Output equation

Write down the state equation and output equation for the spring-mass-damper system shown in Figure 8.1(a). 

For the 2 mass system shown in Figure 8.3, find the state and output equation when the state variables are the position and veloeity of eaeh mass. 

The state-spaee representation in equation (8.33) is ealled the eontrollable eanonieal form and the output equation is... 

Consider a system described by the state and output equations... 

Equation (8.106) provides the output equation and (8.107) the state equation... 

Consider the ease of the measurement of a single state variable x t). The output equation is therefore... 

Find the state and output equations for the positional servomeehanism shown in Figure 8.13 when the state and eontrol variable are... 

Hence, we can view the transfer function as how the Laplace transform of the state transition matrix O mediates the input B and the output C matrices. We may wonder how this output equation is tied to the matrix A. With linear algebra, we can rewrite the definition of O in Eq. (4-5) as... 

If C2 is the only output variable, we define C according to the output equation (E4-20). Matrices A and B remain the same. The respective effects of changes of C0 and Q on C2 can be obtained with... 

The output equation remains the same as in (E4-40). Laplace transform of the model equations and rearrangement lead us to... 

The next task is to make use of the full state estimator equations. Before that, we have to remold Eq. (9-46) as if it were a full state problem. This exercise requires some careful bookkeeping of notations. Let s take the first row in Eq. (9-46) and make it to constitute the output equation. Thus we make a slight rearrangement ... 

The fact that the algorithm used by MATLAB does not return a normalized output matrix C can create problems when we do feedback calculations in Chapter 9. The easy solution is to rescale the model equations. The output equation can be written as... 

Predictive models are built with ANN s in much the same way as they are with MLR and PLS methods descriptors and experimental data are used to fit (or train in machine-learning nomenclature) the parameters of the functions until the performance error is minimized. Neural networks differ from the previous two methods in that (1) the sigmoidal shapes of the neurons output equations better allow them to model non-linear systems and (2) they are subsymbolic , which is to say that the information in the descriptors is effectively scrambled once the internal weights and thresholds of the neurons are trained, making it difficult to examine the final equations to interpret the influences of the descriptors on the property of interest. 

Remember 14.3 Equation (14.22) provides the relationship between the usual electrochemical impedance response and the transfer function for cases where current or potential is the output. Equation (14.26) provides the corresponding relationship for cases where current or potential is the input. Equation (14.22) and equation (14.26) are distinguished by a minus sign resulting from the implicit function derivation. 

In order to maximize power output, Equation (66), the conditions dPIX/du= 0 and dPJA/<1X, =0 are necessary. Also, in order to maximize ecological function, Equation (67), the conditions d ,A/du=0 and 0 u/0Z, =0 are necessary. Hence, one can find the form of Z, for each maximized features function. In case of maximizing power output, the following is obtained ... 

State-space models relate the variation in state variables over time to their values in the immediate past and to inputs with differential or difference equations. Algebraic equations are then used to relate output variables to state variables and inputs at the same time instant. Consider a system of first-order differential equations (Eq. 4.44) describing the change in state variables and a system of output equations (Eq. 4.45) relating the outputs to state variables ... 

State and input coefficient matrices in output equation of... 

Which input-output equation was used to create this table ... 

Step 1 The nonlinear state and output equations are derived from the material and energy balances that model the process. These are expressed in the form ... 

Step 2 The state and output equations are solved at a stationary (steady) state that is defined either in terms of the desired state variable values or those of the input variables ... 

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