State Variable Representation

The eigenvalues of the 4 matrix, will be the openloop eigenvalues and will be equal to the roots of the openloop characteristic equation. In order to help us 

Now suppose a feedback controller is added to the system. The manipulated variables tp will now be set by the feedback controller. To keep things as simple as possible, let us make two assumptions that arc not very good ones, but permit us to illustrate an important point. We assume that the feedback controller matrix consists of just constants (gains). 7 3itd we assume that there are as many manipulated variables m as state variables x. 

Rearranging to put the differential equations in the standard form gives 

This equation describes the closedloop system. Let us define the matrix that multiplies x as the closedloop matrix and use the symbol 4cl  

Thus the characteristic matrix for this closedloop system is the 4cl niatrix. Its eigenvalues will be the close oop eigenvalues, and they will be the roots of the closedloop characteristic equation. 

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Using a state variable representation of a system, the eharaeteristie equation is given by... 

The time-domain differential equation description of systems can be used instead of the Laplace-domain transfer function description. Naturally the two are related, and we will derive these relationships later in this chapter. State variables are very popular in electrical and mechanical engineering control problems which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representations are more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. 

Modern control synthesis procedures are often based on the state variable representation of the plant equations. In applying them to high order systems, such as distillation columns, practical reasons of controller design require order reduction. That means one has to replace the large number of state variables obtained by physical laws, thru the introduction of a smaller set of suitably chosen state variables. 

State variable representation ean be transformed into transfer function representation by Laplace-transforming the set of M linear ordinary differential equations [Eq. (12.23)]. 

In order to prove this point let us compare (11.6), (11.7), (11.10), and (11.13) with (11.14), (11.15), (11.16), and (11.17). These equations are in the Cauchy form of the differential first-order equations, but can be arranged into matrix form for the typical state variable representation of the dynamics of the system. Here we will... 

The focus of the representation so far, has been on giving the form of rules, which enable us to reason about the values of state variables. This, however, is only one part of the overall reasoning task. We must also represent the theoreies we are going to use to derive the new control knowledge. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

To derive the state space representation, one visual approach is to identify locations in the block diagram where we can assign state variables and write out the individual transfer functions. In this example, we have chosen to use (Fig. E4.6)... 

The important message is that there is no unique state space representation, but all model matrixes should have the same eigenvalues. In addition, the number of state variables is the same as the order of the process or system. 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

In the development of a general state-space representation of the reactor, all possible control and expected disturbance variables need to be identified. In the following analysis, we will treat the control and disturbance variables identically to develop a model of the form... 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Tennyson, J. and Henderson, J.R. (1989). Highly excited rovibrational states using a discrete variable representation The Hj molecular ion, J. Chem. Phys. 91, 3815-3825. 

As anticipated in Remark 3.1, the constraints in Equation (3.13) depend on u1. In other words, the slow dynamics of the process cannot be completely characterized (in the sense of obtaining a reduced-order ODE representation of the type (2.48)) prior to defining u1 as a function of the process state variables (or measured outputs) via an appropriate control law. These issues are addressed in the following section. 

Several more recent efforts have provided models that can be used to incorporate dynamic representations of diazotrophs and N2-fixation in N-cycle models, although aU of these models have deficiencies that potential users should be aware of Hood et al. (2001) explicitly included a state variable for Trichodesmium in a 6 compartment N-cycle model (Fig. 33.12). In this model, growth of phytoplankton (Gp) and Trichodesmium (Gr) are specified by the following equations ... 

Figure 14.1 Schematic representation of an electrochemical system with input, output, and state variables. (Taken from Gabrielli and Tribollet. )...

The major challenge in the model formulation is the representation strategy adopted for the tank cycle. Normal operation considers that each tank is filled up completely before settling. After the settling period, the tank is released for clients satisfaction, until it is totally empty. These procedures are usually related to the product quality, where it isn t desired to mix products from several different batches. This implies that they are formulated four states for each tank i) full, ii) delivering product to clients, iii) empty and iv) being filled up with product from the pipeline. Each one of the states has a corresponding state variable, related to tank inventory ID), and has to be activated or deactivated whenever a boundary situation occurs (Eq. 1) the maximum UB) and minimum LB) capacities of the tank are met. For this purpose, the state variable y, binary) will have to be activated whenever both inequalities ( < and > ) hold (Eq. 2) ... 

The Mass-Action representation is clearly a special case of the GMA representation in which all exponents are positive integers. The Michaelis-Menten representation is, in turn, a special case of the traditional Mass-Action representation in which two important restrictions have been imposed (Savageau, 1992). First, it is assumed that the mechanism is in quasi-steady state. The derivatives of the dependent state variables in the Mass-Action Formalism can then be set to zero, thereby reducing the description from differential equations to algebraic equations. Second, it is assumed that complexes do not occur between different forms of an enzyme or between different enzymes. The algebraic equations will then be linear in the concentrations of the various enzyme forms, and one can derive the rational function that is the representation of the rate law within the Michaelis-Menten Formalism. 

In the global equilibrium state, when all constituent subsystems are mutually open, the transformations of infinitesimal displacements (perturbations) of the global and local parameters of state, which determine the Legendre transformed representation under consideration, into differentials of the respective conjugate state-variables (responses) can be summarized in terms of the following matrix integral equations [4,5,8,12,13] ... 

W. H. Miller, J. Chem. Phys. 97 2499 (1992). (c) W. H. Miller and T. Seideman, Cumulative and state-to-state reaction probabilities via a discrete variable representation— absorbing boundary condition Green s function, Time Dependent Quantum Molecular Dynamics Experiments and Theory (J. Broeckhove, ed., NATO ARW. (d) W. H. Miller, Accts. Chem. Res. 26 174 (1993). 

Matrix Properties /12.1.2 Transfer Function Representation / 12.1.3 State Variables... 

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