State-space equations

In order to present this observer, the state space equation referred to the vector jce can be conveniently rewritten as... 

Following Badgwell (1997), consider that the real process behavior is described by the stable state-space equations... 

The gas-lift oil well operation can be described by the following state-space equations [7] ... 

Some disturbances can be measured, but the presence of others is only recognized because of their influence on process and/or output variables. The state-space model needs to be augmented to incorporate the effects of disturbances on state variables and outputs. Following Eq. 4.28, the state-space equation can be written as... 

Assume that the dynamic behaviour of a process is within a neighbourhood of an operating point and can be described sufficiently accurate by a linear time-invariant state space model. Then sensor and actuator faults, e.g. leakage from a tank, are additional external input signals to the process. They are commonly taken into account as additive terms in the state space equations and are classified as additive faults [7, 8]. 

Parametric faults are called multiplicative because they contribute terms to the state space equations that are the product of one of the usually time constant matrices AA, AB, AC, AD with the state vector x(r) or the vector of known inputs (f), i.e. with a time-varying vector [10]. 

Existing software such as CAMP-G/MATLAB supported by the Symbolic Math Toolbox can derive equations from the bond graph and from its associated incremental bond graph and can build the matrices of the state space equations and the output equations for both bond graphs in symbolic form. 

As has been shown by Rosenberg [29], a linear state space equation in terms of Xj and the input vector u can be derived from the equations of the bond graph junction structure and the linear constitutive equations of the storage fields and the dissipative fields by eliminating Xd and other variables. The result is... 

Having determined that a flow input is required and that the system is inherently hard to control due to the RHP pole and zero, the next step is to design a controller. A state-space approach is used here and so a state-space model must first be derived from the bond graph of Fig. 5.2c. This system represents a differential-algebraic equation (DAE) which can, however, be rewritten as a state-space equation. In particular, system state-space equations can be derived from Fig. 5.2c as follows. Defining the angular momenta of the two pendula as h and /12, respectively, the torque eo (which drives the controlled pendulum) is, from the left-hand side of Fig. 5.2e,... 

It is clear that (11.6), (11.7), (11.10), and (11.13) form a set of state space equations in first-order form written in the Cauchy form. These set of equations derived by a conventional method such as applying Newton s equations can be solved also using conventional solutions using MATLAB and its tools tailored to first-order differential equations. These equations can also be arranged in matrix form. That is presented next as we compare the two methods. 

It is as simple as entering the bond graph in graphical form as an input to obtain the equations above. So, the next step is to prove that these equations are in fact equivalent to those state space equations obtained from the set of equations displayed in (11.6), (11.7), (11.10), (11.13). If these are in fact equivalent, the automated process behind the bond graph method has sound advantages that such demonstration establishes because now the manual task of finding the mathematical representation of the system has been automated and free of errors in the derivation. 

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 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... 

When the increments Au and Av in the damping terms of the balanced algorithm (7) are replaced by the increments As and At of the corresponding filtered variables, it takes the form of the state-space equations... 

The energy balance equation of the filter algorithm follows from multiplication of the state-space equations with [Au, —Av ]. 

The linear equation 12 can then be easily included in the state-space formulation. After operating upon the state-space equation of the extended system, i.e. that including the linearized equation, and assuming Gaussian behavior of all the responses, one obtains the following differential equation for the evolution of the covariance matrix of the responses ... 

In order to obtain a discrete state-space equation for computation, the fimctions in Eq. 4 is linearized for each time step t e [Mf, (k + l)Af) as follows ... 

The continuous-time state-space equation of motion of a linear time-invariant system can be written as... 

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