Design of state variable FS2017b mk

Nguyễn Công Phương CONTROL SYSTEM DESIGN The Design of State Variable Feedback Systems Contents I Introduction II Mathematical Models of Systems III State Variable Models IV Feedback Control System Characteristics V The Performance of Feedback Control Systems VI The Stability of Linear Feedback Systems VII The Root Locus Method VIII.Frequency Response Methods IX Stability in the Frequency Domain X The Design of Feedback Control Systems XI The Design of State Variable Feedback Systems XII Robust Control Systems XIII.Digital Control Systems s i tes.google.com/site/ncpdhbkhn The Design of State Variable Feedback Systems Introduction Controllability & Observability Full-State Feedback Control Design Observer Design Integrated Full-State Feedback & Observer Reference Inputs Optimal Control Systems Internal Model Design State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn Introduction u System model xɺ = Ax + Bu x y C yɶ = y − Cxˆ + xˆ ɺ −K xˆ = Axˆ + Bu + Lyɶ − Control law Observer C Compensator Three steps for state variable design: Use a full-state feedback control law Construct an observer to estimate the states that are not directly sensed and available as outputs Connect appropriately the observer to the fullstate feedback control law s i tes.google.com/site/ncpdhbkhn The Design of State Variable Feedback Systems Introduction Controllability & Observability Full-State Feedback Control Design Observer Design Integrated Full-State Feedback & Observer Reference Inputs Optimal Control Systems Internal Model Design State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn Controllability & Observability (1) • A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in a finite time, t0 ≤ t ≤ T • For the system xɺ = Ax + Bu, we can determine whether the system is controllable by examining the algebraic condition: rank[B AB A B … A n −1 B] = n A is n×n & B is nì1 For multi-input systems, B can be n×m, where m is the number of inputs s i tes.google.com/site/ncpdhbkhn Controllability & Observability (2) • A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t0) to any other desired location x(t) in a finite time, t0 ≤ t ≤ T • For a single-input, single-output system, the controllability matrix Pc is: Pc = [B AB A B … A n −1 B] • If the determinant of Pc is nonzero, the system is controllable s i tes.google.com/site/ncpdhbkhn Ex Controllability & Observability (3)   0  xɺ =  0  x + 0  u      −a0 −a1 −a  1  y = [1 0] x + [0] u  A=  −a −a1 0  Pc =  1 − a  Pc = [B AB A 2B … A n−1B ]    0       , B = 0  , AB =   , A 2B =  −a       a − a  −a2  1   −a2  1    −a  → Pc = −1 ≠ a 22 − a1  s i tes.google.com/site/ncpdhbkhn This system is controllable Ex Controllability & Observability (4)  −2  1  xɺ =  x +  u   a −3 0  y = [0 1] x + [0 ] u Pc = [B AB A2 B … A n−1B ] 1 −  Pc = [B AB ] =   a   → Pc = a →a ≠0 s i tes.google.com/site/ncpdhbkhn Controllability & Observability (5) • A system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation history y(t) given the control u(t), t0 ≤ t ≤ T xɺ = Ax + Bu • For the SISO system  ,C is 1×n & x is n×1,  y = Cx  C   CA   we define the observability matrix Po =   ⋮   n−1  CA  • This system is completely observable when the determinant of Po is nonzero s i tes.google.com/site/ncpdhbkhn 10 Optimal Control Systems (6) Ex U (s) X X 1  Y (s)  X1 = s X  sX1 = X s s →   sX = U X = 1U  s  xɺ1 = x2 d  x1  0   x1    → →  = +   u (t ) ↔ xɺ = Ax + Bu    dt  x2  0   x2     xɺ2 = u (t ) u (t ) = −k1 x1 − k x2 R (s ) = U ( s ) X X Y (s) − s s −  xɺ1 = x2 →  xɺ2 = − k1 x1 − k x2 k2  → xɺ = Hx =  − k1  x  − k2  s i tes.google.com/site/ncpdhbkhn k1 56 Optimal Control Systems (7) Ex  xɺ = Hx =  − k1  x  − k2  H P + PH = −I T p P =  11  p12 p12  p 22  U (s) X X1 Y (s) s s R (s ) = U ( s ) X X Y (s) − s s − k2 k1 k1 = 1   −1  0 −1   p11 p12   p11 p12   → + =         1 − k2   p12 p22   p12 p22  − − k2   −1 −2 p12 p11 − p22 − k2 p12  −   ↔ =   p − p − k p p − k p −  11 22 12 12 22     −2 p12 = −1  →  p11 − p22 − k2 p12 =  p − 2k p = −1  12 22 1 → p12 = , p22 = , k2 s i tes.google.com/site/ncpdhbkhn k 22 + p11 = k2 57 Optimal Control Systems (8) Ex 1 p12 = , p22 = , k2 p11 = +2 2k U (s) k 22 J = xT (0)Px(0) R (s ) = U ( s ) X X Y (s) − s s − 1 x(0) =   1 p → J = [1 1]  11  p12 X X1 Y (s) s s k2 k1 p12  1 k22 + k2 + = p11 + p12 + p22 =    p22  1 2k dJ k (2 k + 2) − 2(k 22 + 2k + 4) → = =0 dk (2k ) s i tes.google.com/site/ncpdhbkhn k2 = →  J = 58 Optimal Control Systems (9) Ex U (s) k1 = 1, k = J= k22 X X1 Y (s) s s R (s ) = U ( s ) X X Y (s) − s s − + k2 + 2k k2 22 20 k1 18 16 14 12 10 0.5 1.5 2.5 3.5 4.5 k2 s i tes.google.com/site/ncpdhbkhn 59 Optimal Control Systems (10) Ex 0 xɺ = Hx =  − k 1 x  −k  HT P + PH = −I  p11 P=  p12 p12  p22  U (s) X X1 Y (s) s s R (s ) = U ( s ) X X Y (s) − s s − k k +1 2k + → p12 = , p22 = , p = 11 2k 2k 2k J = x (0)Px (0) k T 1  x(0) =   0  1 → J = 1+ → J = 2k s i tes.google.com/site/ncpdhbkhn k2 60 Optimal Control Systems (11) J = ∫ g (x , u, t ) dt tf J= ∞ ∫ (x T Ix + λuT u)dt u = −Kx →J = = ∫ ∞ ∫ (xT Ix + λ (Kx)T Kx )dt ∞ (xT (I + λ K T K )xdt Q = I + λ KT K →J = ∫ ∞ xT Qxdt T  H P + PH = −Q → T  J = x (0)Px(0) s i tes.google.com/site/ncpdhbkhn 61 Optimal Control Systems (12) Ex  1  0 xɺ =  x + u     0 1  u = −Kx = − [ k U (s) X X1 Y (s) s s  x1  k ]   = − kx1 − kx2  x2   xɺ1 = x2  xɺ1  0   x1    →  = +   (− kx1 − kx2 ) →      xɺ2  0   x2     xɺ2 = − kx1 − kx2  xɺ1   →  =  xɺ2   − k   x1  ↔ xɺ = Hx    − k   x2  s i tes.google.com/site/ncpdhbkhn 62 Optimal Control Systems (13) Ex K = [k k ], 0 H=  −k 1 − k  U (s) X X1 Y (s) s s HT P + PH = −Q  p11 P=  p12 p12  p22   0 k  Q = I + λK K =  + λ   [k   1 k  T − 2kp12  H P + PH =   p11 − kp12 − kp22 T λ k + λk2  k]=   2 λ k + 1  λ k p11 − kp12 − kp22  p12 − 2kp22   −2 kp12 = −λ k −  →  p11 − kp12 − kp 22 = −λ k  2 p − kp = − λ k −1 12 22  s i tes.google.com/site/ncpdhbkhn 63 Optimal Control Systems (14) Ex − 2kp12 = − λ k −   p11 − kp12 − kp 22 = −λ k  2 p − kp = − λ k −1 22  12 2k +  2 p = ( λ k + 1) − λ k  11 2k  λk +  →  p12 = 2k   k +1 λ p = ( k + 1)  22 2 k   p11 P=  p12 p12  p22  J = x (0) Px(0) T U (s) X X1 Y (s) s s → J = (λ k + 1) 2k + − λk2 2k → J = p11 xT (0) = [1 ] s i tes.google.com/site/ncpdhbkhn 64 Optimal Control Systems (15) Ex J= ∫ ∞ U (s) X X1 Y (s) s s ( xT Ix + λ uT u) dtv u = −Kx = − [ k k]x xT (0) = [1 ] → J = (λ k + 1) 2k + − λk2 2k dJ   =  λ −  = → k = dk  k  λ s i tes.google.com/site/ncpdhbkhn 65 The Design of State Variable Feedback Systems Introduction Controllability & Observability Full-State Feedback Control Design Observer Design Integrated Full-State Feedback & Observer Reference Inputs Optimal Control Systems Internal Model Design State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn 66 Internal Model Design xɺ = Ax + Bu   y = Cx R ( s) E − xɺ r = A r x r  r = d r x r K1 s Compensator U − G (s ) x Y (s) K2 r ( n) = α n −1r ( n −1) + α n − r ( n− 2) + ⋯ + α1rɺ + α r r = xr , xɺr = e = y −r → eɺ = yɺ = Cxɺ z = xɺ , w = uɺ  eɺ  0 C   e    → = + w     zɺ  0 A   z  B  w = − K1e − K z → u (t ) = t t ∫ w = −K ∫ e(τ )dτ − K x(t ) s i tes.google.com/site/ncpdhbkhn 67 The Design of State Variable Feedback Systems Introduction Controllability & Observability Full-State Feedback Control Design Observer Design Integrated Full-State Feedback & Observer Reference Inputs Optimal Control Systems Internal Model Design State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn 68 State Variable Design Using Control Design Software (1) Ex  x1  0    d  x2  0 = dt  x3  0     x4  0   x1    −   x2   0.1237  + u 0   x3        100   x4  −1.2621 y = [1 0 0][ x1 x2 x3 x4 ] T • ctrb • obsv • det s i tes.google.com/site/ncpdhbkhn 69 Ex State Variable Design Using Control Design Software (2) Y ( s) = G( s ) = , p1,2 = −1 ± j U (s) s  x1 = y  1 0  → xɺ =  x+  u    0 1   x2 = yɺ P = [ −1 + j −1 − j ] • acker s i tes.google.com/site/ncpdhbkhn 70 ... The Design of Feedback Control Systems XI The Design of State Variable Feedback Systems XII Robust Control Systems XIII.Digital Control Systems s i tes.google.com/site/ncpdhbkhn The Design of State. .. Control Design Observer Design Integrated Full -State Feedback & Observer Reference Inputs Optimal Control Systems Internal Model Design State Variable Design Using Control Design Software s i tes.google.com/site/ncpdhbkhn... tes.google.com/site/ncpdhbkhn → Po = 12 The Design of State Variable Feedback Systems Introduction Controllability & Observability Full -State Feedback Control Design Observer Design Integrated Full -State Feedback & Observer