Parameter sensitivity of transfer functions

Chapter 4 shows how incremental bond graphs enable a matrix-based determination of parameter sensitivities of transfer functions for direct as well as for inverse linear models. The necessary matrices can be generated from a bond graph and its incremental bond graph by means of existing software. Furthermore, incremental bond graphs also support a parameter sensitivity analysis of ARR residuals. 

Abstract Incremental true bond graphs are used for a matrix-based determination of first-order parameter sensitivities of transfer functions, of residuals of analytical redundancy relations, and of the transfer matrix of the inverse model of a linear multiple-input-multiple-output system given that the latter exists. Existing software can be used for this approach for the derivation of equations from a bond graph and from its associated incremental bond graph and for building the necessary matrices in symbolic form. Parameter sensitivities of transfer functions are obtained by multiplication of matrix entries. Symbolic differentiation of transfer functions is not needed. The approach is illustrated by means of hand derivation of results for small well-known examples. 

Keywords Incremental true bond graphs Parameter sensitivities of transfer functions Linear inverse models Fault detection and isolation Parameter sensitivities of the residuals of analytical redundancy relations... 

Advantages of the incremental true bond graph-based approach presented in this chapter are that the matrices can be automatically set up in symbolic form from an original bond graph and its associated incremental bond graph by available software. Parameter sensitivities of transfer functions are then obtained by multiplication of matrix entries which can be performed by software in symbolic form. There is no need for symbolic differentiation of transfer functions. The purpose of determining sensitivities of transfer functions in symbolic form is that, in the design of a robust control, it may be useful to know how sensitive transfer functions are with respect to certain parameter uncertainties. 

The determination of parameter sensitivities of transfer functions from incremental linear inverse bond graph models is considered in Section 4.5. 

Parameter Sensitivities of Transfer Functions from Direct Bond Graph Models... 

Fig. 4.13 Parameter sensitivity of transfer function F21 with respect to Rm...

So far, parameter sensitivities of transfer functions of direct models have been considered. This section presents an incremental bond graph-based procedure to the symbolic determination of parameter sensitivities of transfer functions of linear inverse models given that the latter exist. 

An incremental true bond graph approach to a matrix-based determination of parameter sensitivities of transfer functions of linear MIMO models and of residuals of ARRs in symbolic form has been presented. The approach has the following advantages ... 

W. Borutzky. Parameter Sensitivities of Transfer Functions and of Residuals. In F.E. CeUier and J.J. Granda, editors, 9th International Conference on Bond Graph Modeling and Simulation (ICBGM 2010), volume 42(2) of Simulation Series, pages 4-10, Orlando, FL, April 2010. SCS. ISBN 978-1-61738-209-3. 

If ARRs can be obtained in closed symbolic form, parameter sensitivities can be determined by symbolic differentiation with respect to parameters. If this is not possible, parameter sensitivities of ARRs can be computed numerically by using either a sensitivity bond graph [1 ] or an incremental bond graph [5, 6]. Incremental bond graphs were initially introduced for the purpose of frequency domain sensitivity analysis of LTI models. Furthermore, they have also proven useful for the determination of parameter sensitivities of state variables and output variables, transfer functions of the direct model as well as of the inverse model, and for the determination of ARR residuals from continuous time models [7, Chap. 4]. In this chapter, the incremental bond graph approach is applied to systems described by switched LTI systems. 

This example illustrates that a parameter sensitivity of a transfer function such as 9Fi/9/ 2 (4.31) can be obtained by multiplication of an entry of the transfer matrix... 

A parameter sensitivity of a transfer function out of the multiple possible ones of a linear MIMO model is obtained in symbolic form by multiplication of appropriate matrix entries. This can be performed by computer algebra systems. 

Example 11—Sensitivity of closed-loop stability to small variations in controller parameters. For the stable transfer function... 

The fundamental transfer function of a sensor is determined by the properties of the transducer principles chosen when setting up a signal path. Static sensor performance is defined by the static transfer function that describes the relationship between the output signal Uout and the input signal 0 (Eqs. 3.8 and 3.9) and additional parameters such as the measurement range, attainable sensitivity, resolution, and desired accuracy [8]. 

However, this ideal transfer function is strongly affected by a sensor"s inherent nonideality and by the cross-sensitivity of a sensor to interfering environmental parameters (Yn in Eqs. 3.8 and 3.9). Hence, these cross-sensitivity and interference effects, including those occurring in the sensor device s signal-processing unit... 

According to (4.15), the th output sensitivity function with respect to Oj, dyi/dQj, is a transfer function F j multiplied by the Laplace transform of the output Wj = SjZj of the j th modulated source representing the parameter variation... 

If the value of C2 is increased (decrease of k), we will see immediately that the system will oscillate with greater amplitude and lower frequency, a typical response since the stiffness is reduced. The same thing can be done for any transfer function of the system, a relationship of any force (effort) or velocity (flow) and any of the two inputs as the sliders are displayed for all the inputs. The ability to study the system parameter sensitivity is a very valuable tool for the study of how the system responds to physical parameter changes, a very important study particularly for the design of control systems. 

There are other benefits of this approach. The denominator of this transfer function is the characteristic equation of the system. Figure 11.43 shows it in symbolic form. The coefficients of each power of S are displayed in terms of the physical parameters in symbolic form. This means that each coefficient of S is the result of the physical parameter combined mathematical operations. What this means is that we now know the influence each physical parameter has on the coefficients and their dependencies, which can be used for the study of parameter sensitivities in relation to simulation results. This of course impacts directly the system design to fulfill the functional requirements. 

---