State space transformation

The third alternative to generate the diagonalized form is to use the state space to state space transformation function. The transformation is based on the modal matrix that we obtained earlier. 

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. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

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). 

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. 

A tool that we should be familiar with from introductory linear algebra is similarity transform, which allows us to transform a matrix into another one but which retains the same eigenvalues. If a state x and another x are related via a so-called similarity transformation, the state space representations constmcted with x and x are considered to be equivalent.1... 

That includes transforming a given system to the controllable canonical form. We can say that state space representations are unique up to a similarity transform. As for transfer functions, we can say that they are unique up to scaling in the coefficients in the numerator and denominator. However, the derivation of canonical transforms requires material from Chapter 9 and is not crucial for the discussion here. These details are provided on our Web Support. 

The Jarzynski identity can be used to calculate the free energy difference between two states 0 and 1 with Hamiltonians J%(z) and -A (z). To do that we consider a Hamiltonian -AA iz, A) depending on the phase-space point z and the control parameter A. This Hamiltonian is defined in such a way that A0 corresponds to the Hamiltonian of the initial state, Af(z, A0) = Atfo (z), and Ai to the Hamiltonian of the final state, Ai) = Aif z). By changing A continuously from A0 to Ai the Hamiltonian of the initial state is transformed into that of the final state. The free energy difference ... 

The three state orthogonal transformation to the space of natural solvent coordinates is the 6 x 6 matrix (see BH-I)... 

Geometric control is based in a coordinate transformation defined in the state space. This coordinate change allows the transformation of the affine system (3) into a called normal form, which can be partially or totally linearizable. However, how to know the degree of linearizability of the affine system In other words, how to know if the affine system is partially or totally linearizable Next, some notions are defined in order to answer this question. 

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. 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

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. 

A different approach in the use of orthogonal polynomials as a transformation method for the population balance is discussed in (8 2.) Here the error in Equation 11 is minimized by the Method of Weighted Residuals. This approach releases the restrictions on the growth rate and MSMPR operation, however, at the cost of the introduction of numerical integration of the integrals involved, which makes the method computationally unattractive. The applicability in determining state space models is presently investigated and results will be published elsewere. 

Several identification methods result in a state space model, eithejp by direct identification in the state space structure or by identjLfication in a structure that can be transformed into a state space model. In system identification, discrete-time models are used. The discrete-time state-space model is given by... 

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

There are also some unexpected problems, related to the fact that the stationarity conditions do not discriminate between ground and excited states, between pure states and ensemble states, and not even between fermions and bosons. The IBQ give only information about the nondiagonal elements of y and the Xk, whereas for the diagonal elements other sources of information must be used. These elements are essentially determined by the requirement of w-representability. This can be imposed exactly to the leading order of perturbation theory. Some information on the diagonal elements is obtained from the lCSE,t, though in a very expensive and hence not recommended way. The best way to take care of -representability is probably via a unitary Fock-space transformation of the reference function, because this transformation preserves the -representability. 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

So the function r preserves the absolute value of the bracket and hence complex conjugation is a physical symmetry of the state space. This transformation corresponds to reflecting the sphere in Figure 10.6 in the xz-plane (Exercise 10.16). Complex conjugation does not descend from a (complex) linear transformation however, we have... 

Recall from Section 10.4 that if an observer undergoes a rotation of g (with g e 50(3), the spin-1/2 state space (C ) transforms via the linear operator Pi(g), while the spin-1 state space P(C ) transforms via pi(g). Hence the corresponding transformation of a vector v 0 w in 0 is... 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

In the oxygen VER experiments (3) the n = 1 vibrational state of a given oxygen molecule is prepared with a laser, and the population of that state, probed at some later time, decays exponentially. Since in this case tiojo kT, we are in the limit where the state space can be truncated to two levels, and 1/Ti k, 0. Thus the rate constant ki o is measured directly in these experiments. Our starting point for the theoretical discussion is then Equation (14). For reasons discussed in some detail elsewhere (6), for this problem we use the Egelstaff scheme in Equation (19) to relate the Fourier transform of the quantum force-force time-correlation function to the classical time-correlation function, which we then calculate from a classical molecular dynamics computer simulation. The details of the simulation are reported elsewhere (4) here we simply list the site-site potential parameters used therein e/k = 38.003 K, and a = 3.210 A, and the distance between sites is re = 0.7063 A. 

Our aim is to find the corresponding equation for P x, t), the probability density to find the particle position at x the velocity distribution is assumed equilibrated on the timescale considered. Note that in Section 8.1 we have distinguished between stochastic equations of motion that describe the time evolution of a system in state space (here x), and those that describe this evolution in probability space. We now deal with the transformation between such two descriptions. 

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