By Sheldon Ross

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**Extra resources for A First Course in Probability, 5th Ed scanned + Solutions Manual**

**Example text**

11 Block diagram of system on Fig. 5. 12 shows an electrically heated insulated oven which contains a product to be heated. The temperature of the air in the oven space is Ts, the Insulation qg Tr Ta qa qr Fig. 12 Electrically heated oven production system u Product Tg q Ts Controlled power supply 20 2 State Space Modelling of Physical Systems temperatures of the product, the insulation material and the ambient air are Tg, Tr and Ta respectively. All temperatures are assumed to be uniform. The heating power from the electrical heating element is called q, and the powers entering the product and insulation are qg and qr.

41) are found in C in reverse order. If the numerator polynomial is of lower order than the denominator (in which case the system is said to be strictly proper), bn and maybe some of the elements of C will be zero. The matrix D will be zero in this case. A block diagram of the system based on the state equations is shown on Fig. 14. bn bn – 1 bn – 2 ∫ _ u + + + ∫ xn – 1 .... ∫ x2 ∫ x1 ... b1 + b0 + + + y an – 1 + an – 2 ... .. + + xn + + ... + + + a1 a0 Fig. 14 Block diagram of companion form 1 of a SISO system The states selected here are often called the systems’phase variables and the model is said to be in phase variable form.

If the total heat capacity of the oven air space, the product and the insulation are denoted by Cs, Cg and Cr, expressions for the time derivative of the temperature of the different parts of the system can be formulated. One finds dTs ¼ q À qg À qr ; dt dTg ¼ qg ; Cg dt dTr ¼ qr À qa : Cr dt Cs (2:36) A block diagram of the model can be seen on Fig. 13. It is quite obvious, directly from the differential Eq. 36), that a sensible choice of state variables could be 2 3 2 3 Ts x1 6 7 6 7 x ¼ 4 x2 5 ¼ 4 Tg 5: x3 (2:37) Tr With input u, the disturbance n ¼ Ta and output y ¼ Tg , it is now straightforward to obtain the set of state equations.

### A First Course in Probability, 5th Ed scanned + Solutions Manual by Sheldon Ross

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