Design of H∞ Loop-Shaping Controller for LTI System With Input Saturation: Polytopic Gain Scheduled Approach

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
S. Patra ◽  
S. Sen ◽  
G. Ray
Keyword(s):  
Linear Time ◽  
Design Method ◽  
Input Saturation ◽  
Robust Stabilization ◽  
Linear Matrix ◽  
Linear Time Invariant ◽  
Loop Shaping ◽  
Lti System ◽  
Interpolation Procedure ◽  
The Stability

This paper demonstrates the design of H∞ loop-shaping controller for a linear time invariant (LTI) system with input saturation constraint. The design problem has been formulated in the four-block H∞ synthesis framework, which is equivalent to normalized coprime factor robust stabilization problem. The shaped plant is represented as a polytopic linear parameter varying (LPV) system while saturation nonlinearity is considered. For a polytopic model, the LTI H∞ loop-shaping controllers have been designed at each vertex of the polytope using linear matrix inequalities, and subsequently controllers are scheduled by adopting a certain interpolation procedure. The proposed controller ensures the stability and robust L2-performance of the closed-loop system due to vertex property of the polytopic LPV shaped plant. The effectiveness of the design method has been illustrated through a numerical example.

scholarly journals A multi-model approach to Saint-Venant equations: A stability study by LMIs

2012 ◽  
Vol 22 (3) ◽  
pp. 539-550 ◽  
Author(s):  
Valérie Dos Santos Martins ◽  
Mickael Rodrigues ◽  
Mamadou Diagne
Keyword(s):  
Linear Time ◽  
Dimensional Space ◽  
Stability Study ◽  
Linear Matrix ◽  
Model Concept ◽  
Infinite Dimensional ◽  
Linear Time Invariant ◽  
Saint Venant Equations ◽  
The Stability ◽  
Time Invariant

Abstract This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.

scholarly journals Active Unmatched Disturbance Cancellation and Estimation by State--Derivative Feedback for Plants Modeled as an LTI System

2018 ◽  
Vol 8 (2) ◽  
pp. 237-249
Author(s):  
Halil Ibrahim Basturk
Keyword(s):  
Linear Time ◽  
Position Information ◽  
Linear Time Invariant ◽  
Lti System ◽  
Disturbance Cancellation ◽  
Disturbance Model ◽  
The Stability ◽  
Linear Time Invariant System ◽  
Time Invariant ◽  
Two Degree Of Freedom

We design adaptive algorithms for both cancellation and estimation of unknown periodic disturbance, by feedback of state--derivatives ( i.e.,} without position information for mechanical systems) for the plants which are modeled as a linear time invariant system. We consider a series of unmatched unknown sinusoidal signals as the disturbance.The first step of the design consists of the parametrization of the disturbance model and the development of observer filters.The result obtained in this step allows us to use adaptive control techniques for the solution of the problem.In order to handle the unmatched condition, a backstepping technique is employed. Since the partial measurement of the virtual inputs is not available, we design a state observer and the estimates of these signals are used in the backstepping design.Finally, the stability of the equilibrium of the adaptive closed loop system with the convergence of states is proven.As a numerical example, a two-degree of freedom system is considered and the effectiveness of the algorithms are shown.

Synthesis Of Robust Control Systems With Dynamic Actuators And Sensors Using A Static Output Feedback Method

2020 ◽  
Author(s):  
Bruno Sereni ◽  
Roberto K. H. Galv˜ao ◽  
Edvaldo Assun¸c˜ao ◽  
Marcelo C. M. Teixeira
Keyword(s):  
Output Feedback ◽  
Linear Time ◽  
Robust Stabilization ◽  
Linear Matrix ◽  
Static Output Feedback ◽  
Sensors And Actuators ◽  
Actuator Dynamics ◽  
Linear Time Invariant ◽  
Incomplete System ◽  
Static Output

In this paper, we propose a strategy for the robust stabilization of uncertain linear time-invariant(LTI) systems considering sensors and actuators whose dynamics cannot be neglected. The control problem isaddressed by defining an augmented system encompassing the plant, sensor and actuator dynamics. The centralidea of the proposed method lies in the fact that the actual plant states, measured by sensors, are not available forfeedback, and thus, the problem can be regarded as a static output feedback (SOF) control design. Then, SOFgain matrices are computed through a two-stage method, based on linear matrix inequalities (LMIs). Intendingto illustrate the efficacy and explore the main features of the proposed technique, some computational examplesare presented in an application of the method for the design of a robust controller for the classic benchmarkproblem of the two-mass-spring problem. The examples cover the case of asymptotic stabilization of known anduncertain system model, in addition to decay rate inclusion and incomplete system state information.

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

2020 ◽  
Vol 143 (5) ◽  
Author(s):  
Mohammadreza Kamaldar ◽  
Jesse B. Hoagg
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Linear Time ◽  
Linear Time Invariant ◽  
Harmonic Control ◽  
Lti System ◽  
Higher Harmonic Control ◽  
Higher Harmonic ◽  
The Stability ◽  
Time Invariant

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.

scholarly journals A Method to Determine Oscillation Emergence Bifurcation in Time-Delayed LTI System with Single Lag

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yu Xiaodan ◽  
Jia Hongjie ◽  
Wang Chengshan ◽  
Jiang Yilang
Keyword(s):  
Linear Time ◽  
Oscillation Mode ◽  
Lambert W Function ◽  
Real Eigenvalue ◽  
Linear Time Invariant ◽  
Lti System ◽  
The Stability ◽  
Definition Of ◽  
Complex Phenomena ◽  
Time Invariant

One type of bifurcation named oscillation emergence bifurcation (OEB) found in time-delayed linear time invariant (abbr. LTI) systems is fully studied. The definition of OEB is initially put forward according to the eigenvalue variation. It is revealed that a real eigenvalue splits into a pair of conjugated complex eigenvalues when an OEB occurs, which means the number of the system eigenvalues will increase by one and a new oscillation mode will emerge. Next, a method to determine OEB bifurcation in the time-delayed LTI system with single lag is developed based on Lambert W function. A one-dimensional (1-dim) time-delayed system is firstly employed to explain the mechanism of OEB bifurcation. Then, methods to determine the OEB bifurcation in 1-dim, 2-dim, and high-dimension time-delayed LTI systems are derived. Finally, simulation results validate the correctness and effectiveness of the presented method. Since OEB bifurcation occurs with a new oscillation mode emerging, work of this paper is useful to explore the complex phenomena and the stability of time-delayed dynamic systems.

scholarly journals Suboptimal Disturbance Observer Design Using All Stabilizing Q Filter for Precise Tracking Control

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1434 ◽  
Author(s):  
Wonhee Kim ◽  
Sangmin Suh
Keyword(s):  
Design Method ◽  
Industrial Applications ◽  
Point Of View ◽  
Observer Design ◽  
Linear Matrix ◽  
External Disturbances ◽  
Closed Loop Systems ◽  
Control Target ◽  
The Stability ◽  
Unified Index

For several decades, disturbance observers (DOs) have been widely utilized to enhance tracking performance by reducing external disturbances in different industrial applications. However, although a DO is a verified control structure, a conventional DO does not guarantee stability. This paper proposes a stability-guaranteed design method, while maintaining the DO structure. The proposed design method uses a linear matrix inequality (LMI)-based H∞ control because the LMI-based control guarantees the stability of closed loop systems. However, applying the DO design to the LMI framework is not trivial because there are two control targets, whereas the standard LMI stabilizes a single control target. In this study, the problem is first resolved by building a single fictitious model because the two models are serial and can be considered as a single model from the Q-filter point of view. Using the proposed design framework, all-stabilizing Q filters are calculated. In addition, for the stability and robustness of the DO, two metrics are proposed to quantify the stability and robustness and combined into a single unified index to satisfy both metrics. Based on an application example, it is verified that the proposed method is effective, with a performance improvement of 10.8%.

Coupled Stability of Multiport Systems—Theory and Experiments

1994 ◽  
Vol 116 (3) ◽  
pp. 419-428 ◽  
Author(s):  
J. E. Colgate
Keyword(s):  
Stability Property ◽  
Force Feedback ◽  
Linear Time ◽  
Experimental Studies ◽  
Sufficient Conditions ◽  
Direct Drive ◽  
Property A ◽  
Linear Time Invariant ◽  
The Stability ◽  
Necessary And Sufficient

This paper presents both theoretical and experimental studies of the stability of dynamic interaction between a feedback controlled manipulator and a passive environment. Necessary and sufficient conditions for “coupled stability”—the stability of a linear, time-invariant n-port (e.g., a robot, linearized about an operating point) coupled to a passive, but otherwise arbitrary, environment—are presented. The problem of assessing coupled stability for a physical system (continuous time) with a discrete time controller is then addressed. It is demonstrated that such a system may exhibit the coupled stability property; however, analytical, or even inexpensive numerical conditions are difficult to obtain. Therefore, an approximate condition, based on easily computed multivariable Nyquist plots, is developed. This condition is used to analyze two controllers implemented on a two-link, direct drive robot. An impedance controller demonstrates that a feedback controlled manipulator may satisfy the coupled stability property. A LQG/LTR controller illustrates specific consequences of failure to meet the coupled stability criterion; it also illustrates how coupled instability may arise in the absence of force feedback. Two experimental procedures—measurement of endpoint admittance and interaction with springs and masses—are introduced and used to evaluate the above controllers. Theoretical and experimental results are compared.

Parameter Uncertainty Modeling Using the Multidimensional Principal Curves

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
M. Sepasi ◽  
F. Sassani ◽  
R. Nagamune
Keyword(s):  
Transfer Functions ◽  
Linear Time ◽  
Optimization Technique ◽  
Parametric Uncertainty ◽  
Uncertainty Modeling ◽  
Principal Component ◽  
Linear Matrix ◽  
Linear Time Invariant ◽  
Uncertain Variables ◽  
Uncertainty Model

This paper proposes a technique to model uncertainties associated with linear time-invariant systems. It is assumed that the uncertainties are only due to parametric variations caused by independent uncertain variables. By assuming that a set of a finite number of rational transfer functions of a fixed order is given, as well as the number of independent uncertain variables that affect the parametric uncertainties, the proposed technique seeks an optimal parametric uncertainty model as a function of uncertain variables that explains the set of transfer functions. Finding such an optimal parametric uncertainty model is formulated as a noncovex optimization problem, which is then solved by a combination of a linear matrix inequality and a nonlinear optimization technique. To find an initial condition for solving this nonconvex problem, the nonlinear principal component analysis based on the multidimensional principal curve is employed. The effectiveness of the proposed technique is verified through both illustrative and practical examples.

Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

2006 ◽  
Vol 129 (3) ◽  
pp. 245-251 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Time Delays ◽  
Linear Time ◽  
Characteristic Root ◽  
Invariance Property ◽  
Multiple Time ◽  
Delayed Systems ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Stability Robustness ◽  
The Stability

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernel” surfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

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