1. Consider the two dynamic systems S₁ X₁ = -X₁ + U y = αx₁ where a ER is a parameter. S₁ has state X₁, control u and ou

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1. Consider the two dynamic systems S₁ X₁ = -X₁ + U y = αx₁ where a ER is a parameter. S₁ has state X₁, control u and output y. S₂ has state (X2, X3), control w and output Z. (a) Write down the matrix form of the system equations of S₁ and S₂ and determine whether each system is controllable, observable. W S₂ X2 = 2X2 + X3 + W X3 = X2 - 2x3 + W Z = X₂ + QX3 (b) These two systems are connected in series as seen in the figure below. The resulting system is called S3. Write down the matrix form of the system equations for S3 and determine S₁ S₂ whether S3 is controllable, observable. (c) The systems are now connected in a feedback configuration as shown in the figure below to produce S4. S₂ S₁ Write down the matrix form of the system equations for S4 and determine whether S4 is controllable, observable.