Problem 5. Sensor and Actuator Placement (12 points) In the figure below, each node represents a state of a LTI dynamical system whose dynamics are given by x(t)=Ax(t) The arrows represent relationships between the states. For instance an arrow from node 2 to node 9 implies that state x2 affects state x9. Mathematically this means that a92=0 (However, since there is no arrow from node 9 to node 2,a29=0 ). In this problem we would like to place the minimum number of actuators and sensors to assure controllability and observability of the system. Since no actual values of A are given for this problem, we would like to ensure that we have controllability and observability in "almost all" cases. (Hint: you may perform your investigations using random matrices in MATLAB) (a) Determine the minimum number of actuators required to ensure controllability assuming each actuator can only directly control 1 state. Which states should these actuators control? (b) Determine the minimum number of sensors required to ensure observability assuming each sensor can only measure 1 state. Which states should these sensors measure? Figure 2: State Relationships Problem 6 Problem 5. Sensor and Actuator Placement (12 points) In the figure below, each node represents a state of a LTI dynamical system whose dynamics are given by x(t)=Ax(t) The arrows represent relationships between the states. For instance an arrow from node 2 to node 9 implies that state x2 affects state x9. Mathematically this means that a92=0 (However, since there is no arrow from node 9 to node 2,a29=0 ). In this problem we would like to place the minimum number of actuators and sensors to assure controllability and observability of the system. Since no actual values of A are given for this problem, we would like to ensure that we have controllability and observability in "almost all" cases. (Hint: you may perform your investigations using random matrices in MATLAB) (a) Determine the minimum number of actuators required to ensure controllability assuming each actuator can only directly control 1 state. Which states should these actuators control? (b) Determine the minimum number of sensors required to ensure observability assuming each sensor can only measure 1 state. Which states should these sensors measure? Figure 2: State Relationships Problem 6