State space modeling example

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. 

Example 2.16. Derive the closed-loop transfer function X,/U for the block diagram in Fig. E2.16a. We will see this one again in Chapter 4 on state space models. With the integrator 1/s, X2 is the Laplace transform of the time derivative of x,(t), and X3 is the second order derivative of x,(t). 

In the next two examples, we illustrate how state space models can handle a multiple-input multiple output (MIMO) problem. We ll show, with a simple example, how to translate information in a block diagram into a state space model. Some texts rely on signal-flow graphs, but we do not need them with simple systems. Moreover, we can handle complex problems easily with MATLAB. Go over MATLAB Session 4 before reading Example 4.7A. 

Example 4.6. Derive the transfer function Y/U and the corresponding state space model of the block diagram in Fig. E4.6. 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

The last argument in ss2tf o denotes the i-th input, which must be 1 for our single-input single-out model. To make sure we cover all bases, we can set up our own state space model as in Example 4.1 ... 

We should see that the LTI object is identical to the state space model. We can retrieve and operate on individual properties of an object. For example, to find the eigenvalues of the matrix a inside sys obj ... 

The use of step () and impulse () on state space models is straightforward as well. We provide here just a simple example. Let s go back to the numbers that we have chosen for Example 4.1, and define... 

In this paper, three methods to transform the population balance into a set of ordinary differential equations will be discussed. Two of these methods were reported earlier in the crystallizer literature. However, these methods have limitations in their applicabilty to crystallizers with fines removal, product classification and size-dependent crystal growth, limitations in the choice of the elements of the process output vector y, t) that is used by the controller or result in high orders of the state space model which causes severe problems in the control system design. Therefore another approach is suggested. This approach is demonstrated and compared with the other methods in an example. 

Various extensions of linear state-space approach have been proposed for developing nonlinear models [227, 274]. An extension of linear CVA for finding nonlinear state-space models was proposed by Larimore [160] where use of alternating conditional expectation (ACE) algorithm [24] was suggested as the nonlinear CVA method. Their examples used linear CVA to model a system by augmenting the linear system with pol3momials of past outputs. 

Parametric models are more or less white box or first principle models. They consist of a set of equations that express a set of quantities as explicit functions of several independent variables, known as parameters . Parametric models need exact information about the inner stmcture and have a limited number of parameters. For instance, for the description of the dynamics, the order of the system should be known. Therefore, for these models, process knowledge is required. Examples are state space models and (pulse) transfer functions. Non-parametric models have many parameters and need little information about the inner stmcture. For instance, for the dynamics, only the relevant time horizon shoirld be known. By their stmcture, they are predictive by nature. These models are black box and can be constructed simply from experimental data. Examples are step and pulse response functions. 

If the process conditions vary over a wide range, there may be a need for a non-linear empirical model. In case of a dynamic non-linear model there are a few possibilities for developing such a model, for example a dynamic neural network or a dynamic fuzzy model. One could also develop a Wiener model, in which the process dynamics are represented by a hnear model, such as a state space model. The static characteristics of the process are then modeled by a polynomial, able to represent the non-linearity. 

If the model is linear and static one coirld for example use partial least squares modeling. State space modeling can be used in its linear or non-linear form, depending on the situation. 

In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

In this chapter several model reduction techniques will be discussed. The first method is based on firequency response matching, other methods make use oficonversion ofi the model structure to a state space model and subsequently truncating the states that have a minimum impact on the input-output relationship. The main indicator used fior this purpose is the so-called Hankel singular value. In addition, the model structure is converted to a balanced realization, afiter which the reduction techniques can be applied. Several examples are given on how to apply the dififierent methods. 

Because the state-space model in Eqs. (6-75) and (6-76) may seem rather abstract, it is helpful to consider a physical example. 

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... 

State-space models provide a convenient representation of dynamic models that can be expressed as a set of first-order, ordinary differential equations. State-space models can be derived from first principles models (for example, material and energy balances) and used to describe both linear and nonlinear dynamic systems. 

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