State variable step response

Fig. 8.6 State variable step response and state trajectory for Example 8.5. Hence from equation (8.61)...

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

For the spring-mass-damper system given in Example 8.6, evaluate the transient response of the state variables to a unit step input using... 

The step response of the state variables, together with the state trajectory, is shown in Figure 8.6. 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

Fig. 16. Step response by using auxiliary state variables Set point values AGAref = -0.1 kmolA/m ATref = 1 °C.

It is IVcquctitly usclul to be able to determine the lime constant of a first-order system from experimental step response data. This is easy to do. When time is equal to To in Eq. (2.53), the term ( I - e beeomes ( I - e ) = 0.623. This means that the output variable has undergone 62.3 pereent of the total ehange it is going to make. Thus, the time constant of a first-order system is simply the time it takes the step response to reach 62.3 percent of its new final steady-state value. 

Due to the simplicity of this model the state and output equations are derived by hand and transferred into MATLAB [21]. This step could have been done using a bond graph modeling environment (e.g., 20-Sim [22], CAMP-G [23]) to generate the time responses needed for the Analyze Activity step of MORA (see Fig. 2.2). The dynamic equations are numerically integrated to first produce the time response of the state variables and then the required set of outputs as defined in (2.3). 

In the step response equations, t is the independent variable instead of the input u used earher, and y is the dependent variable expressed in deviation form. Although steady-state gain K appears linearly in both response equations, the time constants are contained in a nonlinear manner, which means that linear regression cannot be used to estimate them. 

The main steps of the described approach are (i) the use of modal analysis to decouple the equation of motion (ii) the determination, in state variable, of the evolutionary frequency response vector functions and of the evolutionary power spectral density function matrix of the structural response and (iii) the evaluatiOTi of the nongeometric spectral moments as weU as the spectral characteristics of the stochastic response of linear systems subjected to stationary... 

Perform a series of steady-state runs to determine the amount of steam required to raise the temperature of the feed water stream to about 200°F (about 80°C). Then, switch to the dynamic mode of operation and perform step response testing by varying the inlet flow rate and feed temperature to determine the process response. Remember to use the strip charts to observe the important process variables. 

As is well known, the steady-state behavior of (spherical and disc) microelectrodes enables the generation of a unique current-potential relationship since the response is independent of the time or frequency variables [43]. This feature allows us to obtain identical I-E responses, independently of the electrochemical technique, when a voltammogram is generated by applying a linear sweep or a sequence of discrete potential steps, or a periodic potential. From the above, it can also be expected that the same behavior will be obtained under chronopotentiometric conditions when any current time function I(t) is applied, i.e., the steady-state I(t) —E curve (with E being the measured potential) will be identical to the voltammogram obtained under controlled potential-time conditions [44, 45]. 

The variability of the process parameters with flow causes variability in load response, as shown in Fig. 8-50. The PID controller was tuned for optimum (minimum-IAE) load response at 50 percent flow. Each curve represents the response of exit temperature to a 10 percent step in liquid flow, culminating at the stated flow. The 60 percent curve is overdamped and the 40 percent curve is underdamped. The differences in gain are reflected in the amplitude of the deviation, and the differences in dynamics are reflected in the period of oscillation. 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

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