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Slide Mode Control (S.M.C.)

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Slide Mode Control (S.M.C.)

  1. 1. Slide Mode Control (SMC) SOLO HERMELIN Updated: 6.03.12 1
  2. 2. Table of Content SOLO Slide Mode Control (SMC) 2 Sliding Mode Control - Introduction Control Statement of Sliding Mode Existence of a Sliding Mode Reachability: Attaining Sliding Manifold in Finite Time Picard-Lindelöf Existence and Uniqueness of a Differential Equations Solutions Uniqueness of Sliding Mode Solutions Sliding Motion Surface Keeping Controller Design Diagonalization Method Other Methods – Relays with Constant Gains Other Methods – Linear Feedback with Switched Gains Other Methods – Linear Continuous Feedback Other Methods – Univector Nonlinearity with Scale Factor Chattering
  3. 3. Table of Content (continue - 1) SOLO Slide Mode Control (SMC) 3 Higher Order Sliding Mode Control Sliding Order and Sliding Set Second Order Sliding Modes The Twisting Controller (Levantosky, 1985) The Problem Statement The Super-Twisting Controller The Super-Twisting Controller (Shtessel version) Sliding Mode Control for Linear Time Invariant (LTI) Systems Regular Form of a LTI Sliding Surface of a Regular Form of a LTI Unit Vector Approach for a Controller of a Regular Form of a LTI Output Feedback Variable Structure Controllers and State Estimators for Uncertain Dynamic Systems System Sliding Surface Sliding Modes and System Zeros Properties of the Sliding Modes Design of a Sliding Surface (Hyperplane)
  4. 4. Table of Content (continue - 2) SOLO Slide Mode Control (SMC) Sliding Mode Control for Nonlinear Systems Continuous Sliding Mode Control Second Order Sliding Mode Control Sliding Mode Observers Sliding Mode Observers of Target Acceleration
  5. 5. Table of Content (continue - 3) SOLO Slide Mode Control (SMC) Slide Mode Control Examples Control System of a Kill Vehicle Equations of Motion of a KV (Attitude) Fu, L-C, et all, Control System of a Kill Vehicle Fu, L-C, et al, Solution for Attitude Control of KV Fu, L-C, et al, Zero SM Guidance of a KV Crassidis , et al -Attitude Control of the Kill Vehicle Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude HTK Guidance Using 2nd Order Sliding Mode
  6. 6. Slide Mode Control (SMC) is a type of Variable Structure Control (VSC) that possess Robust Characteristics to System Disturbances and Parameter Uncertainties. A VSC System is a special type of Nonlinear System characterized by a discontinuous control which change the System Structure when the States reach the Intersection of Sets of Sliding Surfaces. The System behaves independently of its general dynamical characteristics and system disturbances once the controller has driven the System into a Sliding Mode. SOLO Slide Mode Control (SMC) Such a High-Level of Performance requires High-Quality Actuators requiring a Very Fast Responding and Fast Switching Action. This translates to a Very Wide Bandwidth Actuators. 6 Sliding Mode Control - Introduction A Few Examples are presented: • Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude, a solution using Sliding Mode given in 1975 (before the development of the Sliding Mode Method) • Control of a Kill Vehicle (because of the beauty of quaternion mathematics) • Hit-to-Kill (HTK) Guidance Law using a Second Order Sliding Mode Control.
  7. 7. SOLO In control theory, Sliding Mode Control, or SMC, is a form of Variable Structure Control (VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system by application of a high-frequency switching control. The state- feedback control law is not a continuous function of time. Instead, it switches from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward a switching condition, and so the ultimate trajectory will not exist entirely within one control structure. Instead, the ultimate trajectory will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a Sliding Mode and the geometrical locus consisting of the boundaries is called the sliding (hyper)surface. The sliding surface is described by σ = 0, and the sliding mode along the surface commences after the finite time when system trajectories have reached the surface. In the context of modern control theory, any variable structure system, like a system under SMC, may be viewed as a special case of a hybrid dynamical system. Sliding Mode Control (SMC) 7
  8. 8. SOLO Control Statement of Sliding Mode Consider a Nonlinear Dynamical System Affine in Control: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nxmnmn xBtxftutx tuxBtxftx RRRR ∈∈∈∈ += ,,,, , The components of the discontinuous feedback are given by: ( ) ( ) ( ) ( ) ( ) mi xiftxu xiftxu tu ii ii i ,,2,1 0, 0, =     < > = − + σ σ where σi (x) = 0 is the i-th component of the Sliding Surface, and ( ) ( ) ( ) ( )[ ] 0,,, 21 == T m xxxx σσσσ  is the (n-m) dimensional Sliding Manifold The sliding-mode control scheme involves: 1.Selection of a Hypersurface or a Manifold (i.e., the Sliding Surface) such that the system trajectory exhibits Desirable Behavior when confined to this Manifold. 2.Finding discontinuous feedback gains so that the System Trajectory intersects and stays on the Manifold. Sliding Mode Control (SMC) 8
  9. 9. SOLO Control Statement of Sliding Mode Consider a Nonlinear Dynamical System Affine in Control: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nxmnmn xBtxftutx tuxBtxftx RRRR ∈∈∈∈ += ,,,, , ( ) ( ) ( ) ( ) ( ) mi xiftxu xiftxu tu ii ii i ,,2,1 0, 0, =     < > = − + σ σ where σi (x) = 0 is the i-th component of the Sliding Surface, and ( ) ( ) ( ) ( )[ ] 0,,, 21 == T m xxxx σσσσ  is the (n-m) dimensional Sliding Manifold A Sliding-Mode exists, if in the Vicinity of the Switching Surface, σ (x) = 0, the Velocity Vector of the State Trajectory, , is always Directed Toward the Switching Surface. ( )tx Because sliding mode control laws are not continuous, it has the ability to drive trajectories to the sliding mode in finite time (i.e., stability of the sliding surface is better than asymptotic). However, once the trajectories reach the sliding surface, the system takes on the character of the sliding mode (e.g., the origin x=0 may only have asymptotic stability on this surface). Sliding Mode Control (SMC) 9
  10. 10. SOLO Existence of a Sliding Mode The Existence of the Sliding Mode requires Stability of the State Trajectory to the Sliding Surface , σ (x) = 0, at least in a Neighborhood of the Sliding Surface, i.e., the System State must approach the surface at least asymptotically. From a Geometrical point of view, in the Vicinity of the Switching Surface, σ (x) = 0, the Velocity Vector of the State Trajectory, , is always Directed Toward the Switching Surface. ( )tx Sliding Mode Control (SMC) The Existence Problem can be seen as a Generalized Stability Problem hence the Second Method of Lyapunov provides a natural setting for Analysis. Aleksandr Mikhailovich Lyapun 1857 - 1918 10
  11. 11. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Aleksandr Mikhailovich Lyapun 1857 - 1918 Definition A Domain D in a Manifold σ (x) = 0 is a Sliding Mode Domain if for each ε > 0, there is a δ >0, such that any trajectory starting within a n-dimensional δ-vicinity of D may leave the n- dimensional δ-vicinity of D only through the n-dimensional ε- vicinity of the boundary of D. Second Method of Lyapunov 11
  12. 12. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Aleksandr Mikhailovich Lyapun 1857 - 1918 For the (n-m) dimensional domain D to be the Domain of a Sliding Mode, it is sufficient that in some n-dimensional domain Ω ϵ D, there exists a function V (x,t,σ) continuously differentiable with respect to all of its arguments, satisfying the following conditions: 1.V (x,t,σ) is positive definite with respect to σ, i.e., V (x,t,σ) > 0, with σ ≠ 0 and arbitrary x,t, and V (x,t,σ=0) = 0; and on the sphere ||σ|| = ρ, for all x ϵ Ω and any t the relations holds, hρ and Hρ, depend on ρ (hρ ≠ 0 if ρ ≠0) 2.The Total Time Derivative of V (x,t,σ) for the System Dynamics has a negative supremum for all x ϵ Ω except for x on the Switching Surface where the control input are undefined, and hence the derivative of V (x,t,σ) does not exist. Second Method of Lyapunov ( ) ( ) 0,,,sup 0,,,inf >= >= = = ρρ ρσ ρρ ρσ σ σ HHtxV hhtxV 12
  13. 13. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Unfortunately, there are no standard methods to find Lyapunov Functions for Arbitrary Nonlinear Systems. Existence of Sliding Mode Consider a Lyapunov Function candidate: ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) 000 2 1 2 1 2 =⇔=⇒≥== xxVxxxxV T σσσσσσ where ||*|| is the Euclidean norm (i.e. ||σ (x)||2 is the Distance away from the Sliding Manifold where σ (x)=0 ). V (σ (x)) is Globally Positive Definite. ( ) ( ) ( ) ( )tuxBtxftx += , A Sufficient Condition for the Existence of the Sliding Mode is: 0<== td d td d d Vd td Vd T σ σ σ σ in a neighborhood of the surface σ (x)=0. ( ) ( )[ ]utxBtxf xd d td xd xd d td d ,, +== σσσ The feedback control law u (x) has a direct impact on . td d σ 13
  14. 14. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Existence of Sliding Mode (continue – 1) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) 000 2 1 2 1 2 =⇔=⇒≥== xxVxxxxV T σσσσσσ ( ) ( ) ( ) ( )tuxBtxftx += , 0<== td d td d d Vd td Vd T σ σ σ σ Roughly speaking (i.e., for the scalar control case when m = 1), to achieve , the feedback control law u (x) is picked so that σ and have opposite sign, that is ( ) ( )[ ]utxBtxf xd d td xd xd d td d ,, +== σσσ • u (x) makes negative when σ (x) is positive.xd d σ 0< td dT σ σ td d σ • u (x) makes positive when σ (x) is negative.xd d σ 14
  15. 15. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Reachability: Attaining Sliding Manifold in Finite Time ( ) ( ) ( ) ( )tuxBtxftx += , To ensure that the Sliding mode σ (x) = 0 in a Finite Time, dV/dt must be Strongly Bounded Away From Zero. That is, if it vanished to quickly, the Attraction to the Sliding Mode will only be Asymptotic. To ensure that the Sliding Mode is entered in Finite Time α µV td Vd −≤ where μ > 0 and 0 < α < 1 are constant ( ) ( ) ( ) ( )[ ] ( ) ( )0 1 0 00 1 111 0 ttVVd Vtd Vd V S V V −−≤ − −=⇒−≤ − == ∫ µσ α µ α σ σ αα This shows that the time necessary to reach the Sliding Manifold σ [x(ts)] = 0 is bounded by: ( )[ ] ( ) ( ) 100 1 1 0 0 <<> − ≤− − α αµ σ α V ttS 15
  16. 16. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Note that for all single input functions a suitable Lyapunov function is: ( ) ( )xtxV 2 2 1 ,, σσ = which is Globally Positive Definite. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]tutxBtxf xd xd x td xd xd xd x td xd xtxV td d ,,,, +=== σ σ σ σ σ σσ Suppose that we can find u (t) such that in the neighborhood of V (x,t,σ=0)=0 we have: ( ) ( ) ( ) ( ) 0,, <−≤= x td xd xtxV td d σµ σ σσ ( ) ( ) ( ) ( ) ( ) ( ) 0sgn <−≤== µ σσ σ σ σ σ td xd td xd x td xd x x ( )[ ] ( )[ ] ( )00 tttxtx −−≤− µσσ integration This shows that the time necessary to reach the Sliding Manifold σ [x(ts)] = 0 is bounded by: ( )[ ] µ σ 0 0 tx ttS ≤− Reachability: Attaining Sliding Manifold in Finite Time (continue – 1) 16
  17. 17. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Region of Attraction Reachability: Attaining Sliding Manifold in Finite Time (continue – 2) For the Dynamic System given by and for the Sliding Surface σ (x) = 0, the subspace for which the Sliding Surface is Reachable is given by ( ) ( ){ }0: <∈ xxRx Tn σσ  ( ) ( ) ( ) ( )tuxBtxftx += , When Initial Conditions come from this Region, the Lyapunov Function Candidate is a Lyapunov Function and the Space Trajectories are sure to move toward Sliding Mode Surface σ (x) = 0. Moreover, if the Reachable Condition is satisfied, the Sliding Mode will reach σ (x) = 0 in Finite Time. ( )[ ] ( ) ( ) 2/xxxV T σσσ = ( ) ( )10 <<−≤ αµ α VV 17
  18. 18. SOLO Picard-Lindelöf Existence and Uniqueness of a Differential Equations Solutions Sliding Mode Control (SMC) Lipschitz Continuity Condition Charles Émile Picard 1856 - 1941 Ernst Leonard Lindelöf 1870 - 194618 Rudolf Otto Sigismund Lipschitz 1832 – 7 1903 In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. A Function f (x) is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X: ( ) ( ) 1212 xxKxfxf −≤− Picard–Lindelöf Theorem Consider the initial value problem ( ) ( ) [ ]εε +−∈== 0000 ,,,, tttxtxtxf td xd Suppose f is Lipschitz continuous in x and continuous in t. Then, for some value ε > 0, there exists a Unique Solution x(t) to the initial value problem within the range [t0-ε,t0+ε].
  19. 19. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Aleksandr Mikhailovich Lyapu 1857 - 1918 Uniqueness of Sliding Mode Solutions The Nonlinear Dynamical System Affine in Control: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nxmnmn xBtxftutx tuxBtxftx RRRR ∈∈∈∈ += ,,,, , ( ) ( ) ( ) ( ) ( ) mi xiftxu xiftxu tu ii ii i ,,2,1 0, 0, =     < > = − + σ σ with the switching control, do not formally satisfy the classical Picard-Lindelöf Existence and Uniqueness Solutions, since they have discontinuou right-hand sides. Moreover the right-hand sides usually are not defined on the discontinuous surfaces. Charles Émile Picard 1856 - 1941 Ernst Leonard Lindelöf 1870 - 1946 Existence and Uniqueness of Differential Equations with Discontinuous Right-hand Sides is was addressed by different researchers. One of the straightforward approaches is the Method of Filippov (Filippov Aleksei Fedorovich, “Differential Equations with Discontinuous Right Hand Sides”, Kluwer, Dordrecht, the Nederlands) 19
  20. 20. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Second Method of Lyapunov Uniqueness of Sliding Mode Solutions (continue – 1) ( ) ( )utxftx ,,= ( ) ( ) ( ) ( ) ( )    < > = − + 0, 0, xiftxu xiftxu tu σ σ Method of Filippov (Filippov Aleksei Fedorovich) Consider the n-order Single Input System: with the following Control Strategy: The System Dynamics is not defined on σ (x) = 0. Filippov has shown that the solution on the Surface σ (x) = 0 is given by the equation: ( ) ( ) ( ) ( ) 10,,1,, 0 111 ≤≤=−+= −+ ααα nxnxnx futxfutxftx The term α is a function of the System States and can be specified in such a way that the “average” dynamics f0 is tangent to the Surface σ (x) = 0 . 20
  21. 21. SOLO Existence of a Sliding Mode Sliding Mode Control (SMC) Sliding Motion Surface Keeping ( ) ( ) ( ) ( ) 0, = ∂ ∂ + ∂ ∂ = ∂ ∂ = equxB x txf x tx x x σσσ σ  ( ) 0=xσ The Dynamic System will stay on the Sliding Surface σ (x) = 0, if the equivalent control ueq will keep ( )xB x nxm mxn       ∂ ∂ σ If is nonsingular, i.e., the System has a kind of Controllability that assures that we can find a controller to move a trajectory closer to σ (x) = 0, then ( ) ( )txf x xB x ueq , 1 ∂ ∂       ∂ ∂ −= − σσ ( ) ( ) ( )txf x xB x Itx , 1         ∂ ∂       ∂ ∂ −= − σσ The Dynamic System Equation is Note that using ueq any trajectory that starts at σ (x) = 0, remains on it, Since . As a consequence the Sliding Manifold σ (x) = 0 is an Invariant Set. ( ) 0=xσ Return to Chattering 21
  22. 22. SOLO Sliding Mode Control (SMC) Controller Design ( ) ( ) ( ) ( )tutxB x txQtu ,,* 1       ∂ ∂ = − σ We must choose Switched Feedback capable of forcing the Plant State Trajectories to the Switching Surface and maintaining a Sliding Mode Condition. We assume that the Sliding Surface has already been designed. Diagonalization Method The Diagonalization Method converts the multi – input design problem into m single-input design problems. The Method is based on the construction of a new control vector u* through a nonsingular transformation of ueq: where Q (x,t) is an arbitrary mxm Diagonal Matrix with elements qi (x,t), i=1,…,m, such that inf |qi (x,t)| > 0 for all t ≥ 0 and all x. 22
  23. 23. SOLO Sliding Mode Control (SMC) Controller Design (continue – 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tutxQtutxB x tutxB x txQtu *,,,,* 1 =      ∂ ∂ ⇒      ∂ ∂ = − σσ Diagonalization Method (continue – 1) where Q (x,t) is an arbitrary mxm Diagonal Matrix with elements qi (x,t), i=1,…,m, such that inf |qi (x,t)| > 0 for all t ≥ 0 and all x. For existence and reachability of a Sliding Mode is enough to satisfy .0< td dT σ σ ( ) ( ) ( ) ( ) ( ) ( ) mitutxqtxf xd d td d ortutxQtxf xd d td xd xd d td d ii i i ,,1,,,, ** =+      =+== σσσσσ To satisfy the existence and reachability we choose each control u*i as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tutxQtxB x tu xwhentxf x utxq xwhentxf x utxq i i n j j j ii i i n j j j ii *,, 0,, 0,, 1 1 * 1 * − = + = +             ∂ ∂ =⇒        <      ∂ ∂ −> >      ∂ ∂ −< ∑ ∑ σ σ σ σ σ 23
  24. 24. SOLO Sliding Mode Control (SMC) Controller Design (continue – 2) Other Methods A possible structure for the control is: miuuu iNii eq ,,2,1 =+= Where is continuous and uiN is the discontinuous part ( ) ( ) i i txf x xB x u eq         ∂ ∂       ∂ ∂ −= − , 1 σσ ( ) ( )( ) ( ) ( ) ( ) ( ) NNeq Neq uxB x uxB x uxB x txf x uuxB x txf xtd d       ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ = + ∂ ∂ + ∂ ∂ = σσσσ σσσ    0 , , Several Design Methods are applicable 24
  25. 25. SOLO Sliding Mode Control (SMC) Second Method of Lyapunov Controller Design (continue – 3) Other Methods – Relays with Constant Gains A possible structure for the control is: miuuu iNii eq ,,2,1 =+= where is continuous and uiN is the discontinuous part ( ) ( ) i i txf x xB x u eq         ∂ ∂       ∂ ∂ −= − , 1 σσ ( ) NuxB xtd d       ∂ ∂ = σσ ( ) ( )[ ] ( )      = <− > ==>      ∂ ∂ −= − 00 01 01 sgn0sgn 1 xif xif xif x x xxxB x u ii i iiN ασ σ α This controller satisfies the reaching condition since: ( ) ( ) ( ) ( )[ ] ( ) ( ) 00sgn ≠<−=−= xifxxx td xd x iiiiii i i σσασσα σ σ 25
  26. 26. SOLO Sliding Mode Control (SMC) Second Method of Lyapunov Controller Design (continue – 4) Other Methods – Linear Feedback with Switched Gains A possible structure for the control is: miuuu iNii eq ,,2,1 =+= where is continuous and uiN is the discontinuous part ( ) ( ) i i txf x xB x u eq         ∂ ∂       ∂ ∂ −= − , 1 σσ ( ) NuxB xtd d       ∂ ∂ = σσ ( ) [ ]     <<− >> =ΨΨ=Ψ         Ψ      ∂ ∂ = − 00 001 jjij iiij ijij i iN xif xif xxB x u σβ σασ This controller satisfies the reaching condition since: ( ) ( ) ( )( ) 011 <Ψ++Ψ= ninii i i xxx td xd x σ σ σ 26
  27. 27. SOLO Sliding Mode Control (SMC) Second Method of Lyapunov Control Methods (continue – 5) Other Methods – Linear Continuous Feedback A possible structure for the control is: miuuu iNii eq ,,2,1 =+= where is continuous and uiN is the discontinuous part ( ) ( ) i i txf x xB x u eq         ∂ ∂       ∂ ∂ −= − , 1 σσ ( ) NuxB xtd d       ∂ ∂ = σσ ( )xLu nxnN σ−= This controller satisfies the reaching condition since: ( ) ( ) ( ) ( ) ( ) 00 ≠<−= xifxLx td xd x TT σσσ σ σ where Lnxn is a Positive Definite Constant Matrix 27
  28. 28. SOLO Sliding Mode Control (SMC) Second Method of Lyapunov Controller Design (continue – 6) Other Methods – Univector Nonlinearity with Scale Factor A possible structure for the control is: miuuu iNii eq ,,2,1 =+= where is continuous and uiN is the discontinuous part ( ) ( ) i i txf x xB x u eq         ∂ ∂       ∂ ∂ −= − , 1 σσ ( ) NuxB xtd d       ∂ ∂ = σσ ( ) ( ) ( ) ( ) ( ) ( )xxx x x xu T N σσσρ σ σ ρ =>−= 2 2 &0 This controller satisfies the reaching condition since: ( ) ( ) ( ) ( ) 002 ≠<−= xifx td xd xT σσρ σ σ 28
  29. 29. SOLO Sliding Mode Control (SMC) Chattering Due to the presence of external disturbance, noise and inertia of the sensors and actuators the switching around the Sliding Surface occurs at a very high (but finite) frequency. The main consequence is that the Sliding Mode take place in a small neighbor of the Sliding Manifold , which is called Boundary Layer, and whose dimension is inversely proportional with the Control Switching Frequency. The effect of High Frequency Switching is known as Chattering. The High Frequency Switching propagate through the System exciting the fast dynamics and undesired oscillations that affect the System Output To prevent the Chattering Effect different techniques are used. One of the techniques is the use of continuous approximations of sign (.) (such as sat (.) function, the tanh (.) function,..) in the implementation of the Control Law. A consequence of this method is that the Invariance Property is Lost. Invariance Definition 29
  30. 30. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Sliding Order and Sliding Set The Sliding Order r is the number of continuous total derivative, including the zero one, of the function σ = σ (t,x) whose vanishing defines the equations of the Sliding Manifold. The Sliding Set of r – th order associated in the Manifold σ (t,x) = 0 is defined by the equalities ( ) 01 ===== −r σσσσ  which forms an r – dimensional condition on the State of the Dynamic System. The corresponding motion satisfying the equalities is called an r – order Sliding Mode with respect to the Manifold σ (t,x) = 0 . 30
  31. 31. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Problem Statement Consider a Dynamic Single Output System of the form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 ,,,, , nxnn xbtxftutx tuxbtxftx RRRR ∈∈∈∈ += Let σ (t,x) = 0 be the chosen Sliding Manifold, then the Control Objective is to enforce a Second Order Sliding Mode on the Sliding Manifold σ (t,x) = 0 , i.e., in Finite Time. Let analyze the following two cases: Case A: relative degree 01 ≠ ∂ ∂ ⇒= σ u r Case B: relative degree 0,02 ≠ ∂ ∂ = ∂ ∂ ⇒= σσ  uu r 31 ( ) ( ) 0,, == xtxt σσ 
  32. 32. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Problem Statement (continue – 1) ( ) ( ) ( ) ( )tuxbtxftx += , Case A: relative degree 01 ≠ ∂ ∂ ⇒= σ u r ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )tuuxtuxt tuxtbxtfxt x xt t AA   ,,,, ,,,, γϕσ σσσ += + ∂ ∂ + ∂ ∂ = ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )        ∂ ∂ = + ∂ ∂ + ∂ ∂ = xtbxt x xt tuxtbxtfxt x xt t uxt A A ,,:, ,,,,:,, σγ σσϕ   The control u is understand as an internal disturbance affecting the drift term φA. The control derivative is used as an auxiliary control used to steer σ and to 0. Note that affect the dynamics. σu u σ 32 ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  33. 33. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Problem Statement (continue – 2) ( ) ( ) ( ) ( )tuxbtxftx += , Case B: relative degree 0,02 ≠ ∂ ∂ = ∂ ∂ ⇒= σσ  uu r ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( )tuuxtuxt tuxbxtfxt x xt t BB ,,,, ,,, 0 γϕσ σσσ +=         + ∂ ∂ + ∂ ∂ =   ( ) ( ) ( ) ( ) ( ) ( ) ( )        ∂ ∂ = ∂ ∂ + ∂ ∂ = xtbuxt x xt xtfuxt x uxt t uxt B B ,,,:, ,,,,,:,, σγ σσϕ   It is assumed that , which means that the sliding variable has relative degree two. In this case the actual actuator u is discontinuous. ( ) 0, ≠xtBγ 33 ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  34. 34. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Problem Statement (continue – 3) ( ) ( ) ( ) ( )tuxbtxftx += , Case B: relative degree 0,02 ≠ ∂ ∂ = ∂ ∂ ⇒= σσ  uu r Case A: relative degree 01 ≠ ∂ ∂ ⇒= σ u r Both Cases A and B can be dealt with an uniform treatment, because the structure of the System to be stabilized is the same, i.e. a 2nd Order System with Affine relevant control signal (the control derivative in Case A, the actual control u in Case B).u ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     =+⋅= == = xttvxtty xttyty xtty ,, , , 2 21 1 σγϕ σ σ   Case A: relative degree r = 1 ( ) ( ) ( ) ( )   = =⋅ tutv uxtA  ,,ϕϕ Case B: relative degree r = 2 ( ) ( ) ( ) ( )   = =⋅ tutv uxtB ,,ϕϕ Assume ( ) ( ) 21 ,0 GxtG ≤≤< Φ≤⋅ γ ϕ 34 Case A: relative degree r = 1 ( ) ( ) ( ) ( )   = =⋅ tutv uxtA  ,,ϕϕ ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  35. 35. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Twisting Controller (Levantosky, 1985) This Algorithm provides twisting around the origin of the phase plane . This means that trajectories perform rotations around the origin while converging in finite time to the origin of the phase plane. σσ O ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     =+⋅= == = xttvxtty xttyty xtty ,, , , 2 21 1 σγϕ σ σ   ( ) ( ) ( )     ≤>=− ≤≤=− >− = 1;0 1;0 1 211 211 uyyifysignV uyyifysignV uifu tu M m σσ σσ   ( ) ( ) 21 ,0 GxtG ≤≤< Φ≤⋅ γ ϕ 35 Case A: relative degree r = 1 ( ) ( ) ( ) ( )   = =⋅ tutv uxtA  ,,ϕϕ Levant, Arie ( formerly Levantosky, Lev ) ( ) ( ) ( ) ( )tuxbtxftx += , ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  36. 36. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Twisting Controller (Levantosky, 1985) This Algorithm provides twisting around the origin of the phase plane . This means that trajectories perform rotations around the origin while converging in finite time to the origin of the phase plane. σσ O ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     =+⋅= == = xttvxtty xttyty xtty ,, , , 2 21 1 σγϕ σ σ   ( ) ( ) ( ) ( )   >=− ≤=− == 0 0 211 211 σσ σσ   yyifysignV yyifysignV tutv M m ( ) ( ) 21 ,0 GxtG ≤≤< Φ≤⋅ γ ϕ The following conditions must be fulfilled for the finite time convergence ( ) Φ+>Φ− Φ > > > mM m m mM VGVG G V G V VV 21 1 2 0 4 σ 36 Case B: relative degree r = 2 ( ) ( ) ( ) ( )   = =⋅ tutv uxtB ,,ϕϕ ( ) ( ) ( ) ( )tuxbtxftx += , ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  37. 37. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Super-Twisting Controller ( ) ( ) ( ) ( ) ( ) ( )    −= +−= 121 1111 ysigntu tuysigntytu α α ρ  The control is given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     =+⋅= == = xttvxtty xttyty xtty ,, , , 2 21 1 σγϕ σ σ   ( ) ( ) 21 ,0 GxtG ≤≤< Φ≤⋅ γ ϕ ( ) ( ) 5.00 4 1 2 21 22 2 1 2 1 ≤< Φ > Φ− Φ+Φ ≤ ρ α α α α G G G G ( ) ( ) ( ) ( )∫−−= t dtysignysigntytu 0 12111 αα ρ 37 The following conditions must be fulfilled for the finite time convergence ( ) ( ) ( ) ( )tuxbtxftx += , ( ) ( ) 0,,: == xtxtSurfaceSliding σσ  ( ) ( ) ( ) ( )   = =⋅ tutv uxtB ,,ϕϕ
  38. 38. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Second Order Sliding Modes The Super-Twisting Controller (continue) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    ≤− >− =     ≤− >− = += 01111 01101 2 12 1 21 1 1 σα σσα α ρ ρ yifysigny yifysign tu uifysign uifu tu tututu  The control is given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     =+⋅= == = xttvxtty xttyty xtty ,, , , 2 21 1 σγϕ σ σ   ( ) ( ) 21 ,0 GxtG ≤≤< Φ≤⋅ γ ϕ ( ) ( ) 5.00 4 1 2 21 22 2 1 2 1 ≤< Φ > Φ− Φ+Φ ≤ ρ α α α α G G G G 38 The following conditions must be fulfilled for the finite time convergence ( ) ( )utxftx ,,= General Nonlinear System ( ) ( ) ( ) ( )tuxbtxftx += , ( ) ( ) 0,,: == xtxtSurfaceSliding σσ 
  39. 39. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Sliding Order Sliding Modes The Super-Twisting Controller (Shtessel version) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      >−= >=+−= = 0 0, , 21 3/1 122 121 2/1 111 1 αα ασα σ ysigntyty xttyysigntyty xtty   The 2nd Order Sliding Mode is Given by is Finite Time Stable, i.e., is Asymptotically Stable with a Finite Settling Time for any solution and any initial conditions. Proof Let choose the following Lyapunov Function candidate: ( )      == ≠≠>+ = 000 000 4 3 2, 21 21 3 4 12 2 2 21 yandyifonly yandyify y yyV α ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] 6/5 1211 3/1 12221 2/1 111 3/1 12 2211 3/1 122 2 1 1 21 21 , yysigntyyyysignyysigny yyyysignyy y V y y V yyV td d yy ααααα α −=−++−= += ∂ ∂ + ∂ ∂ =        ( ) 00, 1 6/5 12121 ≠<−= yifyyyV td d αα 39
  40. 40. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control alpha1=3, alpha2 =3 alpha1=1, alpha2 =9 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 -2000 -1000 0 1000 X1 X1dot alpha1=1, alpha2 =1 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 -400 -200 0 200 400 X1 X1dot alpha1=3, alpha2 =1 -2000 -1000 0 1000 2000 3000 4000 5000 -400 -200 0 200 X1 X1dot -3000 -2000 -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 X1 X1dot alpha1=3, alpha2 =9 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 1000 X1 X1dot alpha1=9, alpha2 =1 -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 X1 X1dot -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 X1 X1dot alpha1=9, alpha2 =3 alpha1=9, alpha2 =9 -2000 -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 X1 X1dot Time = 100 sec -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000 -1000 -500 0 500 X1 X1dot alpha1=1, alpha2 =3 We can see that to speed-up the Convergence to Origin we must Increase alpha1 and keep alpha2 Small relative to alpha1. 40 The Super-Twisting Controller (Shtessel version)
  41. 41. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control ( ) ( )      == ≠≠>+ = ∫ 000 000 2, 21 21 0 3/1 2 2 2 21 1 yandyifonly yandyifdzzsignz y yyV y α >> [X,Y]=meshgrid(-0.5:0.05:0.5); >> alpha2=1; >> Z=0.5*Y.^2+0.75*alpha2*abs(X).^1.3334+eps; >> mesh(X,Y,Z,'EdgeColor','black') >> contour(X,Y,Z) -0.5 0 0.5 -0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 1y2y ( )21, yyV MATLAB: 1y 2y -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 41 The Super-Twisting Controller (Shtessel version)
  42. 42. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Theorem: Assume a Lyapunov Function that satisfies ( ) ( ) 00,& 000 000 , 121 21 21 21 ≠<    == ≠≠> = yifyyV td d yandyifonly yandyif yyV Assume that in addition exists a Domain , that includes the origin (y1 = 0, y2 = 0), and in this domain ( ) Dyy ∈21, ( ) ( ) ( )10&00,, 2121 <<>≤+ αα kyyVkyyV td d then V (y1, y2) → 0 in a Finite Time ts. Proof: ( ) ( ) 0,, 2121 >−≤ yyV td d yyVk α ( ) ( ) 0 , ,1 21 21 >−≤ α yyV yyVd k td 0>td q.e.d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )100 1 , ,, 1 1 1 211 21 1 0 210 0 0 <<←> − =         − − −≤− − −− α αα α αα k yyV yyVyyV k tt t ttS S  We can see that ts at which V (y1, y2) → 0 is Finite. Let integrate between an initial time t0 to a time ts at which V (y1, y2) → 0. 42 The Super-Twisting Controller (Shtessel version)
  43. 43. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Let find, if in our case, exists a Domain D, that includes the origin, and satisfies: ( ) ( ) ( )10&00,, 2121 <<>≤+ αα kyyVkyyV td d We have: ( )      == ≠≠>+ = 000 000 4 3 2, 21 21 3 4 12 2 2 21 yandyifonly yandyify y yyV α ( ) 00, 1 6/5 12121 ≠<−= yifyyyV td d αα ( ) ( ) 6 5 1 21213 4 12 2 2 21 ,1 4 3 2 , y ktd yyVd k y y yyV αα α α α =−≤        += ( ) 6 5 1 213 4 12 10 2 3 3 4 12 2 2 21 2 3 4 3 2 , 2 1 12 2 2 y k yy y yyV yy αα αα α α α α α ≤      ≤        += << ≤ Define the Domain D that includes the origin by: 1 2 3 1 3 4 12 2 2 <≤ yandyy α α ααα α 1 6 5 1 1 213 4 12 2 3 y k y       ≤or: If since1 6 5 1 1 6 5 <<⇒< α α 11 3 4 ,1 1 6 5 1 <<>< yand α from the Figure, and choosing some k>0, we can see that exists some small |y1s| such that for we have3 4 102 2 2101 2 3 yyandyy α≤≤ ( ) ( ) ( )100,, 2121 <<≤+ αα yyVkyyV td d α α αα α 1 21 2 1 6 5 3 4 10 3 2       =       − k y equality 1y 2y -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 43 The Super-Twisting Controller (Shtessel version)
  44. 44. SOLO Sliding Mode Control (SMC) Higher Order Sliding Mode Control Therefore for: Return to Table of Content ( ) 3 4 10221 2 1 3 4 102202101 2 3 , 2 3 , 0 yyyVandyyyyy t αα =      =≤≤ V (y1, y2) → 0 in Finite Time ts . ( ) ( ) ( ) ( )100 1 , 1 21 0 0 <<←> − ≤− − α α α k yyV tt t S -0.5 0 0.5 -0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 1y2y ( )21, yyV 1y 2y -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 44 The Super-Twisting Controller (Shtessel version)
  45. 45. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Regular Form of a LTI Consider the following LTI: mnRBRARuRxuBxAx nxmnxnmn >∈∈∈∈+= Assume rank (B) = m (i.e., matrix B is full rank) and the pair (A,B) is controllable. Perform the Singular Value Decomposition (SVD) of B: ( ) ( )H B xmmn B Bnxm mxm mxm nxn VUB 1 1 1 0        Σ = − where H means Transpose of a matrix and complex conjugate of it’s elements, and: mB H B H BBnB H B H BB IVVVVIUUUU ==== 11111111 ; ( ) ( ) ( )mxmmnxnn mmB diagIdiagI diagmxm 1,,1,1,1,,1,1 0,,,, 21211   == >≥≥≥=Σ σσσσσσ 45
  46. 46. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Regular Form of a LTI (continue - 1) Consider the following LTI: nxmnxnmn RBRARuRxuBxAx ∈∈∈∈+= Define the Orthogonal Transformation Matrix: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )H nxn xmmnmn mmnmx Bnxn H B mnmxm mnxmmn nxn T I I UTU I I T nxnnxn =         =⇒         = −− −− − −− 0 0 0 0 : 1 1 1 ( ) ( ) ( ) midiagVUB immxmmxm mxm nxn BBBB H B xmmn B Bnxm ,,10,, 0 111 1 1  =≠=Σ        Σ = − λλλS.V.D. of B: ( ) ( ) ( ) ( ) ( ) 0det 000 2 211 1 1 ≠⇒         =         Σ =         Σ = −−− mxm mxmmxmmxm mxm mxm B BV VBT xmmn H BB xmmnH B B xmmn nxmnxn Perform the following Transformation of Variables: xTx nxnr =: 46
  47. 47. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Regular Form of a LTI (continue - 2) Consider the following LTI: nxmnxnmn RBRARuRxuBxAx ∈∈∈∈+= Perform the following Transformation of Variables: xTx nxnr =: We obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0det 0 21 22 1 2221 1211 2 1 1 1 1 1 ≠         +                 =         = −− − −−−− mxm mxmmx xmn mxmmnmx xmmnmnxmn mx xmn Bu Bx x AA AA x x x mx xmmn r r r r r    This is called the Regular Form of the LTI 47
  48. 48. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Sliding Surface of a Regular Form of a LTI Consider the following Regular Form of a LTI: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 22 1 2221 1211 2 1 0 1 1 1 1 mx xmmn r r r r r u Bx x AA AA x x x mxmmx xmn mxmmnmx xmmnmnxmn mx xmn         +                 =         = −− − −−−−    Define a Switching Function s (t) ( ) ( ) [ ] ( ) [ ] 0det0 22211 2 1 211 1 1 ≠=+=         = − − mxm mx xmn mxmmnmx SxSxS x x SSts rr r r S mx     Therefore, on the Sliding Surface: ( ) ( ) ( )txMtxSStx rr M r 111 1 22 −=−= −  ( ) 112112121111 rrrr xMAAxAxAx −=+=Then we have: By analogy to the “Classical” State-Feedback Theory, it can be seen that this is the same problem of finding the State-Feedback matrix M for the Regulator Form where xr2(t) plays the role of the “control” signal. 48
  49. 49. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Sliding Surface of a Regular Form of a LTI (continue - 1) 11 1 212112121111 r M rrr xSSAAxAxAx         −=+= −  Then we have: The Stability and Performance of the System depends on the Controllability of the Regular Form Pair (A11, A12). It can be shown that (A11, A12) is Controllable, if and only if the pair (A , B) is Controllable. Controllability of the Regular Form is [ ] ( )               + +             − −   2 22221221 22121211 2 22 12 2 1 0 B AAAA AAAA B A A B I nrankfullBABAB m xmmn r n rrrr 49
  50. 50. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Sliding Surface of a Regular Form of a LTI (continue - 2) The design of the Sliding Mode Controller must achieve: • The design of the Matrix S=[S1,S2] to obtain the required performance and Stable Dynamics for the Closed-Loop Sliding Mode System. • The design of the Control Law to ensure that the Sliding Surface is Reached and Maintained. For the Sliding Surface Reachability Condition let define the Lyapunov Function: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    === ≠=≠ = 00 00 2 1 : tststsif tststsif tststV T T T The Reachability Condition is: ( ) ( ) ( ) ( ) 0& >∀−≤= µµ someforttststs td tVd T  50
  51. 51. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Sliding Surface of a Regular Form of a LTI (continue - 3) The design of the Sliding Mode Controller must achieve: • The design of the Matrix S = [S1,S2] to obtain the required performance and Stable Dynamics for the Closed-Loop Sliding Mode System. • The design of the Control Law to ensure that the Sliding Surface is Reached and Maintained. ( ) ( ) 0== tsts  Consider the following LTI: ( )[ ] ( ) nxmnxnmn RBRARxtDuRxxtDBuBxAx ∈∈∈∈++= ,,, ξξ ( )xtD ,ξ are the uncertainties in the input On the Sliding Surface ( ) ( ) ( )[ ]{ } 0, =++== xtDBuBxAStxSts eq ξ The Equivalent Control that Maintains the System on the Sliding Surface is ( ) ( ) ( )[ ]xtDBSxASBStueq , 1 ξ+−= − 51
  52. 52. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Sliding Surface of a Regular Form of a LTI (continue - 4) The design of the Sliding Mode Controller must achieve: • The design of the Matrix S = [S1,S2] to obtain the required performance and Stable Dynamics for the Closed-Loop Sliding Mode System. • The design of the Control Law to ensure that the Sliding Surface is Reached and Maintained. Consider the following LTI: ( )[ ] ( ) nxmnxnmn RBRARxtDuRxxtDBuBxAx ∈∈∈∈++= ,,, ξξ ( ) ( ) ( )[ ] ( ) 0, 1 =⇐+−= − tsxtDBSxASBStueq ξ ( )[ ] ( )[ ] ( )[ ]xtDBSBSBIxASBSBIx n P n S , 0 11 ξ       −− −+−= xAPx S= 52
  53. 53. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI Consider the System: where is an unknown but bounded (matched) uncertainty that satisfies: ( )[ ] mn m nxmnxnmn m RRRfRBRARuRxuxtfuBxAx xx,, 1 ∈∈∈∈∈++= ( )uxtfm ,, ( ) ( ) ( )xttukuxtfm ,,, α+≤ By using the Orthogonal Transformation T we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )uxtf B u Bx x AA AA x x x rm xmmn mx xmmn r r r r r mxmmxmmx xmn mxmmnmx xmmnmnxmn mx xmn ,, 00 2 1 22 1 2221 1211 2 1 1 1 1 1         +         +                 =         = −−− − −−−−    The Switching Function can be written: ( ) [ ] 0det 2 2 1 1 1 22 2 1 21 ≠              =      = − S x x ISSS x x SSts r r m Mr r    53
  54. 54. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 1) Define the Sub-System: Let Differentiate this: ( ) ( )uxtf BS u BSs x SASSASMASASMASAS SAMAA uxtf B u Bs x SSS I AA AA SS I s x rm r rm rr ,, 00 ,, 0000 2222 1 1 2222 1 2121222212121111 1 2121211 22 1 1 21 1 22221 1211 21 1       +      +              +−+− − =               +      +            −             =      −− − −−   or: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )             − =                  =      −− ts tx SSS I tx tx tx tx SS I ts tx r r r r r T r S 1 1 21 1 22 1 2 1 21 1 0 & 0  ( )uxtf B u BsS x AAMMAMMAAMA AMAA sS x rm rr ,, 00 22 1 2 1 221212221121 121211 1 2 1       +      +            +−−+ − =         −−   54
  55. 55. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 2) The Sub-System: In order to force s to zero, Φ must satisfy a Lyapunov Equation of the type: ( )uxtf B u BsS x AA AA sS x rm rr ,, 00 22 1 2 1 2221 1211 1 2 1       +      +              =         −−   Choose: NonlinearLinear uuu += ( ) suBSsSASxASs Linearr Φ=++= − 22 1 22221212  ( ) ( )[ ]sSASxASBSu rLinear Φ−+−= −− 1 22221212 1 22 m T IPP −=Φ+Φ 22 ( ) ( ) sP sP BSxtu rNonlinear 2 21 22, − −= ρChoose: and: The Linear Part must keep the System on the Sliding Manifold ( )0== ss The Nonlinear Part must force the System to Reach the Sliding Manifold. 55
  56. 56. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 3) The Sub-System: ( )uxtf B u BsS x AA AA sS x rm rr ,, 00 22 1 2 1 2221 1211 1 2 1       +      +              =         −−   Choose: NonlinearLinear uuu += ( ) ( )[ ]sSASxASBSu rLinear Φ−+−= −− 1 22221212 1 22 ( ) ( ) sP sP BSxtu rNonlinear 2 21 22, − −= ρ The Linear Part must keep the System on the Sliding Manifold ( )0== ss The Nonlinear Part must force the System to Reach the Sliding Manifold. ( ) ( ) ( )[ ] ( ) ( ) ( )uxtfB sP sP BSxtBsSASxASBSBsSAxAsS rmrrr ,,, 2 2 21 222 1 22221212 1 222 1 222121 1 2 +−Φ−+−+= −−−−− ρ ( ) ( )uxtfBS sP sP xtss rmr ,,, 22 2 2 +−Φ= ρ 56
  57. 57. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 4) For the Sub-System: 002 ≠>= sifsPsV T Choose a potential Lyapunov Function that shows the Reachability of the Sliding Surface: ( ) ( )uxtfBS sP sP xtss rmr ,,, 22 2 2 +−Φ= ρ         +−Φ+        +−Φ=+== m T T m TT fBS sP sP sPssPfBS sP sP ssPssPsV td Vd 22 2 2 2222 2 2 22 ρρ ( ) ( ) m TT I TT fBSPssPPs sP sPPs 22222 2 22 2 1 2 +−Φ+Φ= − ρ  m TT fBSPssPss 2222 22 +−−= ρ 57
  58. 58. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 5) For the Sub-System: 002 ≠>= sifsPsV T Choose a potential Lyapunov Function that shows the Reachability of the Sliding Surface: ( ) ( )uxtfBS sP sP xtss rmr ,,, 22 2 2 +−Φ= ρ m TT fBSPssPssV 2222 22 +−−= ρ mm T fBSsPfBSPs 222222 ≤Using Cauchy-Schwarz Inequality: ( )mfBSsPsV 2222 −−−≤ ρ We want to choose ρ (t,x) such that and( )( )xtukBSfBS m ,2222 αρ +≥> ( ) 02 222 <−−−≤ mfBSsPsV ρ 58
  59. 59. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Unit Vector Approach for a Controller of a Regular Form of a LTI (continue – 6) ( )mfBSsPsV 2222 −−−≤ ρ We want to choose ρ (t,x) such that and( )( )xtukBSfBS m ,2222 αρ +≥> ( ) 02 222 <−−−≤ mfBSsPsV ρ 002 ≠>= sifsPsV T ( ) 1 22 1 22 −− +≤+=+≤+= BSuBSuuuuuu LinearLinearNonlinearLinearNonlinearLinear ρρ ( )( ) k xtukBS Linear − + ≥ 1 ,22 α ρ ( ) ( ) ( ) 1 22 1 2222 1 2222 1 22 1 −−−− ≥⇒≤=⇒= BSBSBSBSIBSBSI automatically fulfilled for k > 1 Define η (t,x) > 0 such that 1 22 − +≥+ BSuu Linear ρη ( ) ( ) ( )( ) ( )( )xtBSkukBSxtukBSkfBS Linearm ,, 1 22222222 αραηηρ ++≥++≥+> − 59
  60. 60. SOLO Sliding Mode Control (SMC) where u1 and u2 are the known and unknown inputs, respectively. System Description and Notation 212211 21212211 21 2121 mmppCrankmBrankmBrank RCRyxCy RBRBRARuRuRxuBuBxAx pxn pxnp nxmnxmnxnmmn nxmnxm +≥=== ∈∈= ∈∈∈∈∈∈++= Assumptions: • There exists a known nonnegative scalar function such that( )yt,ρ ( ) tytu ∀≤ ,2 ρ • The pairs (A,B1), (A,B2) are controllable and (A,C) is observable with the matrices B1, B2 and C being of full rank • p ≥ m1+m2, that means that bthe number of output channels is greater or equal then the number of inputs, and rank (CB1) =m1, rank (CB2) – m2 Sliding Mode Control for Linear Time Invariant (LTI) Systems 60 Output Feedback Variable Structure Controllers and State Estimators for Uncertain Dynamic Systems
  61. 61. SOLO Sliding Mode Control (SMC) where u1 and u2 are the known and unknown inputs, respectively. System Sliding Surface xCy uBuBxAx = ++= 2211  Assume that the Sliding Surface is of the type: ( ) xSx 1=σ On the Sliding Surface we must have: ( ) ( ) 0&0 == xx σσ  ( ) ( ) 0221111 =++== uBuBxASxSx σ If (S B1) is nonsingular: ( ) ( )221 1 111 uBxASBSu eq +−= − On the Sliding Surface the dynamics of the System is given by: ( )[ ] 01 1 1 111 = −= − xS xASBSBIx Note that in the Sliding Mode the System is governed by a reduced order of differential equations with the eigenvalues of , and are not affected by the unknown inputs/disturbances. ( )[ ]ASBSBI 1 1 111 − − Sliding Mode Control for Linear Time Invariant (LTI) Systems 61 Output Feedback Variable Structure Controllers and State Estimators for Uncertain Dynamic Systems
  62. 62. SOLO Sliding Mode Control (SMC) The Transmission Zeros of the System are defined as the solutions for λ of: Sliding Modes and System Zeros Consider the System: ( ) 1 1 1 1 1 0 0 0 det mn S BAI rankwhichforor S BAI z nn +<      −− =      −− = λ λ λ λ The solutions are not affected if we multiply the Square Matrix by Non- Singular Square Matrices that are not functions of λ: ( )       −−− =                           −−       −−− 00 00 00 0 1 1 1 1 1 1 1 1 111 TSM NBTTFBAIT rank N T IF I S BAI M T rank xmm nxn m nn mxm nxn λλ ( )       −−− =                     −−       −       −−− 00 0 000 0 1 1 1 1 1 1 1 1 111 TSM NBTTSHAIT rank N T S BAI I HI M T rank xmm nxnn m n mxm nxn λλ xnmm nxmnxnmn RSRxS RBRARuRxuBxAx 11 11 11 1111 ∈∈= ∈∈∈∈+= σσ  Sliding Mode Control for Linear Time Invariant (LTI) Systems 62
  63. 63. SOLO Sliding Mode Control (SMC) The Transmission Zeros of the System are defined as the solutions for λ of: Sliding Modes and System Zeros (continue – 1) Consider the System: ( ) 0 0 det 1 1 =      −− = S BAI z nλ λ The System Zeros are not affected under the following set of transformations: • Nonsingular State Transformation ( )xTx =~ • State Feedback ( A+B F) • Output Injection (A + H S) • Nonsingular Input Control Transformations ( )uNu =~ • Nonsingular Output Signal Transformations ( )σσ M=~ xnmm nxmnxnmn RSRxS RBRARuRxuBxAx 11 11 11 1111 ∈∈= ∈∈∈∈+= σσ  Sliding Mode Control for Linear Time Invariant (LTI) Systems 63
  64. 64. SOLO Sliding Mode Control (SMC) The Transmission Zeros of the System are defined as the solutions for λ of: Sliding Modes and System Zeros (continue – 2) Consider the System: ( ) ( ) ( ) ( ) ( )[ ]1 1 1 1 1 1 1 1 11 1 detdet 0 0 det 0 det BAISAI BAIS BAII IS AI S BAI z nn n nn m nn − − − −−=                 − −−       − =      −− = λλ λ λλλ λ Hence: ( )[ ] ( ) ( )AI z BAIS n n − =− − λ λ λ det det 1 1 1 xnmm nxmnxnmn RSRxS RBRARuRxuBxAx 11 11 11 1111 ∈∈= ∈∈∈∈+= σσ  Sliding Mode Control for Linear Time Invariant (LTI) Systems 64
  65. 65. SOLO Sliding Mode Control (SMC) The Transmission Zeros of the System are defined as the solutions for λ of: Sliding Modes and System Zeros (continue – 3) xnmm nxmnxnmn RSRxS RBRARuRxuBxAx 11 11 11 1111 ∈∈= ∈∈∈∈+= σσ  Consider the System: ( ) ( ) ( )( ) ( )( )[ ] ( )( )[ ]{ }1 11 1111 1 111 1 1 1 11 1 1 111 11 detdet 0 det 0 0 det 0 det BASBSBIISASBSBII S BASBSBII IASBS I S BAI S BAI z nn n m nnn −−− − − −−−−=       −−− =                 −       −− =      −− = λλ λ λλ λ Sliding Mode Control for Linear Time Invariant (LTI) Systems 65
  66. 66. SOLO Sliding Mode Control (SMC) using: Sliding Modes and System Zeros (continue – 4) Let compute: ( ) ( )( )[ ] ( )( )[ ]{ } ( )( )[ ] ( ) ( )( )[ ] ( )11 1 11111 11 111 1 11 1111 1 111 detdetdetdet detdet 1 BSASBSBIIBSASBSBII BASBSBIISASBSBIIz m nn nn −−−− −−− −−=−−= −−−−= λλλλ λλλ ( )( )[ ] 1 11 1111 BASBSBIIS n −− −−λ ( ) ( ) 111111 − ×× − × − ×××× − × − × − ××× +−=+ mmmnnmmmmnnnnmmmmmmnnmmm CBDCBIDCCBDC ( )( )[ ] ( )( )[ ] ( )( ) 0since 1 111111 1 1 1 111 11 11 11 1111 =−= −+=−− −− −−−−− SBSBISBS BSBSBIIISBASBSBIIS nnn λ λλλ  Therefore: ( )( )[ ] ( ) ( )[ ] 1 11 1 111 detdet 1 −− =−− BSzASBSBII m n λλλ We found that the Poles of the System on the Sliding Surface S1x = 0 are defined by the Zeros of the triple (A, B1, S1). Sliding Mode Control for Linear Time Invariant (LTI) Systems 66
  67. 67. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Properties of the Sliding Modes The following are a Summary of the Properties of the Sliding Modes: • The System behaves as a Reduced Order motion which (apparently) does not depend on the control signal u (t). • There are (n-m) States that determines the dynamics of the Closed Loop System. • The Closed-Loop Sliding Motion depends only on the choise of the Sliding Surface. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 1111 112112121111 xmnxmmnmnxmnmxxmmnxmnmnxmnxmn rmnmxrrr xMAAxAxAx −−−−−−−−− −−=+= • The Poles of the Sliding Motion are given by the Invariant Zeros of the System Triple (A, B, S) 67
  68. 68. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Design of a Sliding Surface (Hyperplane) The following “Classical” Methods can be used to obtain the Matrix 1 1 2 SSM − = • Quadratic Minimization • Robust Eigen-structure Assignment • Direct Eigen-structure Assignment 68
  69. 69. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Design of a Sliding Surface (Hyperplane) by Quadratic Minimization Consider the following Regular Form of a LTI: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 22 1 2221 1211 2 1 0 1 1 1 1 mx xmmn r r r r r u Bx x AA AA x x x mxmmx xmn mxmmnmx xmmnmnxmn mx xmn         +                 =         = −− − −−−−    And the following Optimization (Minimization) Problem: ( ) ( )∫∫ ∞∞ ++== SS t r T rr T rr T r t r T r tdxQxxQxxQxtdxQxJ 222221211111 2 2 1 2 1 Let rewrite: ( ) ( ) ( ) ( ) 112 1 22121112 1 22222112 1 222 112 1 22121112 1 2222112 1 222222112 1 222 112 1 2222 1 22121112 1 2222 1 22121112 1 22222222 1 221212222 222211222121 r TT rr T r T r T r r TT rr TT r T rr T r T r r T I T rr T I T rr T I T rr I T rr T r r T rr T rr TT r xQQQxxQQxQxQQx xQQQxxQQQxQQxxQxQQx xQQQQQxxQQQQQxxQQQxxQQQxxQx xQxxQxxQx −−− −−−− −−−−−− −++= −+++= −+++= ++  69
  70. 70. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Design of a Sliding Surface (Hyperplane) by Quadratic Minimization (continue -1) Consider the following Regular Form of a LTI: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 22 1 2221 1211 2 1 0 1 1 1 1 mx xmmn r r r r r u Bx x AA AA x x x mxmmx xmn mxmmnmx xmmnmnxmn mx xmn         +                 =         = −− − −−−−    And the following Optimization (Minimization) Problem: ( ) ( ) ( )[ ]∫ ∞ −−− +++−= St r T r T r T rr TT r tdxQQxQxQQxxQQQQxJ 112 1 22222112 1 222112 1 2212111 2 1 Define: 112 1 22212 1 221211 :&:ˆ r T r T xQQxvQQQQQ −− +=−= Therefore: ( )∫ ∞ += St T r T r tdvQvxQxJ 221111 ˆ 2 1 ( ) vAxQQAAxAxAx r T rrr 12112 1 2212112121111 +−=+= − 70
  71. 71. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Linear Time Invariant (LTI) Systems Design of a Sliding Surface (Hyperplane) by Quadratic Minimization (continue - 2) We ended up with the following Optimization Problem: The “Optimal Control” is: 112 1 22212 1 2212111112 1 221211 :&:ˆ,:ˆ r T r TT xQQxvQQAAAQQQQQ −−− +=−=−= where P1 is given by the Riccati Equation: ( )∫ ∞ += St T r T r tdvQvxQxJ 221111 ˆ 2 1 vAxAx rr 121111 ˆ += ( ) 1112 1 22* rxPAQv − −= 0ˆˆˆ 112 1 2212111 =+−+ − QPAQAPAPPA TT We obtained: ( ) 1112112 1 22112 1 222 * rr TT r T r xMxQPAQxQQvx =+−=−= −− Therefore: ( )TT QPAQSSM 12112 1 221 1 2 +−== −− where S2 is arbitrary. 71
  72. 72. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems Consider a Sliding Manifold σ (x,t). We want to design the Control ueq that keeps the trajectory on the manifold: ( ) ( ) ( )tutftx td d eq+= ,, σσ 72 where f (σ,t) is a known or unknown but bounded function.
  73. 73. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems ( ) ( ) ( )tutxftx td d eq+= ,,σ 73 When f (σ,t) is a known function a solution for ueq is: ( ) ( ) ( ) 0, 2/1 >−−= ρσσρ signtxftueq Continuous Sliding Mode Control We obtain: ( ) ( ) 0, 2/1 =+ σσρσ signtx td d Let choose the following Lyapunov Function: ( )    == ≠> = 00 00 2 1 2 σ σ σσV ( ) 0 2/32/1 ≤−=⋅−=⋅= σρσσρσ σ σ sign td d td Vd ( ) ρσρσσρ α αα <≤+−=+−=+ <=< kforkkVk td Vd 0 2/3 1 4 3 0 22/3 Therefore σ→0 in a Finite Time When f (σ,t) is a unknown we can use an Sliding Mode Observer to estimate it. 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
  74. 74. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems ( ) ( ) ( )tutxftx td d eq+= ,,σ 74 Continuous Sliding Mode Control (continue – 1) Lyapunov Function: ( )    == ≠> = 00 00 2 1 2 σ σ σσV >> [X,Y]=meshgrid(-0.5:0.05:0.5); >> Z=0.5*Y.^2+0.5*X.^2+eps; >> mesh(X,Y,Z,'EdgeColor','black') >> contour(X,Y,Z) MATLAB: -0.5 0 0.5 -0.5 0 0.5 0 0.1 0.2 0.3 0.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
  75. 75. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems ( ) ( ) ( )tutxftx td d eq+= ,,σ 75 When f (σ,t) is a known function a Second Order SM Control for ueq is: ( ) ( ) ( ) ( ) 0,, 100 2/1 1 0 >−−−= ∫ ρρσρσσρ t t eq dtsignsigntxftu Second Order Sliding Mode Control We obtain: ( ) ( ) ( ) 0, 0 0 2/1 1 =++ ∫ t t dtsignsigntx td d σρσσρσ Let choose the following Lyapunov Function: ( ) ( )    === ≠≠> =+=+= ∫ 0,00 0,00 22 , 0 2 0 0 2 z zz dsign z zV σ σ σρσσρσ σ Rewrite: ( ) ( )    = −−= σρ σσρσ signz zsign 0 2/1 1  
  76. 76. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems ( ) ( ) ( )tutxftx td d eq+= ,,σ 76 Second Order Sliding Mode Control (continue -1) ( ) ( )    === ≠≠> =+=+= ∫ 0,00 0,00 22 , 0 2 0 0 2 z zz dsign z zV σ σ σρσσρσ σ Let check: ( ) ( ) ( ) ( ) ( )[ ] 2/1 10 2/1 1000, σρρσσρσρσρσσρσ σ −=−−⋅+⋅=⋅+⋅=      zsignsignsignzsignzzzV td d z For: ( ) 0 2 , 0 0 2 ==≤+= zat C C z zV Max ρ σσρσ ( ) kforkCk Cz kVk td Vd >≤−−=+−≤      ++−=+ <=< 1 2/1 01 2/1 0 12/10 2/1 0 100 2 2/1 10 0 2 ρρρρ ρ ρρσρσρρ α α α α Therefore σ→0 in a Finite Time When f (σ,t) is a unknown we can use an Sliding Mode Observer to estimate it.
  77. 77. SOLO Sliding Mode Control (SMC) Sliding Mode Control for Nonlinear Systems ( ) ( ) ( )tutxftx td d eq+= ,,σ 77 Second Order Sliding Mode Control (continue -2) Lyapunov Function: >> [X,Y]=meshgrid(-0.5:0.05:0.5); >> rho0=2; >> Z=0.5*Y.^2+rho0*abs(X).^1+eps; >> mesh(X,Y,Z,'EdgeColor','black') >> contour(X,Y,Z) MATLAB: ( )    === ≠≠> =+= 0,00 0,00 2 , 0 2 z zz zV σ σ σρσ -0.5 0 0.5 -0.5 0 0.5 0 0.5 1 1.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
  78. 78. SOLO Sliding Mode Control (SMC) 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. 78
  79. 79. 79 Guidance of Intercept 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:
  80. 80. 80 Guidance of Intercept 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
  81. 81. 81 Guidance of Intercept ( ) ( ) ( ) 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
  82. 82. 82 Guidance of Intercept ( ) ( ) ( ) 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
  83. 83. 83 Guidance of Intercept 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
  84. 84. 84 Guidance of Intercept 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
  85. 85. 85 Guidance of Intercept 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
  86. 86. SOLO Slide Mode Control (SMC) 86 Slide Mode Control Examples Control System of a Kill Vehicle Equations of Motion of a KV (Attitude) Fu, L-C, et all, Control System of a Kill Vehicle Fu, L-C, et al, Solution for Attitude Control of KV Fu, L-C, et al, Zero SM Guidance of a KV Crassidis , et al -Attitude Control of the Kill Vehicle Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude HTK Guidance Using 2nd Order Sliding Mode
  87. 87. 87 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Problem: Develop the Midcourse Guidance Law to intercept in a Head On Scenario a Ballistic Missile, by an Vertical Launched Interceptor at altitude above a given value hmin. Solution Method: • Start with a Planar Approximation where the Ballistic Trajectory is a known Straight Line and the Launch Point define the Plane. • Assume a Constant Velocity Bounded Maneuver Interceptor and don’t consider gravitation. • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such that the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. • Define the Interceptor Midcourse Guidance Law as the combination of the Optimal Trajectory that reaches the Sliding Lyapunov Surface followed by the Linear Guidance Law to reach the Ballistic Trajectory in Head On. • Transfer the Planar Guidance Law to a Three Dimensional Law with the same structure. • Simulate the result using a real 6 DOF Interceptor Model.
  88. 88. 88 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Start with a Planar Approximation where the Ballistic Trajectory is a known Straight Line and the Launch Point define the Plane. In this plane choose a Coordinate System xOy, having the x axis on the Ballistic Straight Line approximation pointing toward the Ballistic Target, the origin at the intersection of the straight line with the ground and y axis pointing above the ground. • Assume a Constant Velocity Bounded Maneuver Interceptor and don’t consider gravitation. Under those conditions the Interceptor equation of motion are given by: MAXM M M M M aa V a td xd V td yd V td xd ≤= = = γ γ sin cos where γ is the angle between Interceptor Velocity Vector and the Ballistic Trajectory. To drive the Interceptor to fly on the Ballistic Trajectory we want to bring γ, y and dy/dt to zero, therefore let choose a Interceptor Guidance Law as: MAXMMMM aaVk td yd kyk td d Va ≤−−−== γ γ 321
  89. 89. 89 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. Define: ( ) ( )γγγγ γ cos1sin 0 −==Ψ ∫ MM VdV Choose a Lyapunov Function candidate: ( ) ( ) ( ) 0cos1 2 1 2 1 :, 1 1 2 2 1 2 >−+=Ψ+= k k V y k V yyV MM γγγ We can see that: ( )    ±±==== ≠≠> ,...2,1,0200 000 , jjandy andy yV πγ γ γ MAXM M M M M aa V a td xd V td yd V td xd ≤= = = γ γ sin cos MAXMMMM aaVk td yd kykVa ≤−−−== γγ 321 Guidance Law:
  90. 90. 90 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. Choose a Lyapunov Function candidate: ( ) ( ) 0 ,...2,1,0200 00 cos1 2 1 :, 1 1 2 2 >    ±±==== ≠> −+= k jjandy y k V yyV M πγ γγ ( ) πγγγγ γγγγγ γγ <>≤−−=       −−−−=+= &0,,0sinsin sinsinsinsin , 321 2 1 322 1 2 32 1 1 2 1 2 kkkforV k k V k k kky V k k V Vy td d k V td yd y td yVd MM M M M M We can see that the Lyapunov Function confirms the convergence of the system to y=o, γ =0. MAXM MMMM M M aakky V k k td yd V k y V k V a td d V td yd ≤−−−=−−−== = γγγ γ γ 32 1 3 21 sin sin Guidance Law
  91. 91. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) The contour of the Lyapunov Function in the y, γ plane is defined by ( ) ( ) const k V yyV M ==−+= 2 1 2 2 2 1 cos1 2 1 :, εγγ ( ) ( ) ( )γγγ −−=−= ,,, yVyVyV         ±== ±== − MV k y y 2 sin2,0 0 11 maxmin, maxmin, ε γ εγ The Maximum Contour Ω where the trajectories Converge to y=0, γ = 0 is given for 1 maxmin, 1 maxmin, 22 k V y k V MM ±=±=→=→±= εεπγ Note: when the Guidance Law will assure convergence ( ) ( ) 1 2 1 2 2 2 cos1 2 1 :, k V k V yyV MM ≤−+= γγ • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation.
  92. 92. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) To define the values k1, k2, k3 we use Isoclines method to define the Trajectories behavior: MAXMM M M aaVkky V k td d V td yd ≤−−−= = γγ γ γ 32 1 sin sin The Trajectories in y, γ plane are: The slope (inclination) of the Trajectories in y, γ plane is given by: N Vkky V k V d yd M M M = ++ −= γγ γ γ 32 1 sin sin The curves in y, γ plane for which the slope N is constant are called Isoclines and are given by: ( ) γγγ 3 1 2 1 sin k k V N V k k V y MMM N −      += ( ) ( )           −=→−= −=→∞→ ±±==→= −= ∞→ = γγ γγγ πγ 3 12 3 1 2 1 0 2 sin ,2,1,0,0 k k V y k V N k k V k k V yN kkN M k V N M MM N N M  • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation.
  93. 93. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) O Singular Stable Points * Singular Non-Stable Points ( ) γγγ 3 1 2 1 sin k k V N V k k V y MMM N −      += ( ) ( )           −=→−= −=→∞→ ±±==→= −= ∞→ = γγ γγγ πγ 3 12 3 1 2 1 0 2 sin ,2,1,0,0 k k V y k V N k k V k k V yN kkN M k V N M MM N N M  Isoclines
  94. 94. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) To define the values k1, k2, k3 we use Isoclines. We found that if econvergencyoscillatornonkkk econvergencyoscillatorkkk −⋅≤ ⋅> 321 321 MAXMMMM aaVk td yd kykVa ≤−−−== γγ 321 Guidance Law: • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation.
  95. 95. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Un-Saturated Acceleration Region M MAX MM MAX V a kky V k V a <−−−=<− γγγ 32 1 sin • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. From which: ( ) ( ) 2 1 32 11 32 1 1 :sinsin: y k a kk k V y k a kk k V y MAXMMAXM =−+−>>++−= γγγγ
  96. 96. Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) In can be shown that all the trajectories that enter the non-saturated region for will stay in the un-saturated region, and will finally reach the origin (y=0,γ=0). 11 γγγ <<− • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. where: ( ) ( ) ( ) ( )       +≥ +<         + +         += −− 2 1 2 23 2 1 2 232 1 2 2 31 2 1 2 2 21 1 / / / sin / sin : MAXM MAXM MAXMMAXM aVkkk aVkkk aVkk k aVkk k π γ The unsaturated region around the origin bounded by –γ1<γ<γ1 , defined as Ω1 , is a Capture Zone for the trajectories.
  97. 97. 97 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Define the Straight Line Ballistic Trajectory as the Sliding Curve and define a Linear Guidance Law and a Lyapunov Volume around the Sliding Curve such the Entering Interceptor Trajectory will not escape and will Converge to the Ballistic Trajectory in a Head On situation. Block Diagram of the Linear Guidance Law that assure convergence of the Interceptor Trajectory to the Ballistic Target Trajectory in a Head On situation.
  98. 98. 98 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. Choose the Coordinate System xOy, having the x axis on the Ballistic Straight Line approximation pointing toward the Ballistic Target, the origin at the intersection of the straight line with the ground and y axis pointing above the ground. • Assume a Constant Velocity Bounded Maneuver Interceptor and don’t consider gravitation. ( ) ( ) ( ) ( ) ( ) ( ) ( ) freeTTttt Tzuuz TygivenyyzVy MAX MAXM ==== ==≤= ===⋅= ,001 0,/01 0,0sin 0 0    γγγ γ We want to reach the Ox line, in minimum time T, ( y (T)=0 ), and with angle γ (T)=0. Start without considering the constraint of minimal height hmin. The system equations are: The Hamiltonian of the Optimal Problem is : ( ) 321 sin λλγλ ++⋅= uzVH MAXM  ( ) 0 cos 0 3 12 1 = ∂ ∂ −= ⋅−= ∂ ∂ −= = ∂ ∂ −= t H zV y H y H MAXMMAX λ γγλλ λ    ( ) ( ) ( ) ( ) 133 11 −= ∂ ∂ −== == =Tt t J Tt constTt λλ λλ ( )TtJ uu minmin =
  99. 99. 99 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. Since H is not an explicit function of time, we have H =constant, Therefore: ( ) 1sin 21 −+⋅= uzVH MAXM λγλ  ( )  ( ) 0coscos 211 2 =+⋅−⋅= − uuzVuzV td Hd MAXMMAX z MAXMAXM       λγγλγγλ λ ( ) ( )         =→≤≤= ==→==→=→≤≤= ⇒== constutttu or uzconstzttt u td Hd MAX 21 2212 2 0 0 1 2 00 0      γ π λλ λ The optimum is obtained using: ( )[ ]1sinminargminargminarg 21 * −+⋅=== uzVHJu MAXM uuu λγλ  ( )    <<= ≠− = 212 22* 00 0 ttt sign u λ λλ
  100. 100. 100 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. The trajectories that ends at the origin are given by integrating the equations of state assuming optimal control, and are given by: ( )    <<= ≠− = 212 22* 00 0 ttt sign u λ λλ ( ) ( ) ( ) ( ) ( ) ( ) ( ) freeTTttt Tzuuz TygivenyyzVy MAX MAXM ==== ==≤= ===⋅= ,001 0,/01 0,0sin 0 0    γγγ γ ( ) ( ) ( ) ( ) ( ) ( ) 1&00cos1 1&00cos1 * * −==≤−−= ==≥−= uT V ty uT V ty MAX M II MAX M I γγ γ γγ γ   ( ) ( ) ( ) ( )γγγ γ signusign V ty MAX M III −=−−= * , cos1  or Extremal (non necessarlly Optimal) Trajectories
  101. 101. 101 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. Singular Trajectories: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 2 2 2 0&0 0&0 0&0 ttt ttI ttI ttI nn <<          == == ==    λ λ λ ( ) ( ) ( ) uzVtH tI MAXM 21 sin1 λγλ +⋅+−=     For Singular Trajectories to occur for t1<t<t2, the following conditions must be satisfied: ( ) ( ) ( ) ( ) ( )  ( ) ( ) 0cos 0cos 0 0sin1 12 1 2 1 =⋅−= =⋅= = =⋅+−= MAXMMAX z MAXMAXM MAXM zVt uzVtI t zVtI γγλλ γγλ λ γλ     2 π γ ±=⋅ MAXz  MV/11 ±=λ 0=u
  102. 102. 102 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) The Optimal Trajectories are function of the Initial Conditions y0, γ0. If the constraint of minimal convergence altitude hmin is disregarded, the Optimal Trajectories are given by: • Turn toward the x axis in Ox direction with maximum turn rate. • If γ = +π/2 or - π/2 and the trajectory is toward Ox and the distance is y > R (Turning Radius), than the trajectory will be the straight line normal to Ox. When y = R turn, with maximum turn rate toward Ox direction, on yI,II (t), to reach y (T) = 0 and γ (T) = 0. • If during the first turn we get close to Ox line before reaching γ = +π/2 or - π/2 , we reverse the maximum turn, on yI,II (t), to reach y (T) = 0 and γ (T) = 0. ( )[ ] ( ) ( )[ ]     <<−≠ −+−=−−= 2/2/2/ cos1, * πγππγ γγγ signRysignyysignu III I ( ) ( ) ( ) ( )   −−= −= γγγ γ cos1, * signRy signu III Converging to y=0. γ=0 II ( ) ( )     −−≠ = = γγ πγ cos1 2/ 0* signRy u On Singular Arcs III There are three classes of optimal paths defined by:
  103. 103. 103 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Using Optimal Control Theory define the Interceptor Trajectory from the Launching Point that will enter the Sliding Lyapunov Volume, above the minimal height hmin, in Minimum Time. Block Diagram of the Optimal Law that assure convergence in minimum time of the Interceptor Trajectory to the Ballistic Target Trajectory in a Head On situation.
  104. 104. 104 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Define the Interceptor Midcourse Guidance Law as the combination of the Optimal Trajectory that reaches the Sliding Lyapunov Surface followed by the Linear Guidance Law to reach the Ballistic Trajectory in Head On. • Start with the Optimal Law toward the Sliding Surface until reaching the Captive Volume Ω1. • Switch to Linear Guidance Law that keeps the trajectory inside the Captive Zone Ω1 and converges to the origin y=0 and γ = 0, to the Head On with the Ballistic Target.
  105. 105. 105 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Block Diagram of the Guidance Law that starts with the Optimal Law that assure convergence in minimum time to the Capture Zone, followed by the Linear Law that assre convergence to the Ballistic Target Trajectory in a Head On situation.
  106. 106. 106 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Minimal Altitude hmin of the Trajectory From the Figure we can see that for y0>0 and γ0>0 and for y0<0 and γ0<0 There may be situation when the minimum time trajectory is not feasible and we must change it.
  107. 107. 107 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Minimal Altitude hmin of the Trajectory ( ) ( )αγγα −−+= 00 coscos RRhh ( ) ( )00min cos1 γγα −−=== Rhhh From the Figure Therefore We want to find hmin as function of y0, γ0 ( ) RRyRRdd dRl lh −−=−−= −= = 0001 01 0 cos/cos/ sin cos γγ γ γ ( )000000 sincossincossin γγγγγ +++−= Ryh ( )1sincossincossin 000000min −+++−= γγγγγ Ryh
  108. 108. 108 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) Minimal Altitude hmin of the Trajectory A solution to the altitude problem is to choose on the Ballistic Trajectory the point of the minimum altitude hmin as the point below which convergence of the Intercept Trajectory is not acceptable. By doing this if exists a Singular Arc it will be at |γ| < π/2
  109. 109. 109 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Transfer the Planar Guidance Law to a Three Dimensional Law with the same structure. • Simulate the result using a real 6 DOF Interceptor Model. Define: M – the Interception Position P - the point at the Ballistic Missile Trajectory at altitude hmin. Ix1 – unit vector in the Straight Line Ballistic Trajectory Approximation, pointing up. h  – the vector pointing from P to M. d  – the vector distance from Straight Line Ballistic Trajectory to M, pointing to M. dhxf I  −=:1 MV  – Interceptor Velocity at M. – angle between to .γ MV  Ix1 – angle between to .hγ MV  h          × = − hV hV M M h  1 sinγ         × = − M IM V xV 1 sin 1  γ
  110. 110. 110 Midcourse Intercept of a Ballistic in a Head On Scenario Above a Minimal Altitude SOLO Slide Mode Control (SMC) • Transfer the Planar Guidance Law to a Three Dimensional Law with the same structure. • Simulate the result using a real 6 DOF Interceptor Model. ( ) ( ) ( ) ( ) ( ) MAXC MIM MIM M MM MM C aa VxV VxV Vk dVV dVV dkdka ≤ ×× ×× + ×× ×× +=      1 1 321 γ ( ) ( ) ( ) MAXCMM h h MMM MM hMC aahVV hV k hVV hVV Vka ≤××      = ×× ×× =    γ γ γ sin ( ) ( ) ( ) 321 32 /sin cos11: kkk dVkk Rd SWSWSWM SWSW = +== −∆+= γγ γ When the denominators are close to zero, they will be bounded above zero, to prevent numerical problems. Guidance Law B: Converges to the Ballistic Trajectory in H.O. Scenario Guidance Law A: Reaches the Ballistic Trajectory above Altitude hmin When the Interceptor distance to the Ballistic trajectory is less than dSW (defined bellow) we switch to Guidance Law B.
  111. 111. SOLO Slide Mode Control (SMC) Control System of a Kill Vehicle Assume a Divert Attitude Control System (DACS). The Divert Control Thrusts are located near the Mass Center of the Vehicle, and are aligned with the two axes perpendicular to the longitudinal axis of the Kill Vehicle, so as to generate the Pure but Arbitrary Divert Motion (axB, ayB, azB), where Attitude Control Thrusts are located and aligned such that only Three Pure Rotational Moments about the principal axes are produced (TxB,TyB,TzB). All those Thrusts are Pulse Type, i.e., they only have ON/OFF states with fixed amplitude. Thrust Command Thrust Output See “Kill Vehicle Guidance & Control Using Sliding Mode” Presentation for more details The goal of the Control System is to track a design quaternion and corresponding angular velocity . dq dω 
  112. 112. 112 SOLO Desired Attitude and Angular Rate of the Kill Vehicle (KV) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )           − + −−+ =                       +−−+− −−+−+ +−−−+ = ===           dddd dddd dddd dddddddddddd dddddddddddd dddddddddddd Bd Bd TBd I Bd Bd I Bd I zI yI xI qqqq qqqq qqqq qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq xCxCd d d d 3120 3021 2 3 2 2 2 1 2 0 2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 2 2 0 0 1 22 22 22 111 1 1 1 Kill Vehicle (KV)Control Since the KV roll is free, let choose q1d = 0 We obtain: 0 3 0 20 2 1 , 2 1 , 2 11 q d q q d q d q yI d zI d xI d == + = The Main KV Engine is aligned to xB direction of the KV. The KV Divert Thrusters act normal to to the xB direction. Suppose that we want that the KV xB direction shall follow a given direction and it’s inertial derivative . The rotation position of the KV is free. The Desired KV Attitude is defined as Bd. The following relation must be satisfied: d1 I d td d 1
  113. 113. 113 SOLO Desired Attitude and Angular Rate of the Kill Vehicle (KV) Kill Vehicle (KV)Control We found the quaternion from inertia (I) to Desired Body (Bd) Attitude: 0 3 0 210 2 1 , 2 1 ,0, 2 11 q d q q d qq d q yI d zI dd xI d === + = Taking the derivatives: ( ) td qd q d d td d qtd qd q td qd q d d td d qtd qd q td qd qd td d dtd qd q d d yI yI d d d d d zI zI d d d d dxI xI d d 0 2 00 3 3 0 2 00 2 2 1 1 0 0 2 1 1 2 1 , 2 1 1 2 1 ,0,1 112 1 −      ==−      ====      + ==              = d d d d Bd I q q q q q 3 2 1 0 where: Using those results we can find: ( )                       −− −− −− =← d d d d dddd dddd dddd I IBd q q q q qqqq qqqq qqqq 3 2 1 0 0123 1032 2301 2         ω ( ) ( ) [ ][ ]             ×+−=Θ=← td d td qd Iqq dt d q d d dxdd Bd I Bd I TI IBd ρ ρρω     0 33022
  114. 114. SOLO Kill Vehicle (KV)Control Control System of a Kill Vehicle 114
  115. 115. SOLO Slide Mode Control (SMC) Control System of a Kill Vehicle 115
  116. 116. Equations of Motion of a KV (Attitude) SOLO 1 2 0 2 3 2 2 2 1 2 0 0 4210 =+=+++       =+++==← ρρ ρρ      T B IIB qqqqq q qkqjqiqqq Coordinate Systems Inertial Coordinates (x,y,z), and Body Coordinates at the Body Center of Gravity (xB, yB, zB) The Rotation Matrix from Inertial Coordinates (x,y,z), to Body Coordinates at the Body Center of Gravity (xB, yB, zB), CI B , is defined via the Quaternions: 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ ( ) ( ) ( ) ( ) ( ) ( ) I B B I B I B I B I B IIB qqqqqqqqq qqkqjqiqqq ==⇒=+++= −=−−−+== − − ← *12 3 2 2 2 1 2 0 * 04210 ** 1 ρ ρ      Quaternion Complex Conjugate: kkjjii  −=−=−= *** ( ) ( ) 1ˆˆ2/sinˆ2/cos0 === nnnq T θρθ  116 Kill Vehicle (KV)Control
  117. 117. 117 SOLO Kill Vehicle (KV) Control Product of Quaternions 1−=⋅⋅=⋅=⋅=⋅ kjikkjjii  kijji  =⋅−=⋅ ijkkj  =⋅−=⋅ jkiik  =⋅−=⋅ ⊗ ( ) ( )      ×++ ⋅− =      ⊗      =⊗ BAABBA BABA B B A A BA qq qqqq qq ρρρρ ρρ ρρ    00 0000 [ ] [ ]               ×− − =              ×+ − =      ⊗      =⊗ A A BxBB T BB B B AxAA T AA B B A A BA q Iq qq Iq qqq qq ρρρ ρ ρρρ ρ ρρ      0 330 00 330 000 ( ) ( )32103210 00 BBBBAAAA B B A A BA qkqjqiqqkqjqiq qq qq   +++⋅+++=      ⊗      =⊗ ρρ Using this definition the Product of two Quaternions is given by:BA qq ⊗ Let define: or in Matrix Product Form: Equations of Motion of a KV (Attitude)
  118. 118. 118 SOLO Product of Quaternions ⊗ Let compute: ( ) 1 0 1 0 00 1 2 0 001 =      =                       ×−− + =      − ⊗      =⊗ −         AAAAAA A T AA A A A A AA qq q qq qq ρρρρ ρρ ρρ Equations of Motion of a KV (Attitude) Kill Vehicle (KV) Control
  119. 119. SOLO Coordinate Transformations ( ) ( ) ( ) ( ) [ ] [ ][ ]{ } ( )       ××+×− =      ⊗      ⊗      − =⊗⊗=      = AA B A AB AB B vqI q v q qvq v v  ρρρρρ 22 000 0 00* Given the vector described in (A) coordinates by , and in (B) coordinates byv  ( )A v  ( )B v  [ ] [ ][ ]{ } ( ) ( ) ( ) ( ) ( ) ( )             +−−−+ +−+−− −+−−+ =××+×−= 2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 0 22 22 22 22 qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq qIC B A ρρρ  [ ][ ] ( ) [ ] ( )                 q x T q x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C T Θ Ξ           ×− − ×−−=                 − − − −−−           −− −− −− = ρ ρ ρρ 330 330 012 103 230 321 0123 1032 2301 119 The Rotation Matrix from A to B, CA B , is defined as: Equations of Motion of a KV (Attitude) Kill Vehicle (KV) Control

×