Sliding Mode Observers
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Sliding Mode Observers
- 1. Sliding Mode Observers SOLO HERMELIN Updated: 15.09.10 1
- 2. Table of Content SOLO Sliding Mode Observers 2 Sliding Mode Observers Sliding Mode Observer for a Linear Time Invariant (LTI) System Generic Observer for a Linear Time Invariant (LTI) System Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System Sliding Mode Observers of Target Acceleration References
- 3. SOLO Sliding Mode Observers In most of the Linear and Nonlinear Unknown Input Observers proposed so far, the necessary and sufficient conditions for the construction of Observers is that the Invariant Zeros of the System must lie in the open Left Half Complex Plane, and the transfer Function Matrix between Unknown Inputs and Measurable Outputs satisfies the Observer Matching Condition. Observers are Dynamical Systems that are used to Estimate the State of a Plant using its Input-Output Measurements; they were first proposed by Luenberger. David G. Luenberger Professor Management Science and Engineering Stanford University In some cases, the inputs of the System are unknown or partially unknown, which led to the development of the so-called Unknown Input Observers (UIO), first for Linear Systems. Motivated by the development of Sliding-Mode Controllers, Sliding Mode UIOs have been developed. The main advantage of using Sliding-Mode Observer over their Linear counterparts is that while in Sliding, they are Insensitive to the Unknown Inputs and, moreover, they can be used to Reconstruct Unknown Inputs which can be a combination of System Disturbances, Faults or Nonlinearities. 3 Sliding Mode Observers
- 4. SOLO 4 Diagram of the LTI System and a Sliding Mode Observer pxnnxmnxmnxn pmmn CBBA yuux xCy uBuBxAx RRRR RRRR ∈∈∈∈ ∈∈∈∈ = ++= ,,, ,,, 21 21 21 21 2211 Assumptions: ( ) 222 222 ,, mBCrankBrank pmpCrankmBrank == ≤==1 2 3 tu ∀≤ ρ2 Invariant Zero of the triple (A,B2,C) are in the Open Left-Hand Complex Plane, or equivalently ( ) 0Real 0 2 2 2 2 ≥∀+= − smn C BAIs rank pxmpxn nxmnxnn Sliding Mode Observer for a Linear Time Invariant (LTI) System Observer: ( ) ( ) 2 , ˆˆ ,ˆ,ˆˆˆ 211 mnxp EL xCy yyEBuByyLxAx RR ∈∈ = −+−+= η will stabilize the Observer on the Sliding Surface nxp L R∈ ( ) 0ˆ =− yyF ( ) ( ) ( ) ( ) ( ) xpm F yyFif yyFif yyF yyF yyE 2 ,0 0ˆ0 0ˆ ˆ ˆ ,ˆ, RR ∈∈< =− ≠− − − = η η η Reaching the Sliding Surface using: ( ) 0ˆ =− yyF Sliding Mode Observers LTI System:
- 5. L is obtain by choosing a S.P.D. matrix Q and solving for the S.P.D. matrix P defined by the Lyapunov Algebraic Equation (A.R.E.) SOLO 5 pxnnxmnxmnxn pmmn CBBA yuux xCy uBuBxAx RRRR RRRR ∈∈∈∈ ∈∈∈∈ = ++= ,,, ,,, 21 21 21 21 2211 Sliding Mode Observer for a Linear Time Invariant (LTI) System Observer: ( ) ( ) 2 , ˆˆ ,ˆ,ˆˆˆ 211 mnxp EL xCy yyEBuByyLxAx RR ∈∈ = −+−+= η will stabilize the Observer on the Sliding Surface nxp L R∈ ( ) 0ˆ =− yyF ( ) ( ) ( ) ( ) ( ) xpm F yyFif yyFif yyF yyF yyE 2 ,0 0ˆ0 0ˆ ˆ ˆ ,ˆ, RR ∈∈< =− ≠− − − = η η η Reaching the Sliding Surface using: ( ) 0ˆ =− yyF On the Sliding Surface ( ) 11 ˆˆˆ uByLxCLAx ++−= ( ) ( ) 02 <−=−+− QCLAPPCLA T F is obtain by solving Sliding Mode Observers ( ) 1 22 − =⇒= TTTT CCCPBFPBCF Diagram of the LTI System and a Sliding Mode Observer L.T.I. System:
- 6. SOLO Generic Observer for a Linear Time Invariant (LTI) System pxmpxnnxmnxn pmn DCBA yux uDxCy uBxAx RRRR RRR ∈∈∈∈ ∈∈∈ += += ,,, ,, Observer sxpsxmsxq qxpqxmqxq sxnsq RSM GJF Lwz xLw yRuSzMw yGuJzFz RRR RRR RRR ∈∈∈ ∈∈∈ ∈∈∈ → ++= ++= ,, ,, ,, A Necessary Condition for obtaining an Observer is that (A,C) is Observable. The Observer will achieve if and only ifxLw → =+ =+ −= =− − 0DRS LTGCR DGBTJ CGTFAT valueseigenstablehasF where Tnxn is a Transformation Matrix. L.T.I. System Sliding Mode Observers
- 7. SOLO Generic Observer for a Linear Time Invariant (LTI) System pxnnxmnxn pmn CBA yux xCy uBxAx RRR RRR ∈∈∈ ∈∈∈ = += ,, ,, Observer sxpsxmsxq qxpqxmqxq sxnsq RSG KJF Lwz xLw yRuSzGw yKuJzFz RRR RRR RRR ∈∈∈ ∈∈∈ ∈∈∈ → ++= ++= ,, ,, ,, Taking Laplace Transforms we obtain ( ) ( ) ( ) ( ) ( ) ( )sUBAIsCsYsUBAIssX nn 11 −− −=→−= ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ){ } ( )sUBAIsCRSBAIsCJFIsGsW sUBAIsCJFIssZ nnq nq 111 11 −−− −− −++−+−= −+−= L.T.I. System Sliding Mode Observers
- 8. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System nxqpxnnxmnxn qpmn ECBA dyux xCy dEuBxAx RRRR RRRR ∈∈∈∈ ∈∈∈∈ = ++= ,,, ,,, Observer DesignedbetoMatricesHKTF xz yHzx yKuBTzFz nxpnxpnxnnxn nn RRRR RR ∈∈∈∈ ∈∈ += ++= ,,, ˆ, ˆ ( ) ( ) ( ) ( ) ( ) dECHIuBCHIxACHIyKxCKuBTzFxCHIze nnn KKK n −−−−−−+++=−−= =+ 21 21 The Estimator Error ( ) xCHIzxxe n −−=−= ˆ: An Unknown Input Observer for an LTI System will derive its State Error regardless of the unknown input (disturbance) 0ˆ: asymptotic xxe →−= ( )td ( ) ( )[ ] ( )[ ] ( )[ ] ( ) dECHIuBCHITyHCKACHAKzCKACHAFeCKACHAe nn −−−−+−−−+−−−+−−= 1211 eyHzx −+=Substitute in this equation: L.T.I. System Sliding Mode Observers
- 9. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System nxqpxnnxmnxn qpmn ECBA dyux xCy dEuBxAx RRRR RRRR ∈∈∈∈ ∈∈∈∈ = ++= ,,, ,,, Observer An Unknown Input Observer for an LTI System will derive its State Error regardless of the unknown input (disturbance) 0ˆ: asymptotic xxe →−= ( )td ( ) ( )[ ] ( )[ ] ( )[ ] ( ) dECHIuBCHITyHCKACHAKzCKACHAFeCKACHAe nn −−−−+−−−+−−−+−−= 1211 We can see that if we can make the following relations: ( ) HFK CKACHAF CHIT ECHI n n = −−= −= =− 2 1 0 the State-Estimator Error will be: eFe = We can see that the Observer Error will be zero asymptotically iff all the eigenvalues of F are stable. L.T.I. System Sliding Mode Observers
- 10. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System Observer Lemma 1: This Equation is solvable if: ( ) HFK CKACHAF CHIT ECHI n n = −−= −= =− 2 1 0 but: and a special solution for H is: ( ) ( )[ ] ( ) ( )†1 ECEECECECEH TT nxq sp nxp == − Proof of Lemma 1: nxqpxnnxpnxq ECHE = ( ) ( )nxqpxnnxq ECrankErank ≤⇒ ( ) ( ) ( )[ ] ( )nxqnxqpxnnxqpxn ErankErankCrankECrank ≤≤ ,min ( ) ( ) qErankECrank nxqnxqpxn == Necesity Sufficiency When rank (CE) = rank (E), (CE) is a full column rank matrix, because E is assumed a full column rank matrix, and a left inverse of (CE) exists. ( ) ( ) ( )[ ] ( )TT ECECECEC 1† − = and: ( )† ECEH sp nxp = q.e.d. L.T.I. System ( ) qErankECrank nxqnxqpxn == Observer Matching Condition Sliding Mode Observers
- 11. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System Observer Lemma 2: Let: If s1 ϵ C is an unobservable mode of the pair (C1,A) then: = AC C C1 Then the Detectability for the pair (C,A) is equivalent to that of (C1,A). (A pair (C,A) is Detectable when all the unobservable modes of this pair are Stable). Proof of Lemma 2: n AC C AIs rank C AIs rank n n < − = − 1 1 1 That means that exists a vector α ϵ Cn such that: n C AIs rank C AIs AC C AIs nn n < − ⇒= − ⇒= − 11 1 00 αα s1 is also an unobservable mode of the pair (C,A). If s2 ϵ C is an unobservable mode of the pair (C,A) then: n C AIs rank n < −2 That means that always exists a vector β ϵ Cn such that: 02 = − β C AIs n s2 is also an unobservable mode of the pair (C1,A). ( ) 00 0 0 1 1 1 22 2 = − = − ⇒===⇒ = =− βββββ β β C AIs AC C AIs CssCAC C AIs n n n q.e.d. L.T.I. System Sliding Mode Observers
- 12. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System = ++= xCy dEuBxAx U.I.O. Observer += ++= yHzx yKuBTzFz ˆ Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are: (1)rank (C E) = rank (E) (2)(C, A1) is a Detectable pair, where ( ) ( ) ( )[ ] ( ) ACECECECEAACECEAA TT 1† 1 : − −=−= The condition (2) is equivalent to the condition that the Invariant Zeros for the Unknown Input, i.e., of the triplet (A,E,C) must be stable: qpnsqn C EAIs rank pxqpxn nxqnxnn ≥≥∈∀+= − −C 0 Proof of Lemma 3 (Sufficiency): According to Lemma 1 if rank (C E)= rank (E) exists a solution for H: ( ) ( )[ ] ( ) ( )†1 ECEECECECEH TT nxq sp nxp == − and: ( ) ( )[ ]( ) CKACKACECECECEACKACHAF TT pxnnxnpxn sp nxpnxnnxn nxp 1111 −=−−=−−= We can see that F may be Stabilized by choosing a proper K1, only if the pair (C, A1) is Detectable. L.T.I. System Sliding Mode Observers
- 13. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System = ++= xCy dEuBxAxL.T.I. System U.I.O. Observer += ++= yHzx yKuBTzFz ˆ Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are: (1)rank (C E) = rank (E) (2)(C, A1) is a Detectable pair, where ( ) ( ) ( )[ ] ( ) ACECECECEAACECEAA TT 1† 1 : − −=−= Proof of Lemma 3 (Necessity): ( ) HFK CKACHAF CHIT ECHI n n = −−= −= =− 2 1 0 A General Solution for H is ( ) ( )( )[ ]† 0 † ECECIHECEH mnxp −+= where is an arbitrary matrix andnxm H R∈0 ( ) ( ) ( )[ ] ( )TT ECECECEC 1† − = Since the Observer is a U.I.O. Observer for the L.T.I. System we can solve for H, T, K1, F and K2 ( )[ ] [ ] ( )[ ] [ ] 111 1 011011 1 1 CKA AC C HKA ACECECI C HKACECEICKACHAF C K T m T n −= −= − −−=−−= Since the Matrix F is Stable the pair is Detectable, therefore the pair (C, A1) is also detectable, according to Lemma 2. ( )11, AC q.e.d. Sliding Mode Observers
- 14. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System = ++= xCy dEuBxAxL.T.I. System U.I.O. Observer += ++= yHzx yKuBTzFz ˆ Lemma 3: Necessary and Sufficient Conditions to have an U.I.O. Observer for the L.T.I. System are: (1)rank (C E) = rank (E) (2)(C, A1) is a Detectable pair, where ( ) ( ) ( )[ ] ( ) ACECECECEAACECEAA TT 1† 1 : − −=−= Proof of Lemma 3 (Necessity) (continue): q.e.d. ( ) ( ) ( ) ( ) −∈∀+= − − − = − Csqn C EAIs ECEsCECE I ECEsCECEI rank C EAIs rank pxqpxn nxqnxnn p n pxqpxn nxqnxnn 0 0 0 †† †† ( ) ( ) ( ) ( ) −∈∀+ − − = − +− = − CsErank CAECE C AIs rank ECAECE C ACECEAIs rank C EAIs rank q n nn pxqpxn nxqnxnn nxn † 1 † † 0 0 0 The condition that the pair (C, A1) is detectable, is equivalent to therefore equivalent to the Invariant Zeros of the triplet (A,E,C) being stable −∈∀= − Csn C AIs rank nxnn 1 The Condition that the Invariant Zeros of the triplet (A,E,C) are stable is: Sliding Mode Observers
- 15. SOLO Unknown Input Observer (UIO) for a Linear Time Invariant (LTI) System = ++= xCy dEuBxAxL.T.I. System U.I.O. Observer Design Procedure += ++= yHzx yKuBTzFz ˆ 1 Check if rank (CE)=rank(E). If rank (CE)≠rank(E) go to .10 2 Compute: ( ) ( )[ ] ( ) ATACHTECECECEH pxnnxpnxn TT nxq sp nxp === − 1 1 ,, 3 If (C1, A) Observable a U.I.O. exists, and K1 can be computed using Pole Placements or any other Method. Go to 9 T n T pp 1 ,,1 4 Construct a Transformation Matrix P by choosing n1=rank (WO) (where WO=[C, CA1,…,CA1 n-1 ]) row vectors , together othe n-n1 row vectors to construct the nonsingular T n T n pp ,,11 + [ ]T n T n T n T ppppP ,,11 11 += 5 Perform [ ]0* 0 1 2221 111 1 CPC AA A PAP = = −− 6 Check Detectability of (C,A1). If one of eigenvalues of A22 is unstable, a U.I.O. doesn’t exist and go to 10 7 Select n1 eigenvalues and assign them to using Pole Placement.*1 11 CKA p− 8 Compute where is any (n-n1)xn matrix.( ) ( )[ ]TT p T pp KKPKPK 2111 1 −− == 2 pK 9 Compute HFKKKKCKAF +=+=−= 12111 , 10 Stop Sliding Mode Observers
- 16. 16 Sliding Mode Observers of Target Acceleration Kinematics: ( ) →→ ⋅−⋅+Λ−=Λ tataRR td d MT 11 We want to Observe (Estimate) the Unknown Target Acceleration Component: → ⋅ taT 1 Define: 0:1_ vtaestAt Est T = ⋅= → ( ) mEstEst AvRz td d −+Λ−= 00 The Differential Equation of the Observer will be a copy of the kinematics: mM Ata = ⋅ → :1 Define the Observer Error: EstEstO Rz Λ−= 0:σ Define the Sliding Mode Observers that must drive σO→0: ( ) ( ) ( ) 22 11 1232 201 2/1 0121 1 3/2 10 1.1 5.1 2 vz vz vzsigntv zvzsignvztv zsigntv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ Missile Command Acceleration ( ) ( ) ( ) estAtdRsignRRsignRRNa EstEstRSM EstEstEstEstEstEstEstEstEstEstC _' 2 3/1 2 2/1 1 +ΛΛ+ΛΛ+Λ−= Λ ∫ µαα t1, t2, t3 are Design Parameters Observer 4: Variation of 1 ( ) ( ) ( ) 22 11 122 201 2/1 01 2/1 1 1 3/23/1 0 1.1 5.1 2 vz vz vzsignLv zvzsignvzLv zsignLv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ Observer 1: ( ) ( ) 02 11 3/1 21 1 2/1 10 = = ⋅⋅−= +⋅⋅−= z vz signv zsignv OOO OOO σσα σσα ( ) ( ) 02 11 21 1 2/1 10 = = ⋅−= +⋅⋅−= z vz signv zsignv OO OOO σρ σσρ L is a Design Parameter are Design Parameters21, OO αα are Design Parameters21, OO ρρ Observer 2: Observer 3: Sliding Mode Observers
- 17. 17 Sliding Mode Observer of Target Acceleration: MATLAB Listing % Nonlinear Sliding Mode Target Acceleration Observers At_est=0; v0=0; z0=x1; z1=0; z2=0; Observer=1; %First Observer Parameter L=10; %Second Observer Parameters alphaO1=30; alphaO2=1; %Third Observer Parameters rho1=20; rho2=3; %Fourth Observer Parameters t1=10; t2=3; t3=1; %Second Order Sliding Mode SigmaSM=Range_est*Lamdadot_est; y2 = alpha1*sign(SigmaSM)*abs(SigmaSM)^0.5+x2; x2_dot =alpha2*sign(SigmaSM)*abs(SigmaSM)^(1/3); %Nonlinear Sliding Mode Target Acceleration Observers z0_dot=v0-Rdot_est*Lamdadot_est-Am; SigmaO=z0-SigmaSM; if(Observer==1) v0=-2*L^(1/3)*abs(SigmaO)^(2/3)*sign(SigmaO)+z1; v1=-1.5*L^(1/2)*abs(z1-v0)^(1/2)*sign(z1-v0)+z2; v2=1.1*L*sign(z2-v1); z1_dot=v1; z2_dot=v2; v0_dot=0; At_est=v0; end if(Observer==2) v0=-alphaO1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1; v1=-alphaO2*abs(SigmaO)^(1/3)*sign(SigmaO); z1_dot=v1; z2_dot=0; v0_dot=0; At_est=v0; end if (Observer==3) v0=-rho1*abs(SigmaO)^(1/2)*sign(SigmaO)+z1; v1=-rho2*sign(SigmaO); z1_dot=v1; z2_dot=0; v0_dot=0; At_est=v0; end if(Observer==4) v0=-2*t1*abs(SigmaO)^(2/3)*sign(SigmaO)+z1; v1=-1.5*t2*abs(z1-v0)^0.5*sign(z1-v0)+z2; v2=1.1*t3*sign(z2-v1); z1_dot=v1; z2_dot=v2; v0_dot=0; At_est=v0; end %Missile Acceleration Command and Autopilot Ac=-N*Rdot_est*Lamdadot_est+y2+At_est; N = 3; alpha1 =10; alpha2 = 1; %Nonlinear Sliding Mode Target Acceleration % Observer State Integration z0=z0+z0_dot* delta_time; z1=z1+z1_dot* delta_time; z2=z2+z2_dot* delta_time; v0=v0+v0_dot* delta_time; ( ) ( ) ( ) 22 11 1232 201 2/1 0121 1 3/2 10 1.1 5.1 2 vz vz vzsigntv zvzsignvztv zsigntv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ ( ) ( ) ( ) 22 11 122 201 2/1 01 2/1 1 1 3/23/1 0 1.1 5.1 2 vz vz vzsignLv zvzsignvzLv zsignLv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ ( ) ( ) 02 11 3/1 21 1 2/1 10 = = ⋅⋅−= +⋅⋅−= z vz signv zsignv OOO OOO σσα σσα ( ) ( ) 02 11 21 1 2/1 10 = = ⋅−= +⋅⋅−= z vz signv zsignv OO OOO σρ σσρ Sliding Mode Observers
- 18. 18 ( ) ( ) ( ) 22 11 1232 201 2/1 0121 1 3/2 10 1.1 5.1 2 vz vz vzsigntv zvzsignvztv zsigntv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ t1, t2, t3 are Design Parameters Observer 4: Variation of 1 ( ) ( ) ( ) 22 11 122 201 2/1 01 2/1 1 1 3/23/1 0 1.1 5.1 2 vz vz vzsignLv zvzsignvzLv zsignLv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ Observer 1: ( ) ( ) 02 11 3/1 21 1 2/1 10 = = ⋅⋅−= +⋅⋅−= z vz signv zsignv OOO OOO σσα σσα ( ) ( ) 02 11 21 1 2/1 10 = = ⋅−= +⋅⋅−= z vz signv zsignv OO OOO σρ σσρ L is a Design Parameter are Design Parameters21, OO αα are Design Parameters21, OO ρρ Observer 2: Observer 3: Scenario: R0=10000 m, Rdot=-1000 m/s, alpha1=10, alpha2=1, N=3, Ldot0=0.05 rad/s A step pulse Target acceleration At=100 m/s2 starting at t=3 s and finishing at t=7s With Not Noise 10=L 1,30 21 == OO αα 3,20 21 == OO ρρ 1,3,10 321 === ttt 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -100 0 100 z1 0 1 2 3 4 5 6 7 8 9 10 -10 0 10 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 z1 0 1 2 3 4 5 6 7 8 9 10 -50 0 50 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -10 0 10 z1 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -100 0 100 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 z1 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 SigmaO Sliding Mode Observer of Target acceleration - MATLAB Results Sliding Mode Observers
- 19. 19 ( ) ( ) ( ) 22 11 1232 201 2/1 0121 1 3/2 10 1.1 5.1 2 vz vz vzsigntv zvzsignvztv zsigntv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ t1, t2, t3 are Design Parameters Observer 4: Variation of 1 ( ) ( ) ( ) 22 11 122 201 2/1 01 2/1 1 1 3/23/1 0 1.1 5.1 2 vz vz vzsignLv zvzsignvzLv zsignLv OO = = −⋅⋅= +−−⋅⋅= +⋅⋅⋅= σσ Observer 1: ( ) ( ) 02 11 3/1 21 1 2/1 10 = = ⋅⋅−= +⋅⋅−= z vz signv zsignv OOO OOO σσα σσα ( ) ( ) 02 11 21 1 2/1 10 = = ⋅−= +⋅⋅−= z vz signv zsignv OO OOO σρ σσρ L is a Design Parameter are Design Parameters21, OO αα are Design Parameters21, OO ρρ Observer 2: Observer 3: 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -100 0 100 z1 0 1 2 3 4 5 6 7 8 9 10 -100 0 100 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -10 0 10 z1 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 z1 0 1 2 3 4 5 6 7 8 9 10 -50 0 50 SigmaO 0 1 2 3 4 5 6 7 8 9 10 -200 0 200 Atest 0 1 2 3 4 5 6 7 8 9 10 -1000 0 1000 z0 0 1 2 3 4 5 6 7 8 9 10 -100 0 100 z1 0 1 2 3 4 5 6 7 8 9 10 -20 0 20 SigmaO Scenario: R0=10000 m, Rdot=-1000 m/s, alpha1=10, alpha2=1, N=3, Ldot0=0.05 rad/s A step pulse Target acceleration At=100 m/s2 starting at t=3 s and finishing at t=7s With Lamda_dot Noise Filtered with Time Constant of 200msec 10=L 1,30 21 == OO αα 3,20 21 == OO ρρ 1,3,10 321 === ttt Sliding Mode Observer of Target acceleration - MATLAB Results Sliding Mode Observers
- 20. 20 Sliding Mode Observer of Target acceleration - MATLAB Results Scenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/s No Target acceleration , No Measurement Noises Observer Output 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 Atest 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 0 100 z0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 z1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 0 0.1 SigmaO 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 100 X1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -400 -200 0 200 X1 d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 X2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -500 0 500 Am 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 0 0.1 Lamda d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 Lamda d ot2 Sliding Mode Observers
- 21. 21 Sliding Mode Observer of Target acceleration - MATLAB Results Scenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/s A step pulse Target acceleration At=100 m/s2 starting at t=0.3 s and finishing at t=0.6s Without Noise 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 100 X1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -400 -200 0 200 X1 d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 X2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -500 0 500 Am 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 0 0.1 Lamdad ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 Lamdad ot2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 0 100 Atest 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 0 100 z0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 0 20 z1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 SigmaO Observer Output Sliding Mode Observers
- 22. 22 Return to Table of Content Sliding Mode Observer of Target acceleration - MATLAB Results Scenario: R0=1000 m, Rdot=-1000 m/s, alpha1=30, alpha2=1, N=3, Ldot0=0.05 rad/s A step pulse Target acceleration At=100 m/s2 starting at t=0.3 s and finishing at t=0.6s With Lamda_dot Noise Filtered with Time Constant of 20msec 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 100 X1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -400 -200 0 200 X1 d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 X2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -500 0 500 Am 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 0 0.1 Lamda d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 Lamda d ot2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -200 0 200 Atest 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 0 100 z0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 0 20 z1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 SigmaO Observer Output 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 0 0.1 Lamda d ot 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 Lamdad ot2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.05 0 0.05 Noise L amdadot Lamda_dot Noise Sliding Mode Observers
- 23. References SOLO O’Reilly, J., “Observers for Linear Systems”, Academic Press, 1983 23 Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999 Sliding Mode Observers
- 24. 24 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA
Notes de l'éditeur
- Zak, S.H., Hui, S., “Output Feedback Variable Structure Controllers and Stste Estimators for Uncertain Dynamic Systems”, TR-EE 91-46, November 1991 Floquet,T., Edwards, C., Spurgeon, S.,K., “On Sliding Mode Observers for Systems with Unknown Inputs”, Int. J. Adapt. Control Signal Process, vol. 21, pp. 638-656, 2007 Kalsi, K., Lian, J., Hui, S., Zak, S.,H., “Sliding-Mode Observers for Systems With Unknown Inputs”, August 22, 2008, Draft
- Zak, S.,H., Hui, S., “Output Feedback Variable Structure Controllers and State Estimators for Uncertain Dynamic Systems”, TR-EE 91-46, November 1991 Kalsi, K., Lian, J., Hui, S., Zak, S.,H., “Sliding-Mode Observers for Systems with Unknown Inputs”
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999
- Chen, J., Patton, R., J., “Robust Model-Based Fault Diagnosis for Dynamic Systems”, Kluwer Academic Publishers, 1999