(Solved): Question 2. Control of Quadruple Tank System (35 marks) A quadruple tank is illustrated in Figure ...

Question 2. Control of Quadruple Tank System (35 marks) A quadruple tank is illustrated in Figure 1, which has two input variables \( u_{1} \) and \( u_{2} \) and two output variables \( y_{1} \) and \( y_{2} \). The linearized transfer function model has the following form: \[ G(s)=\left[\begin{array}{cc} \frac{3.7 \gamma_{1}}{62 s+1} & \frac{3.7\left(1-\gamma_{2}\right)}{(23 s+1)(62 s+1)} \\ \frac{4.7\left(1-\gamma_{1}\right)}{(30 s+1)(90 s+1)} & \frac{4.7 \gamma_{2}}{90 s+1} \end{array}\right] \] where \( \gamma_{1}=\gamma_{2}=0.8 \). The following MATLAB program is used to convert the transfer function model to a state space model for the control system design and simulation studies. The system has sampling interval \( \Delta t \) as 1 second. gamma1 \( =0.8 \) gamma2 \( =0.8 \) num \( 11=3.7 * \) gamma1 \( \operatorname{den} 11=\left[\begin{array}{ll}62 & 1\end{array}\right] \) Figure 1: A quadruple tank system 1. Determine the eigenvalues of the quadruple tank system ( 5 marks). 2. Determine the controllability and observability of this system (5 marks). 3. Design a state estimate feedback control system so that both outputs have the capability to follow step reference changes and reject step input disturbances without steady-state errors. We use MATLAB program lqr.m for the design of both controller and observer. In the controller design, we choose \( Q=C^{T} C \), and \( R=I \) where \( C \) is the output matrix for the augmented system. In the observer design, we choose \( Q_{o b} \) as an identity matrix and \( R_{o b}=0.1 I \). In both cases, \( I \) is a \( 2 \times 2 \) identity