Optimal Mini Segway Control Using Non-Minimum Phase Sample and Hold Input

2019 ◽  
Author(s):  
Yingxu Wang ◽  
Guoming G. Zhu
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
System Performance ◽  
Low Cost ◽  
Sampling Period ◽  
High Gain ◽  
Minimum Phase ◽  
Discrete Time System ◽  
Loop Control ◽  
Sample And Hold

Abstract Our early work shows the reduction of feasible sampling period when sample and hold inputs (SHI) are used to convert a continuous-time non-minimum phase (NMP) system to a discrete-time minimum phase (MP) system, comparing to conventional zero-order hold. Consequently, high-gain discrete-time controllers can be designed and used to improve continuous-time NMP system performance since the resulting discrete-time system is MP. This paper demonstrates the performance improvements of a mini Segway robot through experiments utilizing a dual-loop control architecture. An inner-loop continuous-time controller stabilizes the mini Segway robot and the outer-loop discrete-time controller, designed based on the discrete-time MP system, is used to improve the overall system performance. Experimental results show that the mini Segway cart oscillation magnitudes are significantly reduced and its stability is also improved. This study also confirms the feasibility of implementing the SHI into a low cost microcontroller such as Arduino. That is, the additional computational load of SHIs is minimal.

Experimental Study of NMP Sample and Hold Input Using an Inverted Pendulum

2018 ◽  
Author(s):  
Yingxu Wang ◽  
Guoming G. Zhu ◽  
Ranjan Mukherjee
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Inverted Pendulum ◽  
Linear Quadratic Regulator ◽  
Sampling Period ◽  
Minimum Phase ◽  
Discrete Time System ◽  
Linear Quadratic ◽  
Sample And Hold ◽  
Control Configuration

Early research showed that a zero-order hold is able to convert a continuous-time non-minimum-phase (NMP) system to a discrete-time minimum-phase (MP) system with a sufficiently large sampling period. However the resulting sample period is often too large to adequately cover the original NMP system dynamics and hence not suitable for control application to take advantage of a discrete-time MP system. This problem was solved using different sample and hold inputs (SHI) to reduce the sampling period significantly for MP discrete-time system. Three SHIs were studied analytically and they are square pulse, forward triangle and backward triangle SHIs. To validate the finding experimentally, a dual-loop linear quadratic regulator (LQR) control configuration is designed for the Quanser single inverted pendulum (SIP) system, where the SIP system is stabilized using the Quanser continuous-time LQR (the first loop) and an additional discrete-time LQR (the second loop) with the proposed SHIs to reduce the cart oscillation. The experimental results show more than 75% reduction of the steady-state cart displacement variance over the single-loop Quanser controller and hence demonstrated the effectiveness of the proposed SHI.

Stability of real-time hybrid simulation involving time-varying delay and direct integration algorithms

2021 ◽  
pp. 107754632110016
Author(s):  
Liang Huang ◽  
Cheng Chen ◽  
Shenjiang Huang ◽  
Jingfeng Wang
Keyword(s):  
Real Time ◽  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Simulation ◽  
Time Varying ◽  
Discrete Time System ◽  
Time System ◽  
Integration Algorithms ◽  
The Stability ◽  
Continuous Time System

Stability presents a critical issue for real-time hybrid simulation. Actuator delay might destabilize the real-time test without proper compensation. Previous research often assumed real-time hybrid simulation as a continuous-time system; however, it is more appropriately treated as a discrete-time system because of application of digital devices and integration algorithms. By using the Lyapunov–Krasovskii theory, this study explores the convoluted effect of integration algorithms and actuator delay on the stability of real-time hybrid simulation. Both theoretical and numerical analysis results demonstrate that (1) the direct integration algorithm is preferably used for real-time hybrid simulation because of its computational efficiency; (2) the stability analysis of real-time hybrid simulation highly depends on actuator delay models, and the actuator model that accounts for time-varying characteristic will lead to more conservative stability; and (3) the integration step is constrained by the algorithm and structural frequencies. Moreover, when the step is small, the stability of the discrete-time system will approach that of the corresponding continuous-time system. The study establishes a bridge between continuous- and discrete-time systems for stability analysis of real-time hybrid simulation.

CHAOS OF DYNAMICAL SYSTEMS ON GENERAL TIME DOMAINS

2009 ◽  
Vol 19 (11) ◽  
pp. 3829-3832
Author(s):  
ABRAHAM BOYARSKY ◽  
PAWEŁ GÓRA
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
Chaotic Map ◽  
Discrete Time System ◽  
Combined System ◽  
Time Intervals ◽  
Time Dynamics ◽  
Time Component ◽  
Time Domains

We consider dynamical systems on time domains that alternate between continuous time intervals and discrete time intervals. The dynamics on the continuous portions may represent species growth when there is population overlap and are governed by differential or partial differential equations. The dynamics across the discrete time intervals are governed by a chaotic map and may represent population growth which is seasonal. We study the long term dynamics of this combined system. We study various conditions on the continuous time dynamics and discrete time dynamics that produce chaos and alternatively nonchaos for the combined system. When the discrete system alone is chaotic we provide a condition on the continuous dynamical component such that the combined system behaves chaotically. We also provide a condition that ensures that if the discrete time system has an absolutely continuous invariant measure so will the combined system. An example based on the logistic continuous time and logistic discrete time component is worked out.

Robust Adaptive Control Scheme for Discrete-Time System With Actuator Failures

2004 ◽  
Vol 127 (3) ◽  
pp. 520-526 ◽  
Author(s):  
Juntao Fei ◽  
Shuhao Chen ◽  
Gang Tao ◽  
Suresh M. Joshi
Keyword(s):  
Adaptive Control ◽  
Discrete Time ◽  
System Performance ◽  
Linear Time ◽  
Adaptive Controller ◽  
Robust Adaptive Control ◽  
Discrete Time System ◽  
Control Approach ◽  
Actuator Failures ◽  
Robust Adaptive

A robust adaptive control approach using output feedback for output tracking is developed for discrete-time linear time-invariant systems with uncertain failures of redundant actuators in the presence of the unmodeled dynamics and bounded output disturbance. Such actuator failures are characterized by some unknown inputs stuck at some unknown fixed values at unknown time instants. Technical issues such as plant-model output matching, adaptive controller structure, adaptive parameter update laws, stability and tracking analysis, and robustness of system performance are solved for the discrete-time adaptive actuator failure compensation problem. A case study is conducted for adaptive compensation of rudder servomechanism failures of a Boeing 747 dynamic model presented in discrete time, verifying the desired adaptive system performance in the presence of uncertain actuator failures.

scholarly journals Evolutionary dynamics in discrete time for the perturbed positive definite replicator equation

2021 ◽  
Vol 62 ◽  
pp. 148-184
Author(s):  
Amie Albrecht ◽  
Konstantin Avrachenkov ◽  
Phil Howlett ◽  
Geetika Verma
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Evolutionary Dynamics ◽  
Positive Definite ◽  
Genotype Distribution ◽  
Replicator Equation ◽  
System Matrix ◽  
Discrete Time System ◽  
Time System ◽  
Time Dynamics

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation. doi: 10.1017/S1446181120000140

On Zeros of Discrete-Time Models for Collocated Mass-Damper-Spring Systems

2004 ◽  
Vol 126 (1) ◽  
pp. 205-210 ◽  
Author(s):  
Mitsuaki Ishitobi ◽  
Shan Liang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sampling Period ◽  
Sufficient Condition ◽  
Time System ◽  
Time Model ◽  
Discrete Time Model ◽  
Mass Damper ◽  
Approximate Expressions ◽  
Continuous Time System

When a continuous-time system is discretized using the zero-order hold, there is no simple relation which shows how the zeros of the continuous-time system are transformed by sampling. In this paper, for a discrete-time model of a collocated mass-damper-spring system, the asymptotic behavior of the zeros is analyzed with respect to the sampling period and the linear approximate expressions are given. In addition, the linear approximate expressions lead to a sufficient condition for all the zeros of the discrete-time model to lie inside the unit circle for sufficiently small sampling periods. The sufficient condition is satisfied when a damping matrix is positive definite. Moreover, an example is shown to illustrate the validity of the linear approximations. Finally, a comment for a noncollocated system is presented.

scholarly journals Stability Results of a Class of Hybrid Systems under Switched Continuous-Time and Discrete-Time Control

2009 ◽  
Vol 2009 ◽  
pp. 1-28 ◽  
Author(s):  
M. De la Sen ◽  
A. Ibeas
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Linear Time ◽  
Sampling Period ◽  
Switched System ◽  
Time Interval ◽  
Stabilizing Controller ◽  
Switching Law ◽  
Discrete Dynamic System ◽  
Time Control

This paper investigates the stability properties of a class of switched systems possessing several linear time-invariant parameterizations (or configurations) which are governed by a switching law. It is assumed that the parameterizations are stabilized individually via an appropriate linear state or output feedback stabilizing controller whose existence is first discussed. A main novelty with respect to previous research is that the various individual parameterizations might be continuous-time, discrete-time, or mixed so that the whole switched system is a hybrid continuous/discrete dynamic system. The switching rule governs the choice of the parameterization which is active at each time interval in the switched system. Global asymptotic stability of the switched system is guaranteed for the case when a common Lyapunov function exists for all the individual parameterizations and the sampling period of the eventual discretized parameterizations taking part of the switched system is small enough. Some extensions are also investigated for controlled systems under decentralized or mixed centralized/decentralized control laws which stabilize each individual active parameterization.

Discretization of Nonautonomous Nonlinear Systems Based on Continualization of an Exact Discrete-Time Model

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Triet Nguyen-Van ◽  
Noriyuki Hori
Keyword(s):  
Nonlinear Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Van Der Pol Oscillator ◽  
Discrete Time System ◽  
Time System ◽  
Time Model ◽  
Forward Difference ◽  
Discrete Time Models

An innovative approach is proposed for generating discrete-time models of a class of continuous-time, nonautonomous, and nonlinear systems. By continualizing a given discrete-time system, sufficient conditions are presented for it to be an exact model of a continuous-time system for any sampling periods. This condition can be solved exactly for linear and certain nonlinear systems, in which case exact discrete-time models can be found. A new model is proposed by approximately solving this condition, which can always be found as long as a Jacobian matrix of the nonlinear system exists. As an example of the proposed method, a van der Pol oscillator driven by a forcing sinusoidal function is discretized and simulated under various conditions, which show that the proposed model tends to retain such key features as limit cycles and space-filling oscillations even for large sampling periods, and out-performs the forward difference model, which is a well-known, widely-used, and on-line computable model.

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