We assume a plate with a thickness [m] to be heated by a heater through conduction from the plate surface. Consider that the surface temperature u [K] can be controlled by means of a suitable heating-up. The plate layers has thickness d = a / 2n. if the surface temperature is equal on both sides of the temperature distribution symmetrical map plate center and it is enough to start from the 5-layer temperatures indicated in the figure down. Every layer forms a natural state. A modeling of this plate will thus result in a model with 5 state (ie five coupled differential equations). Suppose that the plate material is zinc with the following characteristics: Density: p = 7,14x10 ^ 3 [kg / m 3]. Heat capacity c = 0.39 [kJ / kg.K] .Heat conductivity s = 116 [kJ / m.k.s] For the n layers with temperatures x1, x2, ... x5 in the figure, the following energy-equation per unit area plate: pcd dx1 / dt = s (u-x1) / d -s (x1-x2) / d, pcd dx1 / dt = p (x1-x2) / d -s (x2-x3) / d, pcd dx1 / dt = s (x2-x3) / d -s (x3-x4) / d, pcd dx1 / dt = s (x3-x4) / d -s (x4 x5) / d, pcd dx1 / dt = 2s (x4 x5) / d. 

1) The constant T = (p.c.d ^ 2) / s re- write the equations in a state- form {A, B, C and D matrices}! Assume that the temperature in the middle of the plate (x5),is primarily consider as output. 

2) Simulate the temperature changes in the middle of the plate when the temperature (u) increases the step size by 10 degree. Let the plate thickness be 0,5[m].Use Simulink or MATLAB. In MATLAB use the command >>sys =ss(A,B,C,D) respective >>[y,t,x] = step(sys). 

3) Plot the Step response för all temperature layers. Which state swings fastest respective slowest? And Why? 

C x 2 - x3 x5 ( x ) ( x 3 ) [ x2 ) ( (1)