Discrete control models

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

In this section, a three-dimensional example problem is developed to demonstrate the capabilities of the advective control model. The simulation model domain is 2000 m by 2000 m in the horizontal dimensions and 20 m in the vertical. The domain is discretized into 80 rows, 80 columns, and 5 layers. The rows and columns are 40 m wide along the domain boundaries and 20 m wide near the area of interest. The northern and southern... 

A two-dimensional example problem is also developed to demonstrate the advective control model. The example problem is solved for both confined and unconfined conditions and the solutions are compared. In this example problem, the aquifer is homogeneous and isotropic, with no flow conditions imposed at the top and bottom boundaries and constant head conditions along the left and right boundaries. The head on the constant head boundaries slopes downward toward the bottom of the domain. The domain is 3100 m by 3100 m and is discretized into 49 rows and 58 columns. A river runs through the domain as shown in Figure 6. 

Pattern recognition self-adaptive controllers exist that do not explicitly require the modeling or estimation of discrete time models. These controllers adjust their tuning based on the evaluation of the system s closed-loop response characteristics (i.e., rise time, overshoot, settling time, loop damp-... 

The continuous models (e.g., differential equations in the time domain, or input-output models in the Laplace domain) are not convenient to use to analyze the dynamic behavior of loops with computer control discrete-time models are needed. 

Example 27.5 Discrete-Time Model of a Digital PID Controller... 

Consequently, the control action of a digital PID controller is determined by the following discrete-time model ... 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Saturation of controllers, 247, 257, 637 Scheduling computer control, 33 Secondary loop, cascade control, 395, 397, 398-99, 400-2 Secondary measurements, 16, 16-18 Second-order system, 186-87 Bode diagrams, 328-30 with dead time, 215, 216 discrete-time model, 585-86 dynamic characteristics, 187-93 experimental parameter identification, 233,668... 

Using the same computer control system as in item 1, explain why we need to convert continuous- to discrete-time models, and vice versa. 

In this section we consider the continuous elements of a DDC loop (i.e., the process and the hold as shown in Figure 29.1b). Although both elements are continuous, the input to the hold is a discrete-time signal c(nT), and the output from the process is sampled by a sampler. We would like to relate the sampled output values y(nT) with the discrete control commands c(nT), through a simple input-output model in the z -domain of the form... 

It should be noted that (MIP) problems, and their special cases, may be regarded as steady-state models. Hence, one important extension is the case of dynamic models, which in the case of discrete time models gives rise to multiperiod optimization problems, while for the case of continuous time it gives rise to optimal control problems that contain differential-algebraic equation (DAE) models. 

Crew Station/equipment characteristics The crew station design module and library is a critical component in the MIDAS operation. Descriptions of discrete and continuous control operation of the equipment simulations are provided at several levels of functiontil deteiil. The system can provide discrete equipment operation in a stimulus-response (blackbox) format, a time-scripted/ event driven format, or a full discrete-space model of the transition among equipment states. Similarly, the simulated operator s knowledge of the system can be at the same varied levels of representation or can be systematically modified to simulate various states of misunderstanding the equipment function. 

Klein, P. Pressure and temperature control in molecular dynamics simulation a unitary approach in discrete time. Model. Simul. Mater. Sci. Eng. 6, 405-422 (1998). doi 10.1088/ 0965-0393/6/4/009... 

We do not review this class of research since centralized control prevents internal conflict altogether, between manufacturers and intermediaries or otherwise. Instead, we refer the reader to reviews by Muckstadt and Roundy (1993) for deterministic demand models, Federgruen (1993) for discrete time models with stochastic demands, and Axsater (1993) for continuous time models with stochastic demands. 

The generalized operator/controller model can be found in Fig. la. Notice that only discrete places are needed to represent its input and output conditions. The input place PS(/) and the non-timed transition TS(/) denote, respectively, the status and confirmation action of the ith initiation signal. The output place PC(/) is used to reflect the status of the ith operation command. Notice also that the order in which these commands are issued can be arranged according to a given recipe. 

---