Fa2004 16.333910 Aircraft State Space Control Can now design a full state feed back controller for the dynamics Asn sn+ bnd with desired poles being at wn=3 and s=0.6=s=-1.8+2.41 d(s)=s2+36s+9 Ksp=place(Asp, Bsp, [roots([1 2*0. 6*3 32])']) e Design controller u 0.0264-2.3463 With full model, could arrange it so phugoid poles remain in the same lace just move the ones associated with the short period mode s=-1.8±2.4i.-0.0033±0.0672i ev=eig(A); lo damp short period, but leave the phugoid where it is P1ist=[ roots([12*.6*33^2]))ev([34],1)]; K1=place(A, B(:, 1), Plist) →2=[02-0025-280890 6
� � � � � � Fall 2004 16.333 9–10 Aircraft State Space Control • Can now design a full state feedback controller for the dynamics: x˙ sp = Aspxsp + Bspδe with desired poles being at ωn = 3 and ζ = 0.6 ⇒ s = −1.8 ± 2.4i φd(s) = s2 + 3.6s + 9 Ksp=place(Asp,Bsp,[roots([1 2*0.6*3 3^2])’]) w • Design controller u = −0.0264 −2.3463 q • With full model, could arrange it so phugoid poles remain in the same place, just move the ones associated with the short period mode s = −1.8 ± 2.4i, −0.0033 ± 0.0672i ev=eig(A); % damp short period, but leave the phugoid where it is Plist=[roots([1 2*.6*3 3^2])’ ev([3 4],1)’]; K1=place(A,B(:,1),Plist) ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ u w ⎥ ⎥ ⎥ ⎦ ⇒ u = 0.0026 −0.0265 −2.3428 0.0363 q θ
Fa2004 16.333911 Can also add the lag dynamics to short period model with 0 included 4 Asp. sp+ Bspde, d +4 4x6+46 Aon B 0 0-4 Add s=-3 to desired pole list P1ist=[ roots([12*.6*33^2])),-.25,-3] A2=[Asp2Bsp2(:,1); zeros(1,3)-4];Bt2=[ zeros(3,1);4] Kt=place(At2, Bt2, Plist) step(ss(At2Bt2*Kt2,Bt2,[0010],0),35) =[0001-34617-491240.5273 q No problem working with larger systems with state space tools Main control issue is finding good"locations for closed-loop poles
� � � � � � � � � � Fall 2004 16.333 9–11 • Can also add the lag dynamics to short period model with θ included x˙ sp = A˜spxsp + ˜ δa δa = 4 δc Bsp e ; e s + 4 e → x˙ δ = −4xδ + 4δe c , δa e = xδ x˙ sp = A˜ ˜ sp Bsp xsp 0 + δc ⇒ e x˙ δ 0 −4 xδ 4 • Add s = −3 to desired pole list Plist=[roots([1 2*.6*3 3^2])’,.25,3]; At2=[Asp2 Bsp2(:,1);zeros(1,3) 4];Bt2=[zeros(3,1);4]; Kt=place(At2,Bt2,Plist); step(ss(At2Bt2*Kt2,Bt2,[0 0 1 0],0),35) ⎡ ⎤ ⎢ ⎢ ⎢ ⎣ w q θ xδ ⎥ ⎥ ⎥ ⎦ u = 0.0011 −3.4617 −4.9124 0.5273 0 5 10 15 20 25 30 35 −0.25 −0.2 −0.15 −0.1 −0.05 0 Step Response Time (sec) θ rads • No problem working with larger systems with state space tools • Main control issue is finding “good” locations for closedloop poles
Fa2004 16.333912 Estimators/observers Problem: So far we have assumed that we have full access to the state a (t) when we designed our controllers Most often all of this information is not available Usually can only feedback information that is developed from the sensors measurements Could try "output feedback Kx→ y Same as the proportional feed back we looked at at the beginning of the root locus work This type of control is very difficult to design in general Alternative approach: Develop a replica of the dynamic system that provides an estimate"of the system states based on the mea sured output of the system N ew pl an 1. Develop estimate of (t)that will be called i(t) 2. Then switch from u =-Ki(t to u=-Ki(t) · Two key questions: How do we find i(t? Will this new plan work
Fall 2004 16.333 9–12 Estimators/Observers • Problem: So far we have assumed that we have full access to the state x(t) when we designed our controllers. – Most often all of this information is not available. • Usually can only feedback information that is developed from the sensors measurements. – Could try “output feedback” u = Kx ⇒ u = Ky ˆ – Same as the proportional feedback we looked at at the beginning of the root locus work. – This type of control is very difficult to design in general. • Alternative approach: Develop a replica of the dynamic system that provides an “estimate” of the system states based on the measured output of the system. • New plan: 1. Develop estimate of x(t) that will be called xˆ(t). 2. Then switch from u = −Kx(t) to u = −Kxˆ(t). • Two key questions: – How do we find xˆ(t)? – Will this new plan work?
Fa2004 16.333913 Estimation Schemes Assume that the system model is of the form AT+ Bu, a(O) unknown y Where 1.A.B, and C are known 2. u(t) is known 3. Measurable outputs are y(t) from CtI Goal: Develop a dynamic system whose state (t)=x(t) for all time t>0. Two primary approaches O )pen-loop Closed-loop
� Fall 2004 16.333 9–13 Estimation Schemes • Assume that the system model is of the form: x˙ = Ax + Bu , x(0) unknown y = Cx where 1. A, B, and C are known. 2. u(t) is known 3. Measurable outputs are y(t) from C = I • Goal: Develop a dynamic system whose state xˆ(t) = x(t) for all time t ≥ 0. Two primary approaches: – Openloop. – Closedloop