Discrete-time response

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

Example 29.1 Discrete-Time Response of a Digital PID Controller... 

Figure 29.8 (a) First-order lag without hold element (b) its discrete-time response to unit step input. 

VII.23 (a) Compute the discrete-time responses of the processes given in Problem VII. 21 to a unit step change in the input variable. The sampling period is T =1. 

Chap. 29 Discrete-Time Response of Dynamic Systems... 

Example 29.2 Discrete-Time Response of a First-Order Digital Filter... 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

The calculation procedures described above to evaluate integrals from discrete experimental response data to a pulse input may be modified to take tailing into account. This is done by dividing the data into time periods before and after tT. The data for t < tr are treated as already described, and the data for t > tT are treated analytically based on an exponential relation. We consider various quantities and the corresponding equations and integrals in turn. 

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

To determine the state space model with system Identification, responses of the nonlinear model to positive and negative steps on the Inputs as depicted in Figure 4 were used. Amplitudes were 20 kW for P,, . 4 1/s for and. 035 1/s for Q. The sample interval for the discrete-time model was chosen to be 18 minutes. The software described In ( 2 ) was used for the estimation of the ARX model, the singular value analysis and the estimation of the approximate... 

The computational power and flexibility of the computer is much used now to simulate controllers having characteristics other than the standard P, PI, etc., modes. Controllers are described in the following for which the design algorithm is derived directly from a specification of the discrete time character of the response of the controlled variable to a given change in set point. 

There is a variety of specifications that can be imposed on the system closed-loop response for a given change in set point. These lead to a number of alternative discrete-time control algorithms—the best known of which are the Deadbeat and Dahlin s algorithms. 

Design of a Discrete Time Controller Based Upon a Deadbeat Response... 

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-... 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

Both these models find their basis in network theories. The stress, as a response to flow, is assiimed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. 

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