A Unified Approach for Weighted Sensitivity Design of PID Controllers

2009 ◽  
Author(s):  
Tooran Emami ◽  
John M. Watkins
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Arbitrary Order ◽  
Shift Operator ◽  
Sampling Period ◽  
Time Shift ◽  
Pid Controllers ◽  
Unified Approach ◽  
Time Model ◽  
Graphical Technique

In this paper a graphical technique is introduced for finding all continuous-time or discrete-time proportional integral derivative (PID) controllers that satisfy a weighted sensitivity constraint of an arbitrary order transfer function with time delay. These problems can be solved by finding all achievable PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. The key advantage of this procedure is that this method depends only on the frequency response of the system. The delta operator is used to describe the controllers in a discrete-time model, because it not only possesses numerical properties superior to the discrete-time shift operator, but also converges to the continuous-time controller as the sampling period approaches zero. A unified approach allows us to use the same procedure for discrete-time and continuous-time weighted sensitivity design of PID controllers.

A Unified Approach for H∞ Complementary Sensitivity Design of PID Controllers Applied to a DC Motor With Communication Delay

2009 ◽  
Author(s):  
T. Emami ◽  
J. M. Watkins
Keyword(s):  
Transfer Function ◽  
Frequency Response ◽  
Discrete Time ◽  
Continuous Time ◽  
Dc Motor ◽  
Communication Delay ◽  
Time Shift ◽  
Pid Controllers ◽  
Unified Approach ◽  
Experimental Frequency

In this paper a graphical technique is introduced for finding all continuous-time and discrete-time proportional integral derivative (PID) controllers that satisfy the discrete-time H∞ complementary sensitivity constraint of an arbitrary order transfer function with time delay. These problems can be solved by finding all achievable PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. The key advantage of this procedure is that this method depends only on the frequency response of the system. If the plant transfer function is given, the procedure is still appropriate. The delta operator is used to describe the discrete-time controllers because it not only possesses numerical properties superior to the discrete-time shift operator, but also converges to the continuous-time controller as the sampling period approaches zero. A unified approach allows us to use the same procedure for discrete-time and continuous-time complementary sensitivity design of PID controllers. The method is demonstrated by using the experimental frequency response of a DC motor with communication delay for H∞ complementary sensitivity design of PID controllers.

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 Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

2008 ◽  
Vol 38 (02) ◽  
pp. 399-422 ◽  
Author(s):  
Eric C.K. Cheung ◽  
Steve Drekic
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Size Distribution ◽  
Discrete Time ◽  
Continuous Time ◽  
Risk Model ◽  
Time Model ◽  
Jump Size ◽  
Time Of Ruin ◽  
The Time Of Ruin

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

scholarly journals Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design

2019 ◽  
Vol 9 (16) ◽  
pp. 3220 ◽  
Author(s):  
Ryo Kurokawa ◽  
Takao Sato ◽  
Ramon Vilanova ◽  
Yasuo Konishi
Keyword(s):  
Discrete Time ◽  
Pid Control ◽  
Continuous Time ◽  
Dead Time ◽  
Design Method ◽  
Control Design ◽  
The Other ◽  
Time Model ◽  
Trade Off ◽  
First Order

The present study proposes a novel proportional-integral-derivative (PID) control design method in discrete time. In the proposed method, a PID controller is designed for first-order plus dead-time (FOPDT) systems so that the prescribed robust stability is accomplished. Furthermore, based on the control performance, the relationship between the servo performance and the regulator performance is a trade-off relationship, and hence, these items are not simultaneously optimized. Therefore, the proposed method provides an optimal design method of the PID parameters for optimizing the reference tracking and disturbance rejection performances, respectively. Even though such a trade-off design method is being actively researched for continuous time, few studies have examined such a method for discrete time. In conventional discrete time methods, the robust stability is not directly prescribed or available systems are restricted to systems for which the dead-time in the continuous time model is an integer multiple of the sampling interval. On the other hand, in the proposed method, even when a discrete time zero is included in the controlled plant, the optimal PID parameters are obtained. In the present study, as well as the other plant parameters, a zero in the FOPDT system is newly normalized, and then, a universal design method is obtained for the FOPDT system with the zero. Finally, the effectiveness of the proposed method is demonstrated through numerical examples.

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.

scholarly journals Predicting IBNYR Events and Delays II. Discrete Time

1990 ◽  
Vol 20 (1) ◽  
pp. 93-111 ◽  
Author(s):  
William S. Jewell
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Bayesian Method ◽  
Predictive Accuracy ◽  
Random Delay ◽  
Continuous Time Model ◽  
Practical Situation ◽  
Time Model ◽  
Numerical Example ◽  
Exposure Interval

AbstractAn IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

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