Power Systems Control Prof. Wonhee Kim Ch.3. Controller Design in Time Domain
Stability in State Space Equation: State Feeback t A t B t t C t D t x x u y x u u t Kx t t A t BK t A BK x t x x x K shoul be chosen such that (A-BK) is Hurwitz! 2
Observer Design System x Ax Bu y Cx Observer esign - Aim: ˆx x - Estimation error efine: x x xˆ - Observer yˆ Cxˆ - Estimation error ynamics xˆ Axˆ Bu L y yˆ x x xˆ Ax Bu Axˆ Bu LC x xˆ A LC x L shoul be chosen such that (A-LC) is Hurwitz! 3
Separation Principle System x Ax Bu y Cx Control input using estimate state u Kxˆ System with control input Close-loop System x Ax BKxˆ A BK x BKx x A BK x BKx x A LC x K an L can be separatively esigne 4
Controllability an Observability 5
Controllability an Observability Controllability: for any esire eigenvalues, K exists such that (A-BK) has the esire eigenvalues n Controllability matrix C B BA BA has n rank. con Observability: for any esire eigenvalues, L exists such that (A-LC) has the esire eigenvalues T T T T T Controllability matrix n T O ob C A C A C has n rank. O ob C CA n CA 6
State Feeback Controller Design for Reference Tracking Regulation) xt Ax t Bu t yt Cxt xt Ax t Bu t yt Cxt u t Kxt Jr t Y s T s C si A B U s t A t BK t BKr t A BK x t BJr t x x x Y s T s C si A BK B J R s Using final value theorem, steay-state response with step reference shoul be. where K c is a c gain for C(sI-(A-BK)) - B y lim st s R s limt s lim JK s0 s0 s0 J K c c 7
Tracking Controller Design. System x x x 2 x 2 3 xn ax anxn bu 2. Controller esign - Aim: r x - Tracking error efine: e r x en r x - Tracking Error ynamics e r x e 2 n n en r xn r ax anxn bu - Controller n n u r a x a x a x k e k e b 2 2 n n n n n e 2 n e e k e k e n n 8
Tracking Controller Design Example. System x x 2 x a x a x bu 2 2 2 2. Controller esign - Aim: x r - Tracking error efine: - Tracking Error ynamics e r x e 2 2 2 2 2 e r x e r x 2 2 e r x r a x a x bu 2 2 2 2 - Controller u r ax a2x2 ke k2e2 b Control gain k, k2 shoul be esigne such that Ac = [0 ; k k2] is Hurwitz e e e k e k e 9
Tracking Controller II Design. System x Ax Bu y Cx 2. Desire state ynamics: u is esigne for the esire state x Ax Bu y Cx - Tracking error: e x x - Aim: e x x 0 as t - Tracking error ynamics: e x x Ae Bu Bu - Controller u u Ke Ax Bu Ax Bu e Ae Bu B u Ke Ae Bu Bu BKe A BK e 0
Tracking Controller II Design: DC Motor. System B Ki J i Kt Ri v L 2. Desire constant velocity ω, i an v : K 0 B Ki, 0 Kt Ri v KR i, v Kt Ri Kt J L B B 3. Desire state ynamics: u is esigne for the esire state - Tracking error: e x x - Aim: e x x 0 as t - Tracking error ynamics: e x x - Controller: u u Ke x Ax Bu y Ax Bu Ax Bu Ae Bu Bu K Ri k k i i t 2 Cx e Ae Bu B u Ke Ae Bu Bu BKe A BK e
Control Algorithm Implementation Control Boar DC Motor System i K B ˆ ˆ v K Ri k k i i t 2 ˆ ˆ, i v Input voltage (v) B Ki J i Kt Ri v L y ˆ B ˆ Kiˆ l y yˆ J ˆ i K ˆ ˆ 2 ˆ t Ri v l y y L yˆ ˆ Velocity feeback (y=ω) 2
Control Algorithm Implementation Control Boar DC Motor System Input voltage (v) DC Motor + Encoer Velocity feeback (y=ω) 3
Control Algorithm Implementation Control Boar DC Motor System Input voltage (v) DC Motor + Encoer Loa isturbance Velocity feeback (y=ω) Loa isturbance may egrae control performance of DC motor! 4
Tracking Controller: Position Control of DC Motor DC motor with unknown loa isturbance, for position control: Not normal form! Note that the isturbance can inclue B Ki J the parameter uncertainties as well as i K Ri v the external isturbance Definition of Acceleration state variable With α, DC motor moel: L t B Ki J B Ki J 2 B BK B KKt KR K i i v 2 2 2 J J J JL JL JL Now DC motor moel becomes normal form, thus, tracking controller can be applie! 5
Extene State Observer Design: DC Motor DC motor with unknown loa isturbance, for position control B Ki J i Kt Ri v L Extene state with assumption that is constant B Ki J i Kt Ri v L 0 Extene state observer to estimate full state an isturbance using position feeback ˆ ˆ l ˆ 4 ˆ B ˆ Kiˆ ˆ l ˆ 2 J ˆ i K ˆ ˆ 3 ˆ t Ri v l L ˆ l ˆ 6
Extene State Observer Design: DC Motor Estimation error Estimation error ynamics ˆ, ˆ, i i iˆ, ˆ l 4 B ˆ Kiˆ ˆ l2 J i K ˆ ˆ t Ri l3 L l l 0 0 B K l2 J J J i Kt R i l3 0 L L l4 0 0 0 A l, l 2, l 3 an l 4 shoul be chosen such that A is Hurwitz! 7
Experimental Results Position control with normal observer (without isturbance rejection) Position control with ESO (with isturbance rejection) 8