[3] A single-input, single-output system described by dx dt = is controllable and observable. PID (Proportional...

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[3] A single-input, single-output system described by dx dt = is controllable and observable. PID (Proportional plus Integral plus Derivative) control is to be applied to this system. For this purpose, y is integrated in the controller. The output of the integrator is denoted by x,. We know x, exactly since it is a part of the controller. Notice that the original system and the integrator in the controller may be expressed by the following state and output equations. Ax + Bu, y = Cx d x1 1 0 XI [0 Cx Discuss the controllability and observability of this enlarged system. Note that the transfer function of the original system is C(sI A) B and the transfer function of the integrator is 1/s. CXI X1 #N-DI-BH M-C CH = dt x 0 A X = + [3] A single-input, single-output system described by dx dt = is controllable and observable. PID (Proportional plus Integral plus Derivative) control is to be applied to this system. For this purpose, y is integrated in the controller. The output of the integrator is denoted by x,. We know x, exactly since it is a part of the controller. Notice that the original system and the integrator in the controller may be expressed by the following state and output equations. Ax + Bu, y = Cx d x1 1 0 XI [0 Cx Discuss the controllability and observability of this enlarged system. Note that the transfer function of the original system is C(sI A) B and the transfer function of the integrator is 1/s. CXI X1 #N-DI-BH M-C CH = dt x 0 A X = +