Linear system state space form

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

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. 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

The term direct bond graph model refers to a bond graph model in preferred integral causality that enables to compute the dynamics of the state x and the output y in terms of the input u and known parameters (see also Section 6.2.1.1). In the case of a linear time-invariant (LTI) system, the model equations are of state space form... 

The algorithm with consistent energy dissipation can now be expressed for linear systems including structural damping C. The state space form follows from substitution of the value a, = a into the state-space equations (7). The formulation takes on a particularly compact form when introducing the parameter... 

Many methods for linear system analysis are based on explicit linear ODEs. Constrained linear systems have therefore to be reduced first to an explicit linear ODE. In Sec. 1.4 such a reduction was obtained for tree structured systems by formulating the system in relative coordinates. For general systems this reduction has to be performed numerically. The reduction to this so-called state space form will be the topic of the first part of this chapter. Then, the exact solution of linear ODEs and DAEs is discussed. 

State Space Form of Linear Constrained Systems... 

In this section the numerical generation of the state space form for linear time invariant systems will be discussed. We saw in Sec. 1.5 how the equations of motion (1.5.2) can be linearized to... 

If this equation is then written in first order form, the state space form of the linear mechanical system consisting of 2Uy first order differential equations is obtained... 

Integral invariants can be used to reduce the number of coordinates. The system in reduced coordinates is called the state space form of the ODE with invariants. We will give here a formal definition of a state space form and relate it to the state space form of constrained mechanical systems, which we already studied in the linear case, see Sec. 2.1. 

The analysis so far was related to linear systems with constant system matrices. It suggests that the dynamics of discretized systems is entirely determined by the discretized state space form (see middle part of Tab. 5.3). Having the linear test equation in mind, one might think of stiff DAEs as those systems having a stiff state space form. 

A state space form of this system is just the linear test equation... 

Discretization of the state space form by an implicit BDF method. The resulting system is - in contrast to the discretized form of (5.3.7) - a square linear system with a well defined solution. 

The SSI technique lies in the class of time domain methods and is based on the discrete-time stochastic state-space form of the dynamics of a linear time-invariant system under unknown excitation. 

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. 

Fig. 7.2. Developmental path of the cAMP signaUing system in the parameter space formed by adenylate cyclase and phosphodiesterase activity. The stability diagram is established by linear stability analysis of the steady state admitted by the three-variable system (5.1) governing the dynamics of the allosteric model for cAMP signalling in D. discoideum (see section 5.2). In domain C sustained oscillations occur around an unstable steady state. In domain B, the steady state is stable but excitable as it amplifies in a pulsatile manner a suprathreshold perturbation of given amplitude. Outside these domains the steady state is stable and not excitable. The arrow crossing successively domains A, B and C represents the developmental path that the system should follow in that parameter space to account for the observed sequence of developmental transitions no relay relay oscillations (Goldbeter, 1980).

In the case of linear system models, the combination of CAMP-G, MATLAB, and the Symbolic Math Toolbox can generate state space matrices as well as transfer functions in symbolic form from a bond graph. MATLAB in conjunction with the Symbolic Math Toolbox can also be used for the incremental bond graph approach presented in Chapter 4. 

It is easily seen that damping can be obtained by introducing terms in the diagonal of first matrix of the integrated state space equations (4). For a linear system without structural damping introduction of terms of order 0 h) this leads to the following form,... 

The notion and properties of, and the transformation to minimal models is well developed and understood in the area of linear and nonlinear system theory (Kailath, 1980 and Isidori, 1995). Moreover, a wide class of lumped process models can also be transformed into the form of nonlinear state-space models. Therefore, the case of nonlinear state-space models is used as a basic case for the notion and construction of minimal models. This is then extended to the more complicated case of general lumped process models. 

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

The states in Eqn (25.2) are now being formed as linear combinations of the -step ahead predicted outputs k= 1, 2,. ..). The literature on state space identification has shown how the states can be estimated directly from the process data by certain projections. (Verhaegen, 1994 van Overschee and de Moor, 1996 Ljung and McKelvey, 1996). The MATLAB function n4sid (Numerical Algorithms for Subspace State Space System Identification) uses subspace methods to identify state space models (Matlab 2000, van Overschee and de Moor, 1996) via singular value decomposition and estimates the state x directly from the data. 

A state-space formulation of the equations of motion for a linear MDOF system is useful to describe the response of both classically and nonclassically damped systems. The general (second-order) equations of motion for an -degree-of-freedom linear structure under ground excitation are in matrix form ... 

In terms of linear vector space, Buckingham s theorem (Theorem 2) simply states that the null space of the dimensional matrix has a fixed dimension, and Van Driest s rule (Theorem 3) then specifies the nullity of the dimensional matrix. The problem of finding a complete set of B-numbers is equivalent to that of computing a fundamental system of solutions of equation 13 called a complete set of B-vectors. For simplicity, the matrix formed by a complete set of B-vectors will be called a complete B-matrix. It can also be demonstrated that the choice of reference dimensions does not affect the B-numbers (22). 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

In Reference [35], numerical examples of perturbative Sq - S2 excitation and the S2 IC dynamics for the / -carotene are discussed, too. The absence of reliable potential surfaces for this system motivated the use of a minimal two-dimensional model [66], which utilizes a Morse potential in each dimension. All three electronic surfaces Sq, and S2 involved in this example assume the same 2D potential form however, these potentials are shifted to each other. More importantly, in Ref. [35], each potential has 396 bound states in each electronic state within this model, while additionally the S2 and electronic states are coupled by linear coupling. Thus, the Q-space and P-space, as introduced in the context of the QP-algorithm in Section 1.3.1, consist of the S2 and 5 bound states, respectively. 

This specter should once and for all be laid to rest. The Curie principle as it applies to the system at hand (I will not state it in its most general form) forbids, in the linear regime, coupling between a vectorial process such as a flow and a scalar process such as a chemical reaction in an isotropic space. However, active transport does not occur in an isotropic or symmetrical system. Clearly, the protein constituting the pump is uniquely oriented within the membrane. 

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