(Solved): 19 State-variable representation of system with unstable mode-Part 2. For the system in Part 1, co ...

19 State-variable representation of system with unstable mode-Part 2. For the system in Part 1, consider the state variables \[ v_{1}(t)=\gamma(t) v_{2}(t)=\dot{\gamma}(t)+\gamma(t)-x(t) \] (a) Obtain the matrix \( \mathrm{A}_{2} \) and the vectors \( \mathrm{b}_{2} \) and \( \mathrm{c}_{2}^{T} \) for the state and the output equations that realize the ordinary differential equation in Part 1. (b) Draw a block diagram for the state variables and output realization. (c) How do \( \mathrm{A}_{2}, \mathrm{~b}_{2} \), and \( \mathrm{c}_{2}^{T} \) obtained above compare with the ones in Part 1? (d) In the block diagram in Part 1, change the summers into nodes, and the nodes into summers, invert the direction of all the arrows, and interchange \( x(t) \) and \( y(t) \), i.e,, make the input and output of the previous diagram into the output and the input of the diagram in this part. This is the dual of the block diagram in Part 1. How does it compare to your block diagram obtained in item (b)? How do you explain the duality? Answers: \( \dot{v}_{1}(t)=v_{2}(t)-v_{1}(t)+x(t) ; \dot{v}_{2}(t)=2 v_{1}(t)-x(t) \) 6.18 State-variable representation of system with unstable mode-Part 1. A LTI system is represented by an ordinary differential equation \[ \frac{d^{2} y(t)}{d t^{2}}+\frac{d y(t)}{d t}-2 y(t)=\frac{d x(t)}{d t}-x(t) \] (a) Obtain the transfer function \( H(s)=Y(s) / X(s)=B(s) / A(s) \) and find its poles and zeros. Is this system BIBO stable? Is there any pole-zero cancelation? (b) Decompose \( H(s) \) as \( W(s) / X(s)=1 / A(s) \) and \( Y(s) / W(s)=B(s) \) for an auxiliary variable \( w(t) \) with \( W(s) \) as its Laplace transform. 6.6 Problems Obtain a state/output realization that uses only two integrators. Call the state variable \( v_{1}(t) \) the output of the first integrator and \( v_{2}(t) \) the output of the second integrator. Give the matrix \( \mathbf{A}_{1} \), and the vectors \( b_{1} \), and \( c_{1}^{T} \) corresponding to this realization. (c) Draw a block diagram for this realization. (d) Use the MATLAB's function tf2ss to obtain state and output realization from \( H(s) \). Give the matrix \( \mathrm{A}_{c} \) and the vectors \( \mathrm{b}_{c} \) and \( \mathrm{c}_{c}^{T} \). How do these compare to the ones obtained before. Answers: \( H(s)=1 /(s+2) ; d^{2} w(t) / d t^{2}+d w(t) / d t-2 w(t)=x(t) \); \[ y(t)=d w(t) / d t-w(t) \]