State-space models disturbance

Some disturbances can be measured, but the presence of others is only recognized because of their influence on process and/or output variables. The state-space model needs to be augmented to incorporate the effects of disturbances on state variables and outputs. Following Eq. 4.28, the state-space equation can be written as... 

Let A, B, C, D, E, F, G, K be constant coefficient matrices of appropriate dimensions and let x denote the state vector, u the vector of known inputs, y the vector of measured outputs, fit) additive faults and d (t) disturbances. The dynamic behaviour of a process subject to additive faults can then be described by the linear state space model... 

Note that the state-space model for Example 6.6 has d = 0, because disturbance variables were not included in (6-77). By contrast, suppose that the feed composition and feed temperature are considered to be disturbance variables in the original nonlinear CSTR model in Eqs. 2-66 and 2-68. Then the linearized model would include two additional deviation variables cXi and 7y, which would also be included in (6-77). As a result, (6-78) would be modified to include two disturbance variables, diAcXianddiATi... 

In the development of a general state-space representation of the reactor, all possible control and expected disturbance variables need to be identified. In the following analysis, we will treat the control and disturbance variables identically to develop a model of the form... 

Using the methods presented in Chapter 2, the above formulation can be used to derive a state-space realization of the slow dynamics of the type in Equation (2.48). The resulting low-dimensional model should subsequently form the basis for formulating and solving the control problems associated with the slow time scale, i.e., stabilization, output tracking, and disturbance rejection at the process level. 

One of the basic requirements in performing the operability analysis is that we have a model of the process relating the inputs and the disturbances to the outputs. Many of the process models can be described by the following state-space representation ... 

Hints Use the MATLAB MPC Toolbox, if desired, for this exercise. Two commands are used to produce a linear model of the plant in the representation needed for controller design. First, the dlinmod command obtains a state-space representation (A, B, C, D). To use this command, be sure that the Simuhnk diagram is drawn so that the process manipulated inputs and disturbances correspond to in ports on the top level of the Simulink flow sheet similarly, the outputs must correspond to out ports. Then the ss2mod command produces a model in MPC mod format, specifying inputs that are manipulated variables, measured disturbances, and unmeasured disturbances. The scmpc command simulates control of the hnearized plant with the MPC controller. 

Basic aspects of the spatial relaxation of the electrons in collision-dominated plasmas can be revealed when the evolution of the electrons whose velocity distribution has been disturbed at a certain space position is studied under the action of a space-independent electric field (Sigeneger and Winkler, 1997a Sigeneger and Winkler, 1997b). Sufficiently far from this position in the field acceleration direction of the electrons, a uniform state finally becomes established. Such relaxation problems can be analyzed on the basis of the parabolic equation for the isotropic distribution, Eq. (54), when the initial-boundary-value problem is adopted to the relaxation model. 

Georgakis and co-workers in Chapter A4 cover the issue of process operability analysis. Operability measures quantify the ability of the process to maintain the operating specifications despite the influence of disturbances in an ctcceptable dynamic fashion. The analysis is carried out using static and dynamic process models and irrespectively of the selected feedback control structure. Steady state operability defines the percentage of the desired output space that can be achieved by the available input space. Dynamic operability... 

In order to investigate the regulatory operability of the process, additionally the anticipated ranges of disturbances needs to be specified, that will define the Expected Disturbance Space (EDS). For the steady-state case, the EDS may also reflect the uncertainties in some of the important model parameters employed in the design, such as kinetic constants, heats of reaction, heat-transfer coefficients, etc. The regulatory operability index is defined from the inputs required to compensate for the effect of disturbances while maintaining the plant at its nominal set point, as ... 

The steady-state operability framework uses the steady-state model of the process and aims to calculate whether the input ranges are sufficient to achieve the desired output ranges in the presence of the expected disturbances. A steady-state operability index is thus defined. The steady-state operability characteristics are quantified by comparing two spaces related to the inputs or the outputs of the process. With respect to the input variables, one can compare the available input space versus the desired input space. The latter can be calculated to be large enough to compensate for all the expected disturbances and desired output values. A similar comparison can be made with respect to the output variables. The calculation of operability in this manner represents the inherent operability of the process and is independent of inventory control structure. 

In the next section extensive simulations are carried out in order to determine an operating range and conditions for the use of a linearised model in control system analysis and design. These simulations cover all aspects of vehicle motion under realistic operating conditions considering driver s inputs as well as external disturbances. For that purpose transient and steady-state analysis is performed. However only a few results are present for the reason of space. Full details are given in [2]. 

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