Nonlinear Control Lecture # 28 Robust State Feedback Stabilization - PowerPoint PPT Presentation

Nonlinear Control Lecture # 28 Robust State Feedback Stabilization Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

Sliding Mode Control x = f ( x ) + B ( x )[ G ( x ) u + δ ( t, x, u )] ˙ x ∈ R n , u ∈ R m , f and B are known, while G and δ could be uncertain, f (0) = 0 , G ( x ) is a positive definite symmetric matrix with λ min ( G ( x )) ≥ λ 0 > 0 Regular Form: � � � � ∂T η 0 = T ( x ) , ∂x B ( x ) = ξ I ˙ η = f a ( η, ξ ) , ˙ ξ = f b ( η, ξ ) + G ( x ) u + δ ( t, x, u ) Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

˙ η = f a ( η, ξ ) , ˙ ξ = f b ( η, ξ ) + G ( x ) u + δ ( t, x, u ) Sliding Manifold: s = ξ − φ ( η ) = 0 , φ (0) = 0 s ( t ) ≡ 0 ⇒ η = f a ( η, φ ( η )) ˙ Design φ s.t. the origin of ˙ η = f a ( η, φ ( η )) is asymp. stable Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

s = f b ( η, ξ ) − ∂φ ˙ ∂η f a ( η, ξ ) + G ( x ) u + δ ( t, x, u ) u = ψ ( η, ξ ) + v Typical choices of ψ : ψ = − ˆ G − 1 [ f b − ( ∂φ/∂η ) f a ] ψ = 0 , s = G ( x ) v + ∆( t, x, v ) ˙ � ∆( t, x, v ) � � � ∀ ( t, x, v ) ∈ [0 , ∞ ) × D × R m � ≤ ̺ ( x )+ κ 0 � v � , � � λ min ( G ( x )) � ̺ ( x ) ≥ 0 , 0 ≤ κ 0 < 1 ( Known ) Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

V = 1 V = s T ˙ ˙ 2 s T s ⇒ s = s T G ( x ) v + s T ∆( t, x, v ) s β ( x ) ≥ ̺ ( x ) v = − β ( x ) � s � , + β 0 , β 0 > 0 1 − κ 0 − β ( x ) s T G ( x ) s/ � s � + s T ∆( t, x, v ) ˙ V = ≤ λ min ( G ( x ))[ − β ( x ) + ̺ ( x ) + κ 0 β ( x )] � s � = λ min ( G ( x ))[ − (1 − κ 0 ) β ( x ) + ̺ ( x )] � s � ≤ − λ min ( G ( x )) β 0 (1 − κ 0 ) � s � √ ≤ − λ 0 β 0 (1 − κ 0 ) � s � = − λ 0 β 0 (1 − κ 0 ) 2 V Trajectories reach the manifold s = 0 in finite time and cannot leave it Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

Continuous Implementation  y, if � y � ≤ 1  Sat( y ) = y/ � y � , if � y � > 1  � s � v = − β ( x ) Sat µ � s � ≥ µ ⇒ Sat( s/µ ) = s/ � s � ⇒ s T ˙ s ≤ − λ 0 β 0 (1 − κ 0 ) � s � Trajectories reach the boundary layer {� s � ≤ µ } in finite time and remains inside thereafter Study the behavior of η : η = f a ( η, φ ( η ) + s ) ˙ Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

α 1 ( � η � ) ≤ V 0 ( η ) ≤ α 2 ( � η � ) ∂V 0 ∂η f a ( η, φ ( η ) + s ) ≤ − α 3 ( � η � ) , ∀ � η � ≥ α 4 ( � s � ) � s � ≤ c ⇒ ˙ V 0 ≤ − α 3 ( � η � ) , for � η � ≥ α 4 ( c ) α ( r ) = α 2 ( α 4 ( r )) V 0 ( η ) ≥ α ( c ) ⇔ V 0 ( η ) ≥ α 2 ( α 4 ( c )) ⇒ α 2 ( � η � ) ≥ α 2 ( α 4 ( c )) ⇒ � η � ≥ α 4 ( c ) ˙ ⇒ V 0 ≤ − α 3 ( � η � ) ≤ − α 3 ( α 4 ( c )) Ω = { V 0 ( η ) ≤ c 0 } × {� s � ≤ c } , c 0 ≥ α ( c ) , Ω ⊂ T ( D ) Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

˙ V 0 ( η ) ≥ α ( µ ) ⇒ V 0 ≤ − α 3 ( α 4 ( µ )) ⇒ Ω µ = { V 0 ( η ) ≤ α ( µ ) } × {� s � ≤ µ } is positively invariant In summary, all trajectories starting in Ω remain in Ω and reach Ω µ in finite time and remain inside thereafter V 0 c 0 α ( ⋅ ) α (c) α ( µ ) µ |s| c Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

Theorem 10.1 Suppose all the assumptions hold over Ω . Then, for all ( η (0) , ξ (0)) ∈ Ω , the trajectory ( η ( t ) , ξ ( t )) is bounded for all t ≥ 0 and reaches the positively invariant set Ω µ in finite time. If the assumptions hold globally and V ( η ) is radially unbounded, the foregoing conclusion holds for any initial state Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

Example 10.2 (Magnetic levitation - friction neglected) x 2 = 1 + m o x 1 = x 2 , ˙ ˙ m u, x 1 ≥ 0 , − 2 ≤ u ≤ 0 We want to stabilize the system at x 1 = 1 . Nominal steady-state control is u ss = − 1 Shift the equilibrium point to the origin: x 1 → x 1 − 1 , u → u +1 x 2 = m − m o + m o x 1 = x 2 , ˙ ˙ m u m x 1 ≥ − 1 , | u | ≤ 1 � � ( m − m o ) � ≤ 1 � � Assume � � m o 3 � Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

s = x 1 + x 2 ⇒ x 1 = − x 1 + s ˙ V 0 = 1 2 x 2 1 ˙ V 0 = − x 2 1 + x 1 s ≤ − (1 − θ ) x 2 1 , ∀ | x 1 | ≥ | s | /θ, 0 < θ < 1 α 1 ( r ) = α 2 ( r ) = 1 2 r 2 , α 3 ( r ) = (1 − θ ) r 2 , α 4 ( r ) = r/θ α ( r ) = α 2 ( α 4 ( r )) = 1 2( r/θ ) 2 With c 0 = α ( c ) , Ω = {| x 1 | ≤ c/θ } × {| s | ≤ c } Ω µ = {| x 1 | ≤ µ/θ } × {| s | ≤ µ } Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

Ω = {| x 1 | ≤ c/θ } × {| s | ≤ c } Take c ≤ θ to meet the constraint x 1 ≥ − 1 s = x 2 + m − m o + m o ˙ m u m � x 2 + ( m − m o ) /m � � m x 2 + m − m o � � � � � ≤ 1 � � = 3 (4 | x 2 | + 1) � � � � m o /m m o m o � � In Ω , | x 2 | ≤ | x 1 | + | s | ≤ c (1 + 1 /θ ) � � with 1 x 2 + ( m − m o ) /m � ≤ 8 . 4 c + 1 � � θ = 1 . 1 , � � m o /m 3 � Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

To meet the constraint | u | ≤ 1 limit c to � s 8 . 4 c + 1 � < 1 ⇔ c < 0 . 238 and take u = − sat 3 µ With c = 0 . 23 , Theorem 10.1 ensures that all trajectories starting in Ω stay in Ω and enter Ω µ in finite time Inside Ω µ , | x 1 | ≤ µ/θ = 1 . 1 µ µ can be chosen small enough to meet any specified ultimate bound on x 1 For | x 1 | ≤ 0 . 01 , take µ = 0 . 01 / 1 . 1 ≈ 0 . 009 Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

With further analysis inside Ω µ we can derive a less conservative estimate of the ultimate bound of | x 1 | . In Ω µ , the closed-loop system is represented by x 2 = m − m o − m o ( x 1 + x 2 ) x 1 = x 2 , ˙ ˙ m mµ which has a unique equilibrium point at � x 1 = µ ( m − m o ) � , x 2 = 0 m o and its matrix is Hurwitz t →∞ x 1 ( t ) = µ ( m − m o ) lim , t →∞ x 2 ( t ) = 0 lim m o Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

� � ( m − m o ) � ≤ 1 � � 3 ⇒ | x 1 | ≤ 0 . 34 µ � � m o � For | x 1 | ≤ 0 . 01 , take µ = 0 . 029 We can also obtain a less conservative estimate of the region of attraction V 1 = 1 2( x 2 1 + s 2 ) � � � � 1 + s 2 − m o m − m o 1 + s 2 − 1 ˙ V 1 ≤ − x 2 � � | s | ≤ − x 2 1 − 2 | s | � � m m o � � for | s | ≥ µ Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

� � s 2 2 | s |− 3 s 2 m − m o � | s |− m o ˙ � � V 1 ≤ − x 2 1 + s 2 + µ ≤ − x 2 1 + s 2 + 1 � � m o m 4 µ � for | s | ≤ µ With µ = 0 . 029 , it can be verified that ˙ V 1 is less than a negative number in the set { 0 . 0012 ≤ V 1 ≤ 0 . 12 } . Therefore, all trajectories starting in Ω 1 = { V 1 ≤ 0 . 12 } enter Ω 2 = { V 1 ≤ 0 . 0012 } in finite time. Since Ω 2 ⊂ Ω , our earlier analysis holds and the ultimate bound of | x 1 | is 0 . 01 . The new estimate of the region of attraction, Ω 1 , is larger than Ω Nonlinear Control Lecture # 28 Robust State Feedback Stabilization

s 0.6 Ω 1 0.4 Ω 0.2 0 Ω 2 x 1 −0.2 −0.4 −0.6 −0.8 −0.5 0 0.5 Nonlinear Control Lecture # 28 Robust State Feedback Stabilization