Selecting a Model Structure in the System Identification Process

The model-based control design process involves modeling the plant to be controlled, analyzing and synthesizing a controller for the plant, simulating the plant and controller, and deploying the controller.

For control design engineers, National Instruments provides a powerful set of mathematical algorithms, in the MATRIXx and LabVIEW System Identification tools, that reduce the effort required to develop models for model-based design. Unlike modeling from first principles, which requires an in-depth knowledge of the system under consideration, system identification methods can handle a wide range of system dynamics without knowledge of the actual system physics.

A variety of model structures are available to assist in modeling a system. The choice of model structure is based upon an understanding of the system identification method and insight and understanding into the system undergoing identification. Even then it is often beneficial to test a number of structures to determine the best one.

Parametric Model Structures

Parametric models describe systems in terms of differential equations and transfer functions. This provides insight into the system physics and a compact model structure.

Generally, you can describe a system using the following equation, which is known as the general-linear polynomial model or the general-linear model.



u(n) and y(n) are the input and output of the system respectively.
e(n) is zero-mean white noise, or the disturbance of the system.
 is the transfer function of the deterministic part of the system.
 is the transfer function of the stochastic part of the system.

The general-linear model structure, shown in Figure 1, provides flexibility for both the system dynamics and stochastic dynamics. However, a nonlinear optimization method computes the estimation of the general-linear model. This method requires intensive computation with no guarantee of global convergence.

Figure 1 General-Linear Model Structure

Simpler models that are a subset of the General Linear model structure are possible. By setting one or more of A(q), B(q), C(q) or D(q) polynomials equal to 1 you can create these simpler models such as AR, ARX, ARMAX, Box-Jenkins, and output-error structures. Each of these methods has their own advantages and disadvantages and is commonly used in real-world applications.

For any particular problem the choice of the model structure to use depends on the dynamics and the noise characteristics of the system. Using a model with more freedom or parameters is not always better as it can result in the modeling of nonexistent dynamics and noise characteristics. This is where physical insight into a system is helpful.

AR Model

The AR model structure is a process model used in the generation of models where outputs are only dependent on previous outputs. No system inputs or disturbances are used in the modeling. This is a very simple model that is limited in the class of problems it can solve. Strictly speaking this means that the AR model structure is the model for a signal, not a system. Time series analyses, such as linear prediction coding commonly use the AR model.

Figure 2 AR Model Structure

ARX Model

The ARX model, shown in Figure 3, is the simplest model incorporating the stimulus signal. The estimation of the ARX model is the most efficient of the polynomial estimation methods because it is the result of solving linear regression equations in analytic form. Moreover, the solution is unique. In other words, the solution always satisfies the global minimum of the loss function. The ARX model therefore is preferable, especially when the model order is high.

The disadvantage of the ARX model is that disturbances are part of the system dynamics. The transfer function of the deterministic part G(q–1, è) of the system and the transfer function of the stochastic part H(q–1, è) of the system have the same set of poles. This coupling can be unrealistic. The system dynamics and stochastic dynamics of the system do not share the same set of poles all the time. However, you can reduce this disadvantage if you have a good signal-to-noise ratio.

When the disturbance e(n) of the system is not white noise, the coupling between the deterministic and stochastic dynamics can bias the estimation of the ARX model. Set the model order higher than the actual model order to minimize the equation error, especially when the signal-to-noise ratio is low. However, increasing the model order can change some dynamic characteristics of the model, such as the stability of the model.

Figure 3 ARX Model Structure

ARMAX Model

Unlike the ARX model, the ARMAX model structure includes disturbance dynamics. ARMAX models are useful when you have dominating disturbances that enter early in the process, such as at the input. For example, a wind gust affecting an aircraft is a dominating disturbance early in the process. The ARMAX model has more flexibility in the handling of disturbance modeling than the ARX model.

Figure 4 ARMAX Model Structure

Box-Jenkins Model

The Box-Jenkins (BJ) structure provides a complete model with disturbance properties modeled separately from system dynamics.

Figure 5 Box-Jenkins Model Structure

The Box-Jenkins model is useful when you have disturbances that enter late in the process. For example, measurement noise on the output is a disturbance late in the process.

Output-Error Model

The Output-Error (OE) model structure describes the system dynamics separately. No parameters are used for modeling the disturbance characteristics.

Figure 6 OE Model Structure

State-Space Model

The previous classical parametric system identification methods minimize a performance function which is based on the sum of squared errors. These methods work well in many cases. However, for complex systems characterized by being of high order i.e. having many parameters, with several inputs and outputs, and having a large number of measurements, the classical methods can suffer from several problems. They can experience many local minima in the performance function and thereby a lack of convergence to global minima. The user will need to specify complicated parameterization of system orders and delays. They also may suffer potential problems with numerical instability and excessive computation time to execute the iterative numerical minimization methods needed. In addition, modern control methods require a state-space model of the system. For cases such as these the State-Space (SS) identification method is the appropriate model structure.

The following equations describe a state-space model.

x(n) is the state vector, y(n) is the system output, u(n) the system input and e(n) is the stochastic error. A, B, C, D, and K are the system matrices. The dimension of the state vector x(n) is the only setting you need to provide for the state-space model.

In general, the state-space model provides a more complete representation of the system, especially for MIMO systems, than polynomial models because state-space models are similar to first principle models. The identification procedure does not involve nonlinear optimization so the estimation reaches a solution regardless of the initial guess. Moreover, the parameter settings for the state-space model are simpler. You need to select only the order, or the number of states, of the model. The order can come from prior knowledge of the system. You also can determine the order by analyzing the singular values of the information matrix.

When the model order is high, use an ARX model because the algorithm involved in ARX model estimation is fast and efficient when the number of data points is very large. The state-space model estimation with a large number of data points is slow and requires a large amount of memory. If you must use a state-space model, for example in modern control methods, reduce the sampling rate of the signal in case the sampling rate is unnecessarily high.

The other polynomial models, including the ARMAX, output-error, Box-Jenkins, and general-linear models, involve iterative, nonlinear optimization in the identification procedure. They require excessive computation time, and the minimization can get stuck at a false local minimum, especially when the order is high and the signal-to-noise ratio is low. However, you can use these models when the stochastic dynamics are important because they provide more flexibility for the stochastic dynamics.

Determining Parameters for the Prediction Error Methods

Determining the delay and model order for the prediction error methods, ARMAX, BJ, and OE, is typically a trial-and-error process. The following is a useful set of steps that can lead to a suitable model. This is not the only methodology you can use, nor is this a comprehensive procedure.

1. Obtain useful information about the model order by observing the number of resonance peaks in the nonparametric frequency response function. Normally, the number of peaks in the magnitude response equals half the order of A(q)F(q).

2. Obtain a reasonable estimate of delay using correlation analysis and/or by testing reasonable values in a medium size ARX model. Choose the delay that provides the best model fit based on prediction errors or other fit criterion.

3. Test various ARX model orders with this delay choosing those that provide the best fit.

4. Since the ARX model describes both the system dynamics and noise properties using the same set of poles, the resulting model may be unnecessarily high in order. By plotting the zeros and poles (with the uncertainty intervals) and looking for cancellations you can reduce the model order. The resulting order of the poles and zeros are a good starting point for ARMAX, OE and/or BJ models with these orders used as the B and F model parameters and first or second order models for the noise characteristics.

5. If a suitable model is not obtained at this point attempt to determine if there are additional signals that may influence the output. Measurements of these signals can be incorporated as extra input signals.

If you cannot obtain a suitable model following these steps additional physical insight into the problem might be necessary. Compensating for nonlinear sensors or actuators and handling of important physical non-linearities are often necessary in addition to using a ready-made model.

From the prediction error standpoint, the higher the order of the model is, the better the model fits the data because the model has more degrees of freedom. However, you need more computation time and memory for higher orders. The parsimony principle advocates choosing the model with the smallest degree of freedom, or number of parameters, if all the models fit the data well and pass the verification test.

Conclusion

As we have discussed there are a variety of model structures available to assist in modeling a system. The choice of model structure is based upon an understanding of the system identification method and insight and understanding into the system undergoing identification. The characteristics of both system and disturbance dynamics play a role is the proper model selection.

These system identification methods can handle a wide range of system dynamics without knowledge of the actual system physics, thereby reducing the engineering effort required to develop models. System Identification using MATRIXx or LabVIEW System Identification tools in conjunction with National Instruments hardware provides the control design engineer with a full suite of tools for developing, prototyping and deploying control algorithms.

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