Modern Control Systes (ECEG-46) Instructor: Andinet Negash Chapter Lecture 3: State Space, II
Eaples Eaple 5: control o liquid levels: in cheical plants, it is oten necessary to aintain the levels o liquids. A sipliied odel o such a syste is shown below. Deterine the state equation o the syste. 3-Oct-3 Chapter : slide 3; Inst. Andinet N.
Eaples where: q, q i A, A h R, q, h, R rates o the lowo liquid areas o the cross section o tanks liquid levels lowresistance,controlled by valves There are two energy storage eleents in the syste. The syste is thus a second order syste or which the two state variables ay be chosen as the heads o liquid levels in tanks and, respectively. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 3
Eaples It is assued that: q h h R, q h R They are proportional to relative liquid levels and inversely proportional to low resistances. The changes o liquid levels are governed by: dh A dh qi q or A qi q dh Adh q q or A q q 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 4
Eaples The state equation or the syste can be shown to be: The output equation becoes: assuing the outlow rate o tank is the desired output. 3-Oct-3 5 Chapter : slide 3; Inst. Andinet N. q i A A R R A R A R A R A R y
Eaples To deterine the transer unction, Dierentiating the irst set o equations above, we get: dh dh R R Upon substitution: dq dq dh R dq dq dq q i q A R R R dq 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 6
Eaples Eliinating q and its derivative, we arrive at the dierential equation: A A R R dq q q A R d q A R R A R q t q t Take the Laplace transor o both sides and ind the transer unction. dq i 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 7
Eaples Eaple 6: DC otor- One coponent which is oten used in control systes is the DC otor. There are several types o DC otors. We present here only the separately ecited type, because its characteristics present several advantages over others, particularly with regard to linearity. Separately ecited DC otors are distinguished in two categories: those that are controlled by the stator, which are usually called ield-controlled otors; those that are controlled by the rotor, which are usually called arature-controlled otors. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 8
Eaples Motors Controlled by the Stator 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 9
Eaples For siplicity, we ake the ollowing approiations: a. The rotor s current i a (t) is constant, i.e., i a (t)=i a. b. The agnetic lu (t) between the stator and the rotor is given by the linear relation (t)=k i where K is a constant and i (t) is the stator s current. c. The torque T (t) that is developed by the otor is given by the relation T (t)= K i a (t) where K is a constant. The Kirchho s voltage law or the stator network is L di R i v (t) 3-Oct-3 Chapter : slide 3; Inst. Andinet N.
Eaples The rotor s rotational otion is described by the dierential equation: Where J is the torque inertia, B is the coeicient o riction, is θ (t) the angular position or displaceent, and ω (t) is the angular velocity o the otor. Let the state variables be: 3-Oct-3 Chapter : slide 3; Inst. Andinet N. i 3 d t T B d J ), (
Eaples We, then, arrive at: where 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 3 3 3 y v L T I K K J T a tie constant electrical R L T tie constant echanical B J T
Eaples Motors Controlled by the rotor: try it! 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 3
Eaples Follow a siilar procedure to arrive at the ollowing state-space representation. Where: 3-Oct-3 4 Chapter : slide 3; Inst. Andinet N. 3 3 3 y v L T K L K J T a a a b a i the rotor o tie constant is electrical R L T and a constant I K K K a a a i,
Eaples Eaple 7: An inverted pendulu ounted on a otor-driven cart is shown in below. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 5
Eaples This is a odel o the attitude control o a space booster on takeo. (The objective o the attitude control proble is to keep the space booster in a vertical position.) The inverted pendulu is unstable in that it ay all over any tie in any direction unless a suitable control orce is applied. Here we consider only a two-diensional proble in which the pendulu oves only in the plane o the page. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 6
Eaples Assue that the center o gravity o the pendulu rod is at its geoetric center. Obtain a atheatical odel or the syste. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 7
Eaples bd 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 8
Eaples The equation that describes the rotational otion o the stick about its center o gravity is obtained by applying the rotational version o Newton s second law. Suing the oents about the center o gravity o the pendulu, we obtain: d I Vlsin Hl cos ; l l where I r d is the oent o inertia about the c.g. l 3 We net write the equation that describes the horizontal otion o the center o gravity o the stick. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 9
Eaples Applying Newton s second law along the ais yields d l sin H, or l sin cos The equation that describes the vertical otion o the center o gravity o the stick is obtained by applying Newton s second law along the y ais. d l l cos V g, cos sin or H V g 3-Oct-3 Chapter : slide 3; Inst. Andinet N.
Eaples Finally, we apply Newton s second law to the cart to get selecting the state variables as: 3-Oct-3 Chapter : slide 3; Inst. Andinet N. c H u d M 4 3
Eaples Show that the ollowing state-space equation is achieved: where: 3-Oct-3 Chapter : slide 3; Inst. Andinet N. c u a a la l a a al ag la l la g 4 4 3 cos 4 / 3 / 3 4 cos / 3 4 cos cos 4 / 3 / 3 4 sin / sin cos / 3 4 / sin sin M a
Eaples For the sae but linearized version o the inverted pendulu proble, see Modern Control Engineering, Katsuhiko Ogata under the section: Matheatical Modeling o Mechanical Systes. A related proble can also be ound on Modern Control Systes, Richard C. Dor and Robert H. Bishop, Twelth Edition, page 86. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 3
Correlation b/n SS and TF In what ollows we shall show how to derive the transer unction o a single-input, single-output syste ro the state-space equations. Let us consider the syste whose transer unction is given by: Y s G s Us This syste ay be represented in state space by the ollowing equations: X y A C Bu Du 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 4
Correlation b/n SS and TF The Laplace transors o the state-space equations are given by. sx( s) () Y ( s) A( s) BU( s) CX( s) DU( s) Since the transer unction was previously deined as the ratio o the Laplace transor o the output to the Laplace transor o the input when the initial conditions were zero, sx( s) A( s) BU( s), or si A X( s) BU( s) 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 5
Correlation b/n SS and TF Solving or X(s), we have: By substituting into the output equation, we get Y ( s) G(s) X( s) C si A B si A B D C si A BU( s) This is the transer-unction epression o the syste in ters o A, B, C, and D. D U( s), or G(s) can be written as Q( s) G( s) s I A, whereq(s) is a polynoial in s. 3-Oct-3 Chapter : slide 3; Inst. Andinet N. 6
Correlation b/n SS and TF Consider again the ass-spring-daper syste. State-space equations or the syste are given by Equations. We shall obtain the transer unction or the syste ro the state-space equations. 3-Oct-3 7 Chapter : slide 3; Inst. Andinet N. ) ( ) ( ) ( ) ( ) ( t t t B K t t ) ( ) ( ) ( t t t y
Correlation b/n SS and TF By substituting A, B, C, and D 3-Oct-3 8 Chapter : slide 3; Inst. Andinet N. k bs s b s k s b k s s t y ) (