University of Portsmouth, Department of Electronic and Computer Enginee ring B307 – Discrete and Modern Contro l Systems Design - Lecture 9/Tutorial 3 Branislav Vuksanovic Tutorial 3 – Transfer Functions and Canonical Forms 1. a) For the third-order system with the state space model: 1 1 2 2 3 3 0 1 0 0 0 0 1 0 10 17 8 1 i x x x x e x x ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − − − ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ i i i [ ] 1 2 3 4 4 1 x y x x ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ determine the transfer function relating Y(s) to E i (s) and check if all of the system poles are also the poles of the transfer function. b) If possible, provide a reduced- order state-space rea lisation having the same input- output relationship as the third-order system analysed in the first part of this question. 2. Simulation diagrams are elementary analog computers that can be used to solve and simulate differential equations describing systems dynamics. They are composed of integrators, adders, subtractors, and multipliers which can be physically realised by using operational amplifiers. In addition, function generators can be used to generate input signals. Usually, the number of integrators in a simulation diagram is equal to the order of the differential equation under consideration. There are many ways to draw a simulation diagram for a given dynamic system, and there are also many ways to obtain the state space form from the given simulation diagram. Draw the diagram to simulate the reduced-order system derived in the previous equation. 3. Draw the simulation diagram to simulate the system with the transfer function: ( ) ( ) 3 2 1 2 3 4 s G s s s s − + = + + + 4. Draw the simulation diagram to simulate the system with the following state-space mod el: 1 1 1 2 1 2 5 3 4 6 x x u x x ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ = + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ i i [ ] 1 2 7 8 x y x ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦
