State space modeling

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. 

The block diagram of the system is shown in Figure 9.10. Continuous state-space model From equations (9.77)-(9.81)... 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

Fig. 1 illustrates the identification result, i.e., validation of identified model. The 4-level pseudo random signal is introduced to obtain the excited output signal which contains the sufficient information on process dynamics. With these exciting and excited data, L and Lu as well as state space model are oalcidated and on the basis of these matrices the modified output prediction model is constructed according to Eq. (8). To both mathematical model assum as plimt and identified model another 4-level pseudo random signal is introduced and then the corresponding outputs fiom both are compared as shown in Fig. 1. Based on the identified model, we design the controller and investigate its performance under the demand on changes in the set-points for the conversion and M . The sampling time, prediction and...

The combinatorial problem is represented by a discrete decision process (DDP) (Ibaraki, 1978) where the underlying information in the problem is captured by an explicit state-space model (Nilsson, 1980). 

Equations (41.15) and (41.19) for the extrapolation and update of system states form the so-called state-space model. The solution of the state-space model has been derived by Kalman and is known as the Kalman filter. Assumptions are that the measurement noise v(j) and the system noise w(/) are random and independent, normally distributed, white and uncorrelated. This leads to the general formulation of a Kalman filter given in Table 41.10. Equations (41.15) and (41.19) account for the time dependence of the system. Eq. (41.15) is the system equation which tells us how the system behaves in time (here in j units). Equation (41.16) expresses how the uncertainty in the system state grows as a function of time (here in j units) if no observations would be made. Q(j - 1) is the variance-covariance matrix of the system noise which contains the variance of w. 

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). 

We ll get a better picture in Chapter 4 when we cover state space models. 

With state space models, a set of differential equations is put in the standard matrix form... 

However, you will find that the MATLAB result is not identical to (E4-5). It has to do with the fact that there is no unique representation of a state space model. To avoid unnecessary confusion, the differences with MATLAB are explained in MATLAB Session 4. 

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. 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

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. 

With the state space model, substitution of numerical values in (E4-18) leads to the dynamic equations... 

To complete the state space model, the output equation is... 

When we used root locus for controller design in Chapter 7, we chose a dominant pole (or a conjugate pair if complex). With state space representation, we have the mathematical tool to choose all the closed-loop poles. To begin, we restate the state space model in Eqs. (4-1) and (4-2) ... 

A note of caution is necessary when we let MATLAB generate the state space model from a transfer function. The vector C (from S. c) is [0 0.5], which means that the indexing is reversed such that x2 is the output variable, and xl is the derivative of x2. Secondly, C is not [0 1], and hence we have to rescale the matrices B and C. These two points are further covered in MALTA Session 4. 

This differential equation is used to compute xel, which then is used to calculate xe with (9-50). With the estimated states, we can compute the feedback to the state space model as... 

We further substitute for X = (si -A) BU with the simple state space model to give... 

What we want is to dictate that the transfer function of this estimator is the same as that of the state space model ... 

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 ... 

Now, you may wonder if we can generate the state space model directly from a transfer function. The answer is, of course, yes. We can use... 

This exercise underscores one more time that there is no unique way to define state variables. Since our objective here is to understand the association between transfer function and state space models, we will continue our introduction with the ss2tf () and tf2ss o functions. 

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... 

Continuous Transfer functions in polynomial or pole-zero form state-space models transport delay... 

If this is not enough to convince you that everything is consistent, try step o on the transfer function and different forms of the state space model. You should see the same unit step response. 

Diagnostic observers consist in the definition of a set of observers from which it is possible to define residuals specific of only one failure [8]. Parity relations are relations derived from an input-output model or a state-space model [11] checking the consistency of process outputs and known process inputs. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

For effective control of crystallizers, multivariable controllers are required. In order to design such controllers, a model in state space representation is required. Therefore the population balance has to be transformed into a set of ordinary differential equations. Two transformation methods were reported in the literature. However, the first method is limited to MSNPR crystallizers with simple size dependent growth rate kinetics whereas the other method results in very high orders of the state space model which causes problems in the control system design. Therefore system identification, which can also be applied directly on experimental data without the intermediate step of calculating the kinetic parameters, is proposed. 

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