4) Derivation of system state space equations by computer algebra (symbolic mani

Question

4) Derivation of system state space equations by computer algebra (symbolic manipulations in MATLAB): The three tank system below is slightly different from the one discussed in Handout 7 in that the exit pressures at the linear flow resistances for the first two tanks are not zero. Recall that the equation for a linear flow resistance is: ?p R q where ?p is the pressure difference across the flow resistance and q is the volume flow rate through the resistance. The cross sectional areas of the tanks are A1,A2,A3; the heights of the liquid in the tanks are h1,h2,h3 and the pressures at the bottoms of the tanks are P1, P2-Py The linear flow resistances at the tank exits are R1, R2, R3 and the volume flow rate through them are 01,22,23 respectively. An input flow rate equal to Q enters the first tank, hence the input u -Q. The system outputs are y Q2 and the state variables are the liquid heightsx - h2 A2 ?? 23 2 h3 P

Explanation / Answer

In control engineering , a state space representation is a mathematical model of a physical system of a input, output and state variables related by 1st order differential equations . State variables are variables whose values evoved through time in the way that depends on the values they have any given time and also depends on the externally inposed values of input variables. Output variables values depend on the values of the state variables.

The state space is euclidean space in which the variables on the access or the state variables. the state of the system can be represented as vector within that space

To obstract form the number of inputs outputs and states , these variables are exprssed as vectors. Additionally, if the dynamical system is linear, time-invariant and finite-dimensional, then the differential and alzebric equation may be written in matrix form.The state-space method is charaterized by significant algebraization of general system theory , which makes it possible to use kronecker vector-matrix structures. the capcity of these structures can be efficiently applied to reasearch system with modelation or with out it.Unlike the frequency domain approach,the use of the state space representation is not limited to systems with linear components and zero initial conditions.The state space model is used in many different areas.In econometrics the state space model can be used for forcasting stock prices and numerous other variables.  

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