• Measurement Signals . The signals represent measurements or signals that are directly available to the controller K . Measurements may include a portion of or all of the plant state variables, measurable plant “outputs,” measurable control signals, measurable exogenous signals, etc. In practice, we typically have more exogenous signals than measurements (i.e., n w ≥ n y ). Generally, the more independent measurements we have the better—since, in theory, more useful information can be extracted. Comment 30.1 (Toward a Separation Principle) It is natural to associate the controls u with the regulated signals z . One might argue that the pair implicitly defines a regulation or control problem. This is analogous to the situation addressed in classical LQR problems. In such problems, one trades off control action (size) versus speed of regulation. Similarly, it is natural to associate the exogenous signals w with the measurements y . One might argue that the pair implicitly defines an information extraction or estimation problem. This is analogous to the situation addressed in classical KBF problems. In such problems, one trades off sensor cost (or immunity to noise) versus speed of estimate construction. Such associations suggest that just as in classical LQG problems, our surprisingly general structure may give rise to a natural separation principle. Indeed, this will be the case for the so-called H 2 output feedback problem that we consider. General 2 Optimization Problem The so-called general H 2 optimization problem may be stated as follows: • Find a proper real-rational (finite dimensional) controller K that internally stabilizes G such that the H 2 norm of the closed loop system transfer function matrix T wz ( K ) is minimized: (30.1) where (30.2) (30.3) (30.4) and f is the impulse response matrix associated with the transfer function matrix F . Comment 30.2 (Use of Two Norm: Wide Band Exogenous Signals) Noting that the two norm measures the energy of the response to an impulse and noting that the transform of unit Dirac delta function δ is unity, it follows that the two norm is appropriate when the exogenous signals w are wide band in nature. This can always be justified by introducing appropriate (low pass) filters within G . It should be noted that these ideas have stochastic interpretations as well. Instead of unit delta functions, one instead deals with white noise with unit intensity. Comment 30.3 (Control and Estimation Problems) Although we are seeking an H 2 optimal controller, it must be noted that the generalized plant framework will enable the design of state estimators as well as dynamic and constant gain control laws. Given the above problem statement, it is appropriate to recall the following elementary result: y R n y ∈ HH min K ||T wz K()|| H 2 ||F|| H 2 def = 1 2p trace F H j ω ()Fjw(){}wd ∞– ∞ ∫ trace f H t()ft(){}td 0 ∞ ∫ = || f || L 2 R + () = 0066_Frame_C30 Page 3 Thursday, January 10, 2002 4:43 PM ©2002 CRC Press LLC • Measurement Signals . The signals represent measurements or signals that are directly available to the controller K . Measurements may include a portion of or all of the plant state variables, measurable plant “outputs,” measurable control signals, measurable exogenous signals, etc. In practice, we typically have more exogenous signals than measurements (i.e., n w ≥ n y ). Generally, the more independent measurements we have the better—since, in theory, more useful information can be extracted. Comment 30.1 (Toward a Separation Principle) It is natural to associate the controls u with the regulated signals z . One might argue that the pair implicitly defines a regulation or control problem. This is analogous to the situation addressed in classical LQR problems. In such problems, one trades off control action (size) versus speed of regulation. Similarly, it is natural to associate the exogenous signals w with the measurements y . One might argue that the pair implicitly defines an information extraction or estimation problem. This is analogous to the situation addressed in classical KBF problems. In such problems, one trades off sensor cost (or immunity to noise) versus speed of estimate construction. Such associations suggest that just as in classical LQG problems, our surprisingly general structure may give rise to a natural separation principle. Indeed, this will be the case for the so-called H 2 output feedback problem that we consider. General 2 Optimization Problem The so-called general H 2 optimization problem may be stated as follows: • Find a proper real-rational (finite dimensional) controller K that internally stabilizes G such that the H 2 norm of the closed loop system transfer function matrix T wz ( K ) is minimized: (30.1) where (30.2) (30.3) (30.4) and f is the impulse response matrix associated with the transfer function matrix F . Comment 30.2 (Use of Two Norm: Wide Band Exogenous Signals) Noting that the two norm measures the energy of the response to an impulse and noting that the transform of unit Dirac delta function δ is unity, it follows that the two norm is appropriate when the exogenous signals w are wide band in nature. This can always be justified by introducing appropriate (low pass) filters within G . It should be noted that these ideas have stochastic interpretations as well. Instead of unit delta functions, one instead deals with white noise with unit intensity. Comment 30.3 (Control and Estimation Problems) Although we are seeking an H 2 optimal controller, it must be noted that the generalized plant framework will enable the design of state estimators as well as dynamic and constant gain control laws. Given the above problem statement, it is appropriate to recall the following elementary result: y R n y ∈ HH min K ||T wz K()|| H 2 ||F|| H 2 def = 1 2p trace F H j ω ()Fjw(){}wd ∞– ∞ ∫ trace f H t()ft(){}td 0 ∞ ∫ = || f || L 2 R + () = 0066_Frame_C30 Page 3 Thursday, January 10, 2002 4:43 PM ©2002 CRC Press LLC . F H j ω ()Fjw(){}wd ∞– ∞ ∫ trace f H t()ft(){}td 0 ∞ ∫ = || f || L 2 R + () = 0066_Frame_C30 Page 3 Thursday, January 10, 2002 4: 43 PM 2002 CRC Press LLC • Measurement Signals . The signals represent measurements. F H j ω ()Fjw(){}wd ∞– ∞ ∫ trace f H t()ft(){}td 0 ∞ ∫ = || f || L 2 R + () = 0066_Frame_C30 Page 3 Thursday, January 10, 2002 4: 43 PM 2002 CRC Press LLC . ( K ) is minimized: (30 .1) where (30 .2) (30 .3) (30 .4) and f is the impulse response matrix associated with the transfer function matrix F . Comment 30 .2 (Use of Two Norm: Wide