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Nonlinear Control Theory Introduction to Dynamical Systems Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline o Ordinary Differential Equations o Existence amp uniqueness 0 Continuous dependence on parameters o Invariant sets nonwandering sets limit sets 0 Lyapunov Stability o Autonomous systems 0 Basic stability theorems 0 Stable unstable amp center manifolds Basics of Nonlinear ODE s Dynamical Systems d tfxtt xeR teR nonautonomous ixtfxt xeR teR autonomous A solution on a time interval t e to tl is a function xt 10 II gt R that satis es the ode Vector Fields and Flow 0 We can visualize an individual solution as a graph xl l gt R 0 For autonomous systems it is convenient to think of f x as a vector eld on Rquot f x assigns a vector to each point in R As I varies a solution xl traces a path through R tangent to the eldf 0 These curves are often called trajectories or orbits o The collection of all trajectories in R is called the ow of the vector eld f x Auto at Constant Speed Notice the three i equilibria Van der Pol Damped Pendulum x1 x2 X2 x22 sinx1 Lipschitz Condition The existence and uniqueness of solutions depend on properties of the function f In many applications f x t has continuous derivatives in x We relax this we require that f is Lipschitz in x Def f R gt R is locally Lipschitz on an open subset D C Rquot if each point x0 6 D has a neighborhood U 0 such that fx fxoll s Lllx xOII for some constant L and all x 6 U0 Note C 0 continuous functions need not be Lipschitz C1 functions always are Lipschitz Condition Intuitively a Lipschitz continuous function is limited in how fast it can change a line joining any two points on the graph of this function will never have a slope steeper than its Lipschitz constant L The mean value theorem can be used to prove that any differentiable function with bounded derivative is Lipschitz continuous with the Lipschitz constant being the largest magnitude of the derivative Examples Lipschitz Y EEMMMSM 4ng Local Existence amp Uniqueness Proposition Local Existence and Uniqueness Let f x I be piecewise continuous in l and satisfy the Lipschitz condition llf x I f ya 2 S Lllx yll forallxyeBr xeRquot x x0 lt r and all ZEZOZ1 Then there exists 5 gt 0 such that the differential equation with initial condition at fxt xt0 x0 eBr has a unique solution over ZOJO 5 The Flow of a Vector Field X fx xt0 x0 gt xx0 t this notation indicates 39the solution of the ode that passes through x0 at t 039 More generally let Px t denote the solution that passes through x at t 0 The function W R x R gt Rquot satis es 6 11 xt 2 fLPxt LPx0 x T is called the ow or ow function of the vector eld f Example Flow of a Linear Vector Field x Ax 6Tb A Pxt 2 Pxt eA x Example 0 1 0 x1 cos t x2 sint xeR3 A 1 0 0 Pxt xzcost xlsint 0 0 1 X2 e x3 Invariant Set A set of points S C R is invariant With respect to the vector eld f if traj ectories beginning in S remain in S both forward and backward in time Examples of invariant sets any entire trajectory equilibrium points limit cycles collections of entire trajectories Example Invariant Set d x1 0 1 1 x 1 0 0 dt 2 2 x3 0 1 x3 each ofthe three trajectories shown are invariant sets the X1X2 plane is an invariant set x1 Limit Points amp Sets A point q e R is called an wlimit point of the trajectory Pt p if there exists a sequence of time values tk gt oo such that Wm q q is said to be an alimit point of I t p if there exists a sequence of time values tk gt oo such that 131223016 p q The set of all wlimit points of the trajectory through p is the wlimit set and the set of all alimit points is the alimit set For an example see invariant set Introduction to Lyapunov Stability Analysis Lyapunov Stability xfx f0 0 f D gt R locally Lipschitz The origin is 0 a stable equilibrium point if for each 8 gt 0 there is a 5 8 gt 0 such that we lt 5 3 MWquot lt a W gt 0 o mama if it is not stable and 0 aswipimic3133 stable if 5 can be chosen such that quotx0 lt 5 gt gt 0 Two Simple Results The origin is asymptotically stable only if it is isolated The origin of a linear system X Ax is stable if and only if He S N lt 00 W gt 0 It is asymptotically stable if and only if in addition A H6 t gt0t oo Example Nonisolated Equilibria x1 x2 All points on the x1 axis are 562 IX2 I x1 x2 equilibrium points 9 K X1 Positive Definite Functions A function V Rquot gt R is said to be 0 positive de nite if V0 0 and Vx gt 0 x 7 0 0 positive semide nite if V0 0 and Vx 2 0 x 7 0 0 negative semi de nite if Vx is positive semi de nite 0 radially unbounded if Vx gt 00 as gt 00 For a quadratic form V x ITQX Q QT the following are equivalent 9 is positive definite 0 the eigenvalues of Q are positive C the pi ineipal minors of Q are positive Lyapunov Stability Theorem V x is called a Lyapunov function relative to the ow of x f x if it is positive de nite and nonincreasing with respect to the ow VOO Vxgt Ofor xiO Theorem Ifthere exists a Lyapunov function on some neighborhood D of the origin then the origin is stable If V is negative de nite on D then it is asymptotically stable Example Rotating Rigid Body x y 2 body axes wxayaz angular velocities in body coord39s diag1x1ylz1x 21y 212gt0 inertia matrix 12 1y wzwy awzwy I note abc gt 0 wxwz baxaz I y 12 wxz I IX Z wyz I Z J 1x Rigid Body Cont d Note that a state 5x 5y 072 is an equilibrium point if any two of the angular velocity components are zero ie the wxayaz axes are all equilibrium points Consider a point 5x0 0 Shift wx gt wx 5x ax awzwy my 2 bax a7xaz 02 caxaxay Rigid Body Cont d Energy does not work for Q at 0 Obvious So how do we nd Lyapunov function We want V0000 Vaxayazgt 0 if wxayaz 000 and wxayazeD V s 0 Lets look at all functions that satisfy V 0ie that satisfy the pde 2 awzwy b a 5xaz Cwx axay 0 All solutions take the form bag2 2waa7x aw cax2 Zowxaz awz2 2a 2a swag5 VaxayazcAbBcA bB2 2 wx2cwy2bwzzjhot a Rigid Body Cont d Clearly VO O V gt O on a neighborhood D of O V 0 gt spin about xaXis is stability This is one approach to finding candidate Lyapunov functions The first order PDE usually has many solutions The method is connected to traditional first integral methods to the study of stability in mechanics 0 Same method can be used to prove stability for spin about 2 axis but spin about yaxis is unstable why First Integrals De nition A rst integral of the differential equation 5c f x t is a scalar function 0x t that is constant along trajectories ie 6 xt 6 xt 0xt TfxtT Observation For simplicity consider the autonomous case 5c f Suppose 01x is a rst integral EO and 02 x an x are arbitrary independent functions on a neighborhhod 0f the point x0 ie lt0195 det3 3 0 a x lt0 x Then we can de ne coordinate transformation x gt 2 Via 6 x z xz39 fx 3 2391 0321 Econstant xq z The problem has been reduced to solving n l differential equations Chetaev s Method Consider the system of equations xfxt f0t 0 We Wish to study the stability of the equilibrium point x 0 Obviously if 0 xt is a first integral and it is also a positive definite function then V x t 0 x t establishes stability But suppose 0 x t is not positive definite Suppose the system has k first integrals 01 xtgpk xt such that at 0t 0 Chetaev suggested the construction of Lyapunov functions of the form VW 211 am xat ELM xi LaSalle Invariance Theorem Theorem Suppose V Rquot gt R is C1 and let QC denote a component of the region x e R Vx lt 0 Suppose QC is bounded and Within QC Vx S 0 Let E be the set of points Within QC where Vx 0 Let M be the largest invariant set Within E 3 every sol39n beginning in QC tends to M as t oo Example LaSalle s Theorem Lagrangian Systems d 6Lx39c 6Lx39c T E 65c 6x Q x e R generalized coordinates 5c 2 dx dt generalized velocities L R2 gt R is the Lagrangian Lx39c Tx39c Ux kinetic energy Tx39c XTM xx total energy Vx Tx Ux Example 1 2 1 xl2 2 x1x2Tx72 U 9 1 32 x2 1x122 239 4 72 0 x22 x12 x1 V V S 0 f 0 x 2 1x12 x 0x2 orcgt Notice that o the level sets are unbounded for V x constant 21 0 V x is not radially unbounded Chetaev Instability Theorem Let D be a neighborhood of the origin Suppose there is a function Vx D gtR and a setD1 CD such that 1 Vx is C1 onD 2 the origin belongs to the boundary of D1 6D1 3 Vx gt 0 V39x gt 0 on D 4 on the boundary ofD1 inside D ie on 6D1 nD Vx 0 Then the origin is unstable Example Rigid Body Cont d Consider the rigid body with spin about the yaXis intermediate inertia 5 0 5y 0T 03x 2 awz my ay Shifted equations by baxaz bl cay 6ywx Attempts to prove stability fails So try to prove instability Consider Vaxa a aa y z x z LetBr axa a w2 a2a2 ltr2 andD1a a 0 EB la gtOa gt0 y I X y 2 x y z r x z Z sothatVgtOonD1andVOon6D1 Vaw12ltwy 5ycay aya wy ayaaz2 ca We can take r2 lt 0 in which case V gt O on D1 3 instability Stability of Linear Systems Consider the linear system at Ax Choose V x xTPx gt Vx xT ATPPAx 2 xTQx a if their exists a positive de nite pair of matrices P Q that satisfy AT P PA Q Lyapunov equation the origin is asymptotically stable b if P has at least one negative eigenvalue and Q gt 0 the origin is unstable c if the origin is stable then for any Q gt 0 there is a unique solution P gt 0 of the Lyapunov equation Second Order Systems Consider the system MxCxKx0MTMgt0CTCgt0KTKgt0 Exx xTMx xTKx d EEO39Qx xTMx39 xTKx xT Cx Kx xTKx d Exx xTCx dt Some interesting generalizations 1C20 2CT C 3KT K The antisymmetric terms correspond to circulatory forces transfer conductances in power systems they are non conservative The antisymmetric terms correspond to gyroscope forces they are conservative Example 0 Assume uniform damping 0 Assume e0 o Designate Gen 1 as swing bus 0 Eliminate internal bus 4 9391 we 2A3 b13sin61 blzsin61 62 9392 we 2M2 2312 511161 62 b23 sin62 6152 5162 253 512Apl PzPIAPz 2133 131 Example Cont d o This is a Lagrangian system with U6162 A19161 AP262 b13 cos61 b12 cos61 62 b23 cos62 Tw1aw2w12w22a Q701 702 0 To study stability choose total energy as Lyapunov function V Ta1a2U9192 V a12 a22 S 0 Note T00 0 and Ta1a2 gt 0 Va1a2 7t 0 3 Equilibria corresponding to U 91 92 a local minimum are stable G V Arononvich and N A Kartvelishvili quotApplication of Stability Theory to Static and Dynamic Stability Problems of Power Systemsquot presented at Second Allunion Conference on Theoretical and Applied mechanics Moscow 1965 Example Cont d Since 7r 3 61 lt 7r and 7r 3 62 lt 7r we should consider U6162 as a function on atorus U J gt R Q 4N wsww quot1212552 14 Q a a o w u 0 q I O 393 Example Cont d iziiz ia quotquot Z 5 l Q 9 h gnf Q QQQ ss sws W Wzomw s a Qs mxm3om 003d 7 to to 0 5 quotgov o i a wmw u I540 0 00 O 1Q ail5 go 2 3 amp 535m quotd m3om no mx hh hth39hht 00 03 t 0 O i it h h h h a hang aw w 2 Q D Q iguana 33 M030 m HH Imuwmw w I II PM Ht 5 mumvml Ta Variable Structure Control Mechanical Systems Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 Mechanical Systems 0 Stabilization o Trajectory Tracking 0 Path Tracking 0 Example wheelset Mechanical Systems Stabilization q Vqp kinematics MqPFPQGPqu dynamics Ak uil lg39silihll l Vq Gp q are square amp invertible F p q 0 Prniolem Stabilize the equilibrium point p 0 q 0 Step 1 Design controller for kinematics try Lyapunov LQqTQqL qTQVQp choose p 7V7 qQq 3 L iqTQVqVT qQq lt 0 for qi 0 Mechanical Systems 2 Step 2 Design switching controller with surface spq p 7 VT qQq TakeL 7sTRs sL39 p qQqq L STRM71qGpquSTRCDpq u iUsgnsx sx GTM 1Rp7VT qQq Another choice for control xi LSTRM71qGpquCDpq Let u um udu L STRM 1qGpqum umCDpq Choose um 7G391MCD um 7 Usgns s GTMquotRp7VT qQq Example Double Pendulum Vq17 M 7 23mmi2 m22222mic0562 22m222cos z q 7 22m2222 c0562 gm7 7g m sin 7g m sin 66 T Fm quota 2 H 2 832 39251 5 2T2 Mechanical Systems Path Tracking q Vqp kinematics MW FPaqGpqu dynamics qEQNRgJaEfPNRquot Given a path qg RR Problem nd u t such that qt qg Example Auto q6 X Y pm vx H 0 1 0 Body mmc V 0 c056 ism v 0 sin cos 6 39 J o o M 0 m 0 0 0 m y rame yzphng Vx V X 211K 0 Z 5 0 0 2 0 F V a 2K 0 0 AF imvxm4K 72aibx Vx vx f Qpnce hume X AutoPath following pulh We can control vx with F so set vx must and drop eq evy dmm J 2abKV i 2112 b2x 2ax5 memm Vx vx V 77W imvxm4KV i 2aibx 2K5 Vx vx Step 1 Choose a 7 15 B 712 in sliding Y 1 Step 2 Chooses7vy 42 damii dvy Ze7mD mm mm x R x H H H b w R x Q Q R V S b k W LL u A a d 65 h E 39 3 LL 391 8 a O 94 h quot A Mquot M mquot 1 quotNquot pua39 v39v o39t quotquotquot quot iuulv quot quot39quotquotquotquotquotquot mww o39ow v 0991 E o g w 39 390 039 039 A unom www WMquot 0 WM Wa l 39 Gradient Tracking Obstacle Avoidance Create an obstacle security circle centered at D with radius R R RD 1 H riDcosgp rrZDcosgDJr D2 14 U w v r v Wheelset Setup x vX cos 9 y VX Sin 9 body xcdfi alne 99 MfX NvxaF J KVX aw T 5acefianw Mmin ltM ltMmax Jmin ltJ ltJmax Nvxalt Nmax X KvxaltKmax Wheelset Sliding Mode For perfect tracking With Ucy1n bcZ yz we need speed vx Vxy direction 5 arctaHEUyUJUXUJJ Switching surfaces slvxiv sZ ABE 8 a 97 arctanEUV 8 1 s2 cwL97 arctan yj What is the motion like in sliding Solve kinematics along With vx V a i aiarctant yn Wheelset Reaching Vslzszz V SISI szsZ FiN sls1 sl VXVX c056v v sm6 M V X 7JTK w vx ycoseixsin6 szszsZ T x y F7Frmxsgns1 T7TmaxsgnsZ Wheelset Simulations Y nus 02 m 01 Dim J 702 701 01 02 I x 701 01 V y 1xy 390392 In all cases 600 Wheelset Simulations in nus Controllability Observability amp Local Decompositions Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 Distributions o Controllability o Controllability Distributions o Controllability Rank Condition 0 Examples 0 Observability o Observability Codistributions o Observability Rank Condition 0 Examples 0 Local Decompositions Distributions V1 vr is a set of vector elds on M AP span vp1vrp is a subspace of 7MP Definition A smooth distribution A on M is a map which assigns to each point peM a subspace of the tangent space to M at p APCTMp such that AP is the span of a set of smooth vector fields v1vr evaluated at p We write Aspanv1vr Definition An integral submanifold of a set of vector fields v1vr is a submanifold NCM whose tangent space TNp is spanned by v1pvrp for each peN The set of vector fields is completely integrable if through every point peM there passes an integral submanifold lnvolutive Distributions Definition A system of smooth vector fields v1vr on M is in involution if there exist smooth real valued functions ckiip peM and ijk lr such that for each ij r ij VIVj chvk kl Proposition Froebenius Let v1vr be an involutive system of vector fields with dim spanv1vrk on M Then the system is integrable with all integral manifolds of dimension k Proposition Hermann Let v1vr be a system of smooth vector fields on M Then the system is integrable if and only if it is in involution Example y 22x MR3 Aspanvw v x w 2yz 0 221 x2 y2 vw E 0 so the distribution A is completely integrable The distribution is singular because dimA 2 everywhere excth on the zaXis x 032 0 and on the circle x2 y2 212 0 where dimA 1 The z axis and the circle are onedimensional integral manifolds All others are the tori To xyzeR3 x2 yz12x2 y2 z2 1c gt2 Example Invariant Distributions Definition A distribution A V1V on M is invariant with respect to a vector field f on M if the Lie bracket fai for each 139 1 r is a vector field of A Notation fA spanf7vi7i 1r that A is invariant with respect to f may be stated AC A In general ASpanfvi7i1r397r SpanV1Vw vla vr lnvolutive Closure 1 0 Problem 1 find the smallest distribution with the following properties 0 It is nonsingular o It contains a given distribution A It is involutive o It is invariant wrt a given set of vector fields T1 Tq ltTl239q Agt Algorithm Algorithm for Problem 1 A0 A Ak Ak l 2171 9 Ak 1 stop When Ak 2 AH Affine Systems at fx mm fx gi ltxgtui yhx xER yERpuERm Controllability 0 xi is Ureachable from XO ifgiven a neighborhood U of XO containing Xf there exists tgt0 and uz on 01 such that XO goes to Xalong a trajectory contained entirely in U o The system is locally reachable from XO if for each neighborhood U of XO the set of states Ureachable from XO contains a neighborhood of X0 If the reachable set contains merely an open set the system is locally weakly reachable from XO o The system is locally weakly controllable if it is locally weakly reachable from every initial state Controllability Distributions ixltflgpgmkpanflgpagmgt AC0 ltfg1gm spanglgmgt AC AC0 satisfy ACO spanfgAC x a regular point of AC0 span f 3 AC0 x span fx AC x if AC0 and AC0 span f are of constant dim then mAC mA l Controllability Rank Condition Proposition A necessary and suf cient condition for the system to be locally weakly controllable is dim AC x0 n on e R A necessary and suf cient condition for the system to be locally controllable is dim AC0 x0 n on e R Example Linear System Controllability 5cAxBuxER uERm fltxAxgiltxbii1m abi Ax a bi Abl 8x 8x Ax bi 1911910 AxAxo Example Linear System Con nued A0 span B A1 span B AB Ak span B AB Ak lB CH Thm gt AC0 span B AB A HB A0 span AxB AC span AxB ABA 1B Example Bilinear System 0 0 14 1 2 4 x 0 0 0 x 0 2 0 xu 0 0 19 0 0 3 1 0 0 AC 2 span 0 1 0 gt weakly locally controllable 0 AC span 0 2x2 gtnot locally controllable Example Parking bodyixed ame t i u1 v u2 a drive y x cos 6 0 space ame i y 2 0 ul x dt 15 sin 6 0 242 6 0 1 b steer Example Parking new directions from Lie bracket sin6 0056 cosQ 0 wriggle steer drive 2 sin slide 2 wriggle drive 2 cos 0 Example Parking implementatign More Controllability Distributions AL spanfadj gi13i SmOSkSn l ALO spanadj gi13i SmOSkSn 1 adv0 w w advk vadf1w weak local controllabilily ltgt AC n lt AL n TT TT TT local controllabilily ltgt AC0 n lt ALO n Example Linear Systems Revisited fx Ax Gx B AC AL spanAxBABA lB AC0 ALO span BAB An lB Controllability Hierarchy AC ltfg1gmspanfg1gmgt AGO ltfg1gmlspang1gmgt AL spanfadfgi1 i Sm0 k n l AL0 spanad39gi1Si Sm0 k n 1 weak local controllabilily C dimAC x0 n C dimAL x0 n TT TT local controllabilily C dimACO x0 n C dimALO x0 n H 11 linear controllabilily C dimB AB Aquot IBn Observability 0 Consider an open set U in R X1X2 are Udistinguishable if there exists a control ut whose trajectories from both X1X2 remain in U such that ytX1u i ytX2u OthenNise they are U indistinguishable o The system is strongly locally observable at X0 if for every nbhd U of every state in U other than X0 is Udistinguishable from X0 It is locally observable at X0 if there exists a nbhd Wof X0 such that for every nbhd U ofx0 contained in Wevery state in U other than X0 is Udistinguishable from X0 0 The system is strongly locally observable if it is strongly locally observable at X0 for every X0 in R Observability Codistributions 90 ltfagpugm spandigdhpgt A0 9 9L spcmLI thJSiSp SkSn l The distribution A0 is invariant wrtfg1 gm and and it is contained in the kernel of spandh1 dhp If it is nonsingular it is also involutive Observability Rank Condition Proposition If 20 equivalently A0 is of constant dimension on some open set U then the system is locally observable on U if and only if dim 20 n or equivalently dim A0 O Example Linear System Observability XAxBuyCx fx Axgt gixbigtdhj 26139 gt LAch39 ch Lbicj O C C 20 spanC Ql spanCAQk span C CA CH T hm gt rank 11 CA 1 Example Role of Input x1 x2 0 x2 2 x3 xl 7quot y 2 x2 x3 0 0 o The linearized system is not observable The system with gXO yields Q 2 0 0 1 O span0 1 o On the other hand for this system 1 0 0 20 span 0 l 0 001 Observability Hierarchy locally observable lt dim 90 x0 n TT TT zero input observable lt dim QL x0 n TT II C linearly observable ltgt dim E n A HC Local Decompositions AC0 20 AC0 2 all of constant dimension on U U 4 Px such that 51 fKCpCza sa G1 1 243 4 5 2 f2 2C4 3452354 43 jglt 33 l4 0 394 1244 0 y h 2 44 restncted to 3 044 0 locally controllable restncted to I 043 0 locally observable Variable Structure Control Motor Control Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 Models of ac Motors o Synchronous motors o Induction motors o Brushless dc motors 0 VS Control formulation o Adjustable speed synchronous motor o Position control of brushless dc motor Nomenclature 3 phase synchronous motor 6 ll Vvivq vyiyqyjl23 q ij123 N 17 ZVil23 rotor angle in inertial frame rotor angular Velocity eld winding Voltage current charge q Iial stator winding Voltage current charge mechanical torque load inertia matrix in inerial frame inertia matrix in rotating Blondel frame mechanical rotating inertia dissipation matrix stator inductances self i j and mutual i j field winding self inductance fieldstator winding mutual inductance Lagrange Equations Lltqqgtrltqqgtivltq 1 9 41 112 q tiff qw i1 i2 139 if generalized coordinates Lagrangian kinetic energy Tq39q Tq39rgdqm potential energy Uq 0 generalized force Q 8D8q39 where potential functionDq39q 247qu 1 v1 v2 v3 vflq39 UIQCI SHQ Synchronous Motor J 0 0 0 0 0 1 71 7L3 L5 c0519 0 12 L 7L3 1450350972771 WM 27 0 7L3 71 1 Lscost9T 0 L5 c0519 1450350972771 Lscost927 I ZI1ZJ 7111 Lscos lf2 Lscos izT jZ3 Lscost927 R diag0rrrr Transform to Rotating Frame ripi iwiwin vv2v3gtvdvqvn we cow 3 Va 75 45 J 0 0 0 0 a 0 0 iLfdi 0 0 a 7r 0 L 0 0 d id 0 7 mL 0 0 id vd 0 0 L 0 0 E ii 7 Ldi 7m r 0 0 ii Va 0 0 0 Ln 0 in 0 0 0 r 0 in VB 0 L 0 0 L i 0 0 0 0 r if v L 11 14 I1 411 ELI Synchronous speed control We Want to design a speed controller that maintains a desired speed 5 1 a gt 5 exponentially yl M607 5 L ifiq 7 r1w7 6 A gt 0 2 zero daxis current yZ id 3 balanced operation y3 I Z Ev I 4 constant eld current y4 if Procedure if reduce to regular form check zero dynamics design sliding surfaces design reaching controller ulate Synchronous speed control Now we have the system in the form xrw id id id id 8 I v vd vq v Vf xfxGxvB yhx y y1 yz y yd Reduce to normal form r1 1 r2 1 r3 1r4 1gt zx hx and dimzero dynamics 3 In fact zero dynamics are r r w7iwtniL tn8w I Note that sliding dynamics are the zero dynamics The standard choice forK gtk 1139 1234 sz z Synchronous speed control Now set up 77 STQS ZTQZ 3 75 ZZTQZ Fu hermore since Z h x 2 ahaxHxfxHxGxvB Z39HxfxH xGx xv 75 ZZTQZ 2h7x QHxfxHxGx xv 3 v1 7Vmaxyx sgns 3x HGETQhx Synchronous speed control Synchronous speed control mqu 39r quot r WWWWM Induction The induction motor has two field windings They are closed circuits so that the field currents are induced as the coils move through the stator field J 0 0 0 0 L 0 0 0 0 L 0 M 0 0 0 Ln 0 L 0 0 0 0 L 0 a a 1d 1d d i 1 quot 7C quot dt 1D In 1 11 If2 If2 0 0 0 L 0 119 0 L L i i1 C 1d 11 0 0 0 L 0 0 0 0 L TL quot q i d L jif2 r raj1 a ooox ox 0000 00000 Induction Motor Nomenclature rotor angular velocity field winding voltage statorwinding voltages dq0 stator Blondel voltages 2 field winding currents statorwinding currents q0 stator Blondel currents mechanical torque load electrical torque mechanical rotating inertia stator and field winding resistances stator d amp q axis inductances stator zero sequence axis inductance field winding self inductance fieldstator mutual inductance Ind uction Motor Speed Control Again we Wish to regulate speed a to the desired value 5 Define z w 2 11 rotor ux magnitude 1 41112 y22 111 L i Lif y2 L i Li2 Choose r12r21r32 ylhlwidiqi0i i aka 21 W600 7 7 22 L iZidJrLdi i 71 y2 ihz a1d1q101 12 71 3 23 Z q yhza gt1dgt1qgtlogt1AJ2gtZ3l l 24 wig10 25 rVIZl 7 rV212 2dim Zero Dynamics 2 Z Z 133 11J111J2 L1J0 zero dynamics manifold Brushless dc Motor In the brushless dc motor the eld coils are replaced by permanent magnets thereby removing one coordinate T q 9 111 qz 13 i2 i3 T 0 a 0 0 0 d id 7 0 r 0 dt iq Jy Zke wLS L0 to 0 0 in Zke 0 a wLS 0 id r 0 iq r 10 Brushless dC 392 6 gooM 3 1 d w Lien 7L9 gt T011 6kgquot I V Zq L15 inq Lsid 7 V0 LnU A L Brushless dc position control We want to design a position controller that maintains a desired position g l 8 gt g exponentially y1 a 1097 j gt 0 2 constant stator current y2 i i 7 I or zero daxis y2 id 3 balanced operation y3 i0 Procedure reduce to regular form check zero dynamics design sliding surfaces design reaching controller simulate Brushless dc position control ylhlamidiqiomag 5 VFW zl au 6 6 yzhz6midiqi0iji Ij 2 2 2 2 Z i 1 15 y3h3t9aidiqi0i0 22 The zero dynamics are z 9 Variable Structure Control Basics Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 A preliminary example 0 VS systems sliding modes reaching 0 Basics of discontinuous systems 0 Example undersea vehicle 0 Design based on normal form Example icsz iSZZLIXI bu abgt0 05x1 x1sgt0 scx1x2 u 05x1 x1slt0 acgt0 slidingmodescx1x2EOXICC10 condition for existence 3339 lt 0 Setup System icfxu xeRquotu ER fsmooth Control ui discontinuous across switching surface si x 0 06 Sixgt0 quot x ux sixlt0 i1 m a a x u x smooth Basics of Discontinuous Systems 1 X F x t F smooth in x Ck kgt0 for each I except on m codimensionone surfaces de ned by st x0 i1 m What is a solution Basics of Discontinuous Systems 2 If there is a motion in the switching surface it is called a sliding motion f How is it defined Basics The Filippov Solution e 1xt t QCOHVFS5 xt 5 xt t S6x y eRquoty x lt 6 5 x sub set of measure zero on which F is not de ned Lyapunov Lemma VxeC1 51 womb 5 eFocam x Resultl VS plt0VZpgt0onP A3 V39s 3lt0V3923gt0onP ResultZ VS psxquotpgt00nP A3VS psxquotpgt00nP Sliding Domain D5 CMX xeRquot sx 0 is a sliding domain if trajectories beginning in a 6Vicinity of Di remain in an gVicinity until reaching GDX D5 does not contain entire trajectories of the 2 associated continuous systems Proposition Vx 6 C1 Vx Z 0 if SOC 0 gt 0 otherwise D 3 DJ is an open connected subset of Rquot V S pquotsx lt 0 on D MJ U DJ is a sliding domain Proposition Vx lt7sx2 039 gt O on a 5Vicinity of D D D D is an open connected subset of Rquot V S pquotsx lt O on D Mx U trajectories beginning in a 5Vicinity of D5 reach D5 in finite time Setup System x fog x e Rquot u e R39 f smooth Control ui discontinuous across switching surface s x 0 06 Sixgt0 quot x ux sixlt0 i1m a x ui x smooth Example Choose 2 Vx ST xsx 3 mg 2ST mg m Suppose 237 x6363fxu lt psxquot onD MS where D is an open set containing D5 CMS 3 D5 is a sliding domain and it is reached in nite time from any initial point in D Example Linear SISO fxu Axbu sx cx V quots X 2 3 V 2sxch 2sx cbu We can only affect the second term Choose u kxsgnsxcb kx gt OVx V 2sxch cbkx sgns x cb S ps provided cbkx ch 2 p Vx k x p chxl This insures that s is a sliding domain Equivalent Control 1 tx S x gt 0 X ft x u ui ui tx Sixlt0 Suppose MK 2 x e Rquot sx 0 contains a sliding domain Equivalent Control 2 sx g ftxueq20 ifa solution exists for ueq then on s 0 s E 0 and i6 ftxueq tx lt2 dynamics in sliding Equivalent Control Special Cases Systems 39linear in control39 icfxGxu SxfxSxGxueq E 0 detSG 02ueq 7SxGxlSxfx xIicltxgtsltxgtcltxgt lsltxgtjfltxgt Linear Systems xAxBu sxCx ueq CB ICAx assuming detCB i 0 XIiBCB71CAx Geometry o A subset S of the linear space over eld 1F is a linear subspace of if Vxlx2 e S and V6102 e F 01x1 czxx e 5 01fo e z39 lk then spanxlxk is a subspace of 0 925 C then 92ltSrsre 9336 5 f2rm5xxefEampxe S 0 Two subspaces 925 are independent if 92 m S 0 Geometry 2 o If J 139 lk are independent subspaces then the sum m is called an indirect sum and may be written m The symbol EB presuposes independence 0 Let 327 16 ED 5 For each x e 32 there are unique r e 46 s e 5 so that x r s This implies a unique function x gt r called the projection on 46 along 5 Geometry 3 0 The projection is a linear map Q gt such that ImQ 73 and kerQ 5 and Q57J I Q71quot 0 Note that I Q is the projection on 5 along 373 Thus Q1Q0 Q2 Q 0 Conversly for any map Q gt such that Q2 Q Im Q G kerQ ie Q is the projection on ImQ along kerQ ImB Geometry 4 kerC Q BCB 1C Q2 BCB 1CBCB 1C BCB 1C ImQ ImB kerQ kerC X ImB B kerC BCB 71 C is the projection on ImB along kerC I BCB71C is the projection on kerC along ImB Linear Example 1 sxcx SEODSEO cfc ch cbuzq 0 3 um cb71 ch 71 xI bltcbgt ciAx De ne a matrix V whose columns span ker c ie V VI VH cv 0 Notice that b e kerc andX Imb EB ker 0 De ne a state transformation xI gtwz xVwbz weRquot39lzeR Vwb2 1 bcb 1cAVw b2 Linear Example 2 V b1 bcb lcAV MB V br5 5ng 32W ZPbltcbgtcw WW am w UAV UAb w sliding 2 0 0 z WUAVw Designing the Sliding Surface Consider the system 5c fxGxu rank G x m around x0 sat1s es controllab111ty rank cond1tlon Transform to regular form Strategy x1 f1x1 x2 1 choose xZ 7sn x1 so that Jkzf1xiaxzJVGZOCNXZ x1flxl7v xl x1 6 Rnim x2 6 Rm det GZ i 0 around x0 has desired behavior 2 choose u to enforce sliding on 5xpxz Sn x1xz Li near Case 5c Ax Bu rankB m AB controllable reorder states to obtain B I with 32 eRWdeth 0 B Bl transform to regular form 1 B 3 1 ZTxT quot39m 1 2 0 71 0 2 All A J z u the pair 14111412 is controllable A21 A22 1 Linear Case 2 To shape sliding dynamics 22 K21 3 2391 All A12KZ1 Choose K by any means pole placement LQG etc Now 32 iKzl 22 To design the control u take 1 2 1 Vltzgt3sltzgtllgEswgsm QTQgt0 V39z sTQ s39 27 K M TQ K ImAz sTQu uI 7K1 Zsgns 3 Z Qsz KZ gt 27 K 1m7 Q K M142 Example Underwater Vehicle mj cxljcl u mc uncertain u e 7UU x v c 1 v i vlvl u m m Choose a sliding surface s v 1x Why Because 3 E 0 3 x 71x v 71x stabilizes rst eq Choose control based on Vxv sTs sz V 1 7U sgt0 V231 1 c 23 u3u i vlvl m U slt0 m Control Based on Normal Form 5 F 522 Z39 AZ EOt z p zu y CZ Recall Brunovsky structure of AE Choose sx such that sx 0 cgt Kzx 0 ulx slxgt0 u x ux slxlt0 Sliding Dynamics sx 0 Cgt Kzx 0 U 2 KAZ KEax pxueq 0 KE 1 U p1ltxm2xgt pxax U 2 1 EKAz Kzt0 0 Choosing K One choice ltl gt cl 0 0 K lt IA gt 0 0 k Eigenvalues of AEK are 0 maIeOandrimaIe 7 7am 7am Reaching Consider the positive de nite quadratic form in s Vx sTQs Upon differentiation we obtain 1 7V 2sTQs 2KAz aTQKz zu p QKz t Ifthe controls are bounded Iulx S 7 gt 0 0 gt Ummx S ux S Umxvx gt 0 then choose Um sxxgt0 x T u i 39 011m s x p VSC Summary Robust Control Design via Feedback Linearization Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 Perturbations of feedback linearizable systems 0 Triangular forms 0 Lyapunov Redesign 5c fx Gxu y hx xeR ueRmyeRP fx G06 300 83106 M96 smooth Setup Z39Azbv ycz 0 1 0 0 0 1 0 E A b E 01 0 0 1 0 Perturbations of SISO Feedback Linearizable Systems X fx 006 gx 7JCLl y hx Assumption the nominal system is feedback linearizable What happens when the nominal system transformation is applied to the actual system Matched Uncertainty etc I 5 spangad OSiSn 1 De nition Suppose the system is of relative degree r We say that the pelturbation satis es 0 D 39 The triangularity condition if adzp gi E 91419 0 S l S r 3 The strict triangularity condition if ad i 6 17 0 S i S I 2 The extended matching condition if Q E The matching condition if 6 5 Triangular Forms r n 0 triangularity gt 39 zl tzl dz lgignil leastrestrietive 2 altxltzgt n lt21 azawmznu 0 strict triangularity gt z39tzm zl z lgignil 2 axz it 21 rgtznpxzu 0 extended matching gt U 2Z1 1Si n72 znrl mail 217 39yzn 239 axz it 21 rznpxzu Omatehing gt 2 z 1 lgi gnil most restrictive 2 7 axz it 21y uz pxzu Example Example Cont d 0 0 0 0 0 0 ad quot 0 ad 0 0 ml 0x4 0A2 0x3 0A2 0x4 0 0 0 ad 93 0A1 0x2 0 0 2 0 0 0 ml 0x3 0A20x4 strict triangularity Example Cont d amp Zzzxz Zsx1x2x3 4amp m 23 A1 Z ZA Ea xzzzz x3 Zl FZZ FZ3 m444 Lyapunov Redesign Process 0 Begin with an exactly feedback linearizable nominal system with matched uncertainty 0 Apply the nominal control uz to actual system and test stability using a Lyapunov function 0 Add a new component to nominal control uz uz pz391uz to enhance stability 0 Choose uz via Lyapunov design Problem Setup 0 A state linearizable system with matched uncertainty is transformable to z39 AzEaz Azut pzu 0 Choose a control based on nominal system MO 2 p 1z az K2 with A EK stable Nominal Closed Loop System z39AEKz A EK stable 3 Lyapunov function VZ ZTPZ VZ ZTQZ quotZquot2 PAEKAEKTP Q 1 QQT gt0 along trajectories of Actual Closed Loop System 0 Apply u to actual system Try Vz Vz zTQz iz2 2zTPEA 0 Assume Alt7zlia 720 V 3 lm Q1Z2 ZVIIPEIIIIZIIZ stability if 7lt 1mm Q12quotPEquot This implies some inherent robustness Redesign we can do more 1 u u p ly 0 Assume the uncertainty satis es A00t 0 v Azu p391yt S azzkull 0 S k lt1 039 smooth 0 The actual closed loop dynamics are 2 AEKZEJAZu p39lyj o The time derivative ofV along trajectories is V zTQz z 2 ZZTPE u A Redesign 2 0 Set W zTPE and try to achieve z2w7yAso 0 Notice that wwA s wwuwnuAn s Wwlltaltzgtllzllk o For a smooth control set u wzlt KZ gt0 Z2 W M S w2 K1kw0ltzgtllzllZ2 1 KZWU 2 2 2 2 Ilzll WW W itllwll 02 0 Wzllillz tllwlla llzll 2 Redesign 3 oSoany Kgt 1 41 02 will do k 0 In particular 0392 z 2 UO41 EIEETPZ 60 gt0 Redesign 4 0 There are other possibilities recall V zTQz z 2 wT u A 0 An W wa S W wlllaltzgtllzllkll lll 0 So choose 77z1 lk llwll 772 gt 0zz Example 1 x2 0 3 x2702x2xf21um Hm KE01 ae70101 a701x2x1x13201 7 1 x1 7 3 u 70 70571 72 x 772x1719x27x12 2 x3 AKxfa72xfjil9xljau a20076111025q4ka 3 u u u 502114x1 7x717030625x15 7492114x2 7030625x14x2 Example Results x K 1a 01 x m 12 1 09 05 00 02 3 2 A a a 09 1 12 a nom systemnom control actual systemrobust control actual systemnom control x2 x2 02 0A 06 03 1 12 A t Setup for Backstepping SISO system 0 r n 0 strict tn39angularity assumption 5c 2 xi1Aix1xit 13139 S n l 5cquot 2 ax pxu An xt det p0 2 0 A0t 0 lAi X t lt aXXa20 Xix1 xi Step 1 A 1 V1 1x17t gty1V1A1y17tf1yl y1x1 V1yf L Vlylvl51ywt choosew1 7 ler1y1y U LAV 71y127K1y1y12y131 takeK1y1gtUlzx1x1i0 Step 2 59 x2 A1x1t x2 vzA2 xvxpt yz xz 7V1x1 xl xl gty1y2 U yl v1y1y231y1t yz v2 Kzy1y2t ahafineV2 Vly choosewz 7y17k2y27K2y1y2y2 Kzy1yzgtc7 ypyz3L2Vz k1 2y12 k2 1y fly1y2 Manifolds Vector Fields amp Flows Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline 0 From Flat to Curved State Space Why 0 Manifolds 0 Maximum rank condition 0 Regular manifolds o Tangent space amp tangent bundle 0 Differential map 0 Vector elds amp ows 0 Lie bracket From Flat to Curved State Space 0 Global models may require it o Mechanical systems spatial rotation a Electric power systems DAE description 0 Correct local approximations at least require acknowledging it 0 Even if not required it may have conceptual benefits 0 Computation often involves flat local approximations but this may not be necessary eg quaternions vs Euler angles Representations of Surfaces o Explicit ygx o Implicit fxy0 o Parametric xhlsyh2sseUcR y yiJl x2 0 x2y2 1 xcossysinsse027z Exam ples Parametrically defined manifolds costsinu cos I3cos u f sintsinu te027rue07r f sinl3cosu IE027ru E027r cos u sinu Definition Manifold An mdirnensional manifold is a setM together with a countable collection of subsets Ux C M and oneto one mappings onto open subsets ome p Ux gt K with the following properties 0 the pair U1 p1 is called a coordinate chart 0 the coordinate chartes coverM 0 on the overlap of any pair of charts the composite map is a smooth function fawn1 ltU mUHMU mU o if p 6 UI and q E U J are distinct points of M then there are neighborhoods WofgpxpEl andUof JqEVJ suchthat 403 WWW 109 Definition Manifold Example Planet Earth Example Circle The unit circle S1 Xy x2y21 can be viewed as a one dimensional manifold with two coordinate charts Define the charts U1S110 and U2S110 Now we define the coordinate maps by projection as shown in the figure R1 R1 W 61 ixzyz1gt any i xzyz1gt M 10 10 Submanifold amp Immersion Definition Let F Rm R be a smooth map The rank ofF at X06 R is the rank of the Jacobian Df at x0 F is of maximal rank on SC R if the rank ofF is maximal for each XOES Definition A smooth submam39fold of R is a subset MCR together with a smooth oneto one map HCR gtM which satisfies the maximal rank condition everywhere where the parameter Space is H andM ll is the image of If the maximal rank condition holds but the mapping is not oneto one thenM is an immersion Note that a submanifold of the space R is a parametrically defined surface Pathologies Regular Submanifold Definition A regular submam39fold N of R is a submanifold parameterized by a smooth mapping such that maps homeomorphically onto is image ie for eacthN there exism neighborhoods U of x inR such that 1U N is a connected open subset of the parameter space N R 1 R2 p PE wt regular Submanifold mum on Submanifold imbedding Implicitly Defined Regular Manifolds Proposition Consider a smooth mapping F R quot gtRquot nSm IfF is of maximal rank on the set S x Fx0 then S is a regular mn dimensional submanifold of R quot Example fxy x2 H12 Dy Df 2xy 71x2 3y2 singular points il0 0 ilkg The Tangent Space Definition Let p RgtM be a Ck k 21 map so that pt is a curve inM The tangent vector v to the curve pt at the point p0 pto is defined by v pltz0gtgg Z The set of tangent vectors to all curves in M passing through p0 is the tangent Space to M at p0 denoted TM p0 R Tangent Space Implicit Manifold If M is an implicit submanifold of dimension In in Rm ie F RmtkaRk M xERm rk Fx0 and DF satisfies the maximum rank condition on M Then TM is the ker DXF x translated of course to the point X That is TMp is the tangent hyperplane to M at p Tangent Vectors De nition The components of the tangent vector v to the curve pt inM in local coordinates U p are the m numbers V1Vm where V d ptdt Consider the map F M a R Lety fx x e U C R denote the realization of F in the local coordinatesU p Again pt denotes a curve inM with xt its image in R39 Then the rate of change of F at a point p on this curve is 1 m Tangent Vectors as Derivations The tangent vector vv1 vm is uniquely determined by the action of the directional derivative operator called a derivation 0 vv1 vm 0x1 0x m Natural Basis De nition The set of partial derivative operators constitute a basis for the tangent space TMp for all points pEUCM which is called the natural basis The natural coordinate system on Tlp induced by Up has basis vectors that are tangent vectors to the coordinate lines on M passing through p Tangent Bundle De nition The union of all the tangent spaces to M is called the tangent bundle and is denoted TM TM TM pEM p Remark The tangent bundle is a manifold with dim TM 2 dim M A point in TM is a pair Xv with XEM veTl i If X1xm are local coordinates on M and v1vm componenm of the tangent vector in the natural coordinate system on TM then natural local coordinates on TM are x1xmvlvm x1xmiclic Recall the natural unit vectors on TM are v188x1 vm Baxm Summary at 1m momma gt9 xmzm 0 Regular Manifold a Parametrically defined 0 Implicitly defined 0 Tangent Space 21 Tangent Vector Tangent Bundle a a MXER3f X XZx30 rank a QlonM Mechanical System State Space A mechanical system is a collection of mass particles which interact through physical constrains or forces A con guration is a specification of the position for each of its constituent particles The con guration Space is a set M of elements such that any con guration of the system corresponds to a unique point in the setM and each point inM corresponds to a unique configuration of the system The configuration space of a mechanical system is a differentiable manifold called the con guration manifold Any system of local coordinates q on the configuration mani o d are called generalized coordinates The generalized velocitiesq are elemenm of the tangent spaces to M m The State Space is the tangent bundle M which has local coordinates q q Example Pendulum Differential Map an arbitray cuwe M on M passing through point p maps into 472 on N passing through point Fp 50 FWD R Given the map F 3M 9N the di krential map is the induced mapping E TMF gtTNFF that takes tangent vectors into tangent vectom Differential Map local coordinates In local coordinates the chain rule yields 53 3 g EV dt ax dt 6x 0 The map Fl is also denoted 1F 0 The Jacobian is the representation of the differential map in local coordinates Vector Fields De nition A vector eld v onM is a map which assigns to each pointpEM a tangent vector VpETMp It is a Ck vector eld if for each pEM there eXist local coordinates Uq0 such that each component vix ilm is a Ck function for each x6 AU De nition An integral curve of a vector field v on M is a parameterized curve p t tEt1t2CR whose tangent vector at any point coincides with v at that point Integral Curves In local coordinates Uq0 the image of an integral curve 361 ltP I satisfies the ode dx dt vx Flow De nition Let v be a smooth vector field on M and denote the parameterized maximal integral curve through pEM by LI tp and LI 0pp LI tp is called the ow generated by v Properties of ows d 0 satisfies ode ETIPVLPLP LP0PP semigroup property k11t2 WOLF k111 14391 2917 Exponential Map We will adopt the notation V e p i t p The motivation for this is that the ow satisfies the three basic properties ordinarily associated with eXponentiation 7 from properties of I t p eo39vp p boundary condition e v v equot differential e uation dt 17 p q etl12v 11v 12v p e e p semigroup property Series Expansion Along Trajectory Suppose xtsatis esjc vx x0 x0 Letf Rm gtRP 1 fxt fxofxtt0t fxtt0t2 Series Representation of Exp Map For f a scalar or vector we can derive the Taylor expansion of fxt about F0 00 k t N k fe x 25v xxx 0 Choose xx to obtain k N e Vx Vquot xx A 00 k0 Example scalar linear fields v113 txxt 3 2 Ptxetax 1t3lt26 2 xxt 0x 2 0x vx3 x3Ttxetx dt 6 so k LI tx euax vk xx H x e x Example general linear field 11 a a a vxAx 2 xiva1xanx 0x1 Ox 5 1 v2xVAxAvxA2x Vquot Example Affine Field a1 b1 1 vxAxb a x 52 2va1xblmanxbna an bx 0 vxa1xbl anxbn 0 Axb 0 1 x VAk 1x Ak Zb Ak 1vx Akx AMI w k w k xx Zr AkarZI Ak lbj eA ereA A lb k0 k k0k vk Examples Cont d Vx 7x3 ltgt 5c 7x3 Exact solution 7 x0 7 2 1242 2063 35 84 63105 x t7 2717x0tx0t iyxot yx0t iyxot xo 1 12x0t Via exponential map 33 X xt e a xo 17x02t4rx3tZ 7xgt3 x t4 7x ot5 x0 Lie Derivative De nition Let vX denote a vector field on M and FX a mapping from M to R both in local coordinates Then the Lie derivative of order 0 k is a L1H L3 F F LiF a Vv x With this notation we can write vk Fx L Fx Vk x VAk 1x Ak lvx Li X Nonlinear Field Example Exponential Map of a Lie Bracket De nition If vw are vector elds on M then their Lie bracket vw is the unique vector eld defined in local coordinates by the formula 3w 3v 1 w v w 8x 8x Property dw I tx dt The rate of change of x w alon the flow of v vw1 0 Lie Bracket Interpretation Let us consider the Lie bracket as a commutator of ows Beginning at point x inM follow the ow generated by v for an infinitesimal time which we take as J for convenience This takes us to point y 6Xp Vx Then follow w for the same length of time then v then w This brings us to a point ygiven by Max eiJEw 64 erw eJva Lie Bracket Interpretation Continued Summary 0 De nition of regular manifold o Implicitly de ned amp parametrically defined 0 Local coordinates o Tangent space vector eld integral curve 0 Differential map exponential map 0 Lie derivative 0 Lie bracket Observer Design Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Nonlinear Control Theory N Lecture 6 Outline Problem definition Observability hierarchy I Singular inputs Local exponential observers I Constant Gain Observers Global observers I High Gain Observer I Detectable Systems I Compensating for uncertain nonlineaiity Linearization up to output injection H 011ml for gti observer Nonlinear Control Theory N Lecture 6 Problem De nition X fX GOO fuX y hX xeRquotueR quotyeRP fx0 0 hx0 0 Given a system Construct an observer A ie an estimator Xl yT LIT T E 0 ID Such that xt 0 gt 0 as t gt oo Nonlinear Control Theory N Lecture 6 Observability Hierarchy dimltf g1 gm span alhp alhp n locally observable dim span Ll th II S 139 S p 0 S k S 71 1 7 zero input observable II C0 COAO d1m lmearly observable C0146quot1 Nonlinear Comrol Theory N Lecture 6 Observers that Mirror Svstem Dynamics system at fxu y hx observer 3 f u1lt y h error e39 fxu fx eu1ltx ehx hx e e 0gte39 0Vurxrif1lt x0 20 K must drive et gt 0 Nonlinear ConLrol Theory N Lecture 6 Drexel Exponental Detectability A system is exponentially detectable at xu if there exists a function yxy Y xiii 0 ov me MM 0 x is an eXp stable ep ofg 21 a y Exponential detectability U 3 fAcu W Jay is a local observer Nonlinear Control Theory N Lecture 6 Constant Gain Observer X fxu Ax Bu h0t y Cx h0t AC detectable choose K fay 2 L010 y Re A LC lt O L SCTV 1 SAT AS SCTVICS W S 2 0 3 r xel Nonlinear Comrol Theory N Lecture 6 Global High Gain Observer fxu Fx Gxu o locally uniformly observable for all inputs 2 A C obsewable o g x globally Lipschitz 2 for inputs u unifonnly bounded amp 9 sufficiently large fc 2 F0 Gfcu L C y is an observer where L SCTV 1 A 91s SA 91T SCTVICS W s 2 0 Gauthier Hammouri Othman 1992 f 5 Nonlinear Control Theory N Lecture 6 rexe Global Observer for Detectable Systems fxu Axf2xu y Cx AC detectable 3 EIL P gt 0 Q gt 0 such that A LCP PA LCT Q o f 2x u globallyy Lipschitz in x uniformly Vu XMQ gt 2 P max Than 1973 3 g Afg f2fcuLCfc y is an observer Raghavan amp Hedrick 1994 Rajamani 1998 V L 7SCTV391 AS SAT 7 SCTV39ICS 7W s 2 0 A7SCTV391CS SA7SCTV391CT eWeSCTVlCS s eP WSCTV1CS a Q Nonlinear ConLrol Theory N Lecture 6 Global Robust Observer fxuAxf2xuyCx o AC detectable 3 EIL P gt 0 Q gt 0 such that ALCPPALCT Q o Eln R XR39 gt RP such that f2xu P 1CTnxu olhxu 3 frAfcLltCfc ySCfc y pis an observer s pP1CT p2H peem 92 x2 s ae b b km me P lt ptuVxu Moreover Walcott amp Zac 1987 Dawson Qu Carroll 1992 f 392 351 Nonlinear Control Theory N Lecture 6 Observer Design Via Linearization t0 Output Iniection Krener Isidori Respondek Xfx Z39AZltPy 2 EL y hx y C2 Hammouri Gauthier Kinnaert Z X fxGxu 3 2L 2 Autz p yut y hx y C2 AA t t PtCT C f a zrltpy u T z y AutPtCTCe P PA ut AutP PC CP Q A2p yLC2y ALCeez 2 r Nonlinear Control Theory N Lecture 6 Drexel Example Krener amp Respondek x1 x2 X1 d 2 2 x2 x1 x1 x2 2 dt x3 X1953 x3 x2x3 1 x1 21 xlz2 1x1x3z3 x13 3x23 d 21 0 0 1 21 yf3 Z 0 0 0 22 0y1 dt 2 00 0 23 y1 r Nonlinear Comrol Theory N Lecture 6 rexel Example Krener amp Respondek d xz d 62 x31ex1u yx1 I x3 3X12X22 X13X3 1x1xzu 4 3 zl xlzz x1 4x22Z3 xlxzx3 1 4 d 21 0 0 21 I d 22 0 0 2 22 21eyu t 23 0 u 0 z3 1y 1eyy3y44u f F Nonlinear Comrol Theory N Lecture 6 Drexel Examnle Hammouri amp Kinnaert Modi ed XI 8 1ux d x2 ax 1uex3 x2 e x2 x12 x1 E 953 e 21x13 ux12 y x4 x4 x5 x5 x1 X1XZ X1X3 zl x1zz x4z3 e 24 x525 e Z1 z3 1uzl2 0 0 1 0 0 21 1y12u d Z2 Z4 0 0 0 0 0 22 y2 E 23 u1 25 0 0 0 0 u 23 u Z4 Z1 0 0 0 0 0 Z 3 3 w ZS 21 0 0 0 0 0 ZS y1 Nonlinear Comrol Theory N Lecture 6 Drexel Introduction to Part 2 Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Part 1 Summary 0 Lecture1 Introduction 0 Lecture 2 Manifolds Vector Fields amp Flows 0 Lecture 3 Distributions amp Frobenious Thm 0 Lecture 4 Controllability Observability 0 Lecture 5 Control via Feedback Linearization 0 Lecture 6 Observer Design 0 Lecture 8 Robust Control 0 Lecture 9 Adaptive Control Part II Outline Lecture 9 Intro to Discontinuous Dynamics o Examples 0 Simulation Tools 0 Solution concepts Lecture 10 Variable Structure Control Basics o Sliding domain equivalent control 0 Lyapunov analysis of discontinuous systems 0 Special Cases linear dynamics normal from Lecture 11 Mechanical Systems 0 Stabilization tracking path following Lecture 12 Electrical Machines Lecture 13 Regulation amp Disturbance Rejection via VSC Lecture 14 SwitchedHybrid Systems Challenge Problem burly xed frame Recall the robot analyzed in the Fall project Design a feedback controller that will steer the vehicle to and wmc mnc asymptotically stabilize a prescribed quot target in the state space G Oriolo A De Luca M Vendit teli WMR Control via Dynamic Feedback Linearization Design Implementation and Experimental Validation IEEE Trans On Control Systems Technolgy Vol 10 No 6 pp833 852 2002 V Sankaranarayanan A D Mahindrakar Switched control of a Nonholonomic Robot Communications in Nonlinear Science and Numrical Simulation Vol 14 pp 23192327 2009 Controllability Observability Compressor Stall Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline Jet engine compressors and the MooreGrietzer model Equilibria and bifurcation in compressors surge and rotating stall Linear controllability and observability at the bifurcation point Nonlinear controllability Nonlinear observability Example Jet Engine Compressor Lowmssuni w nuumssou a 99in t y39 mi 5 5m 39 5M5 a w a gt M g w 7 gww mwpnzssumunamn 7 gt mmm mm sum mm mm with In wan m m AMI m2 cuuvmsou Example MG3 Model v Pc Il A V 13Tl 2 I A Pc 7 D2 PC Asinsds A a A I AJ Pc Asin5sin5d5 2 9 the annulusaveraged ow coefficient llthe pressure rise coefficient A the amplitude of the rst harmonic of annular ow distribution TC The function is the compressor characteristic CDTthe throttle characteristic Example Computational Data MMHltIJ t JJ 1 A i 3AZ 1 2 3 2 41150 2 A AAZ 4 2 Example Axisymmetric Equilibria A0 RD 3 e uns le stab a 25 2 15 l 05 15 2 25 3 Example NonAxisymmetric Stall Linearization at Bifurcation Point 0727113R7 3 6R 1 140845 7 7 R 716R72 R RA22 bifurcation point 211 272R 07 121268 0 71 73 0 A 140845 70517813 0 3 723228 C1 0 0 0 0 0 0 Linearization fails to be controllable or observable Nonlinear Controllability set up equations MW Egshitt a men n m a 2 Phi Fri 2 Juan any nus mg n uw 7 01 m quotI um m 1 quotmy xx A pm Psi ma t Hatrixi um UleMIhmnxrarm ifkm au u 23 a gt3A391Ph1A2Ph1 3ij 4P31 n M22 11 Hatrixi nm DmL l lMmnrfam u Nonlinear Controllability Generic distribution around 000 n2739 Cunuomistrihuunnrt g x Includebx i aFalse 1 Matrixi unn Contromistrihutinn g x o u a Includebrittai alse 1 mtrixronn ommimamxsomz 1 I D Distribution evaluated at 000 o 1 u 0 n 1 UulIZslilhrianom 0 1 0 001 The implication of the singularity of the controllability distribution is that a non smooth control will be required Nonlinear Observability mm 1 n1 One measurement case Ubservabilitycudistrihution g In x Dhservabilitycudistrihution ll ag h x Uhservabilitycadistrihution g h x II a x MWF 1 U U 0 1 U U U 1 MNF 1 393 0 U 1 U 0 U 1 MIMF 33 14PM 0 1 393 1 UH mm In Psi Phi Bhservabilitycudistrihution g h x Two measurement case Dhservabilitycudistrihution ll mg n x Uhservahilitycudistrihution g h x 0 a x MW 1 0a 0 0 1 U U 0 1 Ml47l 1 0 0 0 1 U U 0 1 3mi481 H 39339 1 3 l 0 Switching amp Hybrid Systems Harry G Kwatny Department of Mechanical Engineering amp Mechanics Drexel University Outline Hybrid System 0 composed of a continuous dynamical system interfaced with a discrete event system May be viewed as a switching system with a general switching structure Simple Examples of Hybrid Systems 0 Bouncing ball 0 Heating system c Gearboxcruise control Simulating Hybrid Systems Simulink with Stateflow o StateflowSimulink o Bouncing ball 0 Power conditioning system c Inverted pendulum Simple Examples Bouncing Ball Free fall j 7g yltrgtyltr yz 701 06 01 Collision 7 M 7 x 70Ax2lt03x2 rim Bouncing Ban V VD gt 1 2 XVutgtz 0ni 2 g 1 z cpjx c g g g I T ZN t 2 54 2 CH 2i 17 c TlmeforN N 11 x 11 g g F g 17 c transmons ForOSclt11jmTN2i 1 N g 17c II Zeno phenomenon oo transitions in nite time Heater Off mode x7x50 continuous state x E R 50 xfx39q39 fx39qi100 Z discrete state qE omo 0n qk1 mkv Mm 0n 017 on qojfx73 qojfxgt73 q0nx277 q0nxlt77 On mode x7x100 Automobile Gearbox Control Problem 1 Automated gearbox coordinate automatic gearshift With throttle command Problem 2 Cruise control automate throttle and gearbox to maintain speed Background a R W Brockett quotHybrid Models for Motion Controlquot in Perspectives in Conztggzl L Trentelman and J C VWIems Ed Boston Birkhauser 1993 S 39Hedlun39d and A Rantzer quotO timal Control of Hybrid Systemsquot resented at Conference on Decision an Control Phoenix AZ pp 397239 7 1999 F D Torn39si and A Bemporad quotHYSDELA Tool for Generating Computational Hybrid Models for Anallysis and Synthesis Problemsquot IEEE Transactions on Control Systems Tec nology vol 12 pp 235249 2004 Transmission R R Mq x39z j nfmg wu7 Fb 7 cv2 inehgsina wheel u 6 01 throttle position a engine speed a u en 39ne tor ues eed characteristic feng g1 q P 1 E 111 112 113 I4 115 transmission state Rani 1 5 corresponding gear ratios rear gear ratio wheel radius brake force Cruuse Control Continuous control throttle u and brake FA are chosen so that R R n I 2mg fling 01in CV2 MwhgsmaM 16190 7Vk1JV7Vdti a standard feedback linearized design with PI controller 1 n notice that control depends on the discrete state Discrete control ad hoc gear shift strategy Cruise Control Issues 0 Choice of shift thresholds o V de spread implies large speed deviation before shift o Narrow spread opens possibility of excessive shifting even chattering 0 Does not explicitly consider throttle and brake limits o It must be veri ed that the engine does not stall or exceed red line Simulating Hybrid Systems with StateflowSIMULINK Stateflow o State ow is a Simulink toolbox 0 Provides a graphical means to incorporate discrete event process into Simulink 0 Based on the concept of statecharts o Harel D statecharts A Visual Formalism for Complex Systems Science of Computer Programming 1987 8 p 231274 0 Has evolved to represent an implementation of UML