that readily computable and explicit lower bounds can be found by computing the real eigenvalues of a constant matrix, and LTI controllers, potentially of a low order, can be synthesized to achieve the bounds based on the H∞ control theory. Second, we show that these results can be coherently extended to systems with time-varying delays, also resulting in bounds on the delay range and delay variation that ensure simultaneous stabilization. Finally, we investigate the delay margin problem in the context of PID control. For first-order unstable plants, we obtain bounds achievable by PID controllers. It is worth noting that unlike its continuous-time counterpart, the discrete-time delay margin problem is fundamentally more difficult, due to the intrinsic difficulty in simultaneously stabilizing several systems. Nevertheless, while previous works on the discrete-time delay margin led to largely negative results, the bounds developed in this paper provide instead conditions that guarantee the simultaneous stabilization of multiple delay plants, or ranges within which the delay plants can be robustly stabilized.

CAD-based Approach for Identification of UAVs

The paper presents an approach for the identification of the parameters of an Unmanned Aerial Vehicle (UAV), specifically a quadrotor using experiments in a CAD environment. A linearized control model is proposed to create a global model that operates in all operation range. Usually, in this type of systems a linear time invariant (LTI) controller is used, and then a lineal control model around each operating point is needed to the design of the controller. In this study, a linear control model suitable in each operating point is developed and identified. This serial of LTI control model can be combined and provides a global control model. Therefore, through this global control model can be designed a global controller suitable for all the operating range.