Fall 2001 16.3115-7 LQR Notes 1. The state cost was written using the output y y, but that does not need to be the case e We are free to define a new system output z=Cza that is not based on a physical sensor measurement LQR a(t)(C2C2)c(t)+ru(t) 2] dt Selection of z used to isolate the system states you are most concerned about. and thus would like to be regulated to zero 2 Note what happens as r a oo- high control cost case a()a(-s+rb(s)b(s)=0= asa(-s)=0 So the n closed-loop poles are Stable roots of the open-loop system(already in the LHP. Reflection about the jw-axis of the unstable open-loop poles 3 Note what happens as rm0-low control cost case alsa-s)+r b(s b(s=0= b(s)b(-s=0 sume order o of b(s)b(s)is 2m 2n So the n closed-loop poles go to The m finite zeros of the system that are in the Lhp(or the reflections of the systems zeros in the RHP) The system zeros at infinity(there are n-m of these)Fall 2001 16.31 15—7 LQR Notes 1. The state cost was written using the output yT y, but that does not need to be the case. • We are free to define a new system output z = Czx that is not based on a physical sensor measurement. ⇒ JLQR = Z ∞ 0 £ xT (t)(CT z Cz)x(t) + r u(t) 2 ¤ dt • Selection of z used to isolate the system states you are most concerned about, and thus would like to be regulated to “zero”. 2. Note what happens as r ; ∞ — high control cost case a(s)a(−s) + r−1 b(s)b(−s)=0 ⇒ a(s)a(−s) = 0 • So the n closed-loop poles are: — Stable roots of the open-loop system (already in the LHP.) — Reflection about the jω-axis of the unstable open-loop poles. 3. Note what happens as r ; 0 — low control cost case a(s)a(−s) + r−1 b(s)b(−s)=0 ⇒ b(s)b(−s) = 0 • Assume order of b(s)b(−s) is 2m < 2n • So the n closed-loop poles go to: — The m finite zeros of the system that are in the LHP (or the reflections of the systems zeros in the RHP). — The system zeros at infinity (there are n − m of these)