Lecture Notes on Sliding Mode Control - Nonlinear Control Mechanism System | AOE 5344, Study notes of Aerospace Engineering

Download Lecture Notes on Sliding Mode Control - Nonlinear Control Mechanism System | AOE 5344 and more Study notes Aerospace Engineering in PDF only on Docsity! Lectures 12: Sliding Mode Control Recall that we are considering systems of the standard form ( η̇ ξ̇ ) = ( f(η, ξ) + δη(η, ξ) fa(η, ξ) + Ga(η, ξ) (u + δξ(η, ξ,u)) ) . In the previous lecture, we defined u = ueq + G −1 a (η, ξ)v (1) where the “equivalent control” ueq = G −1 a (η, ξ) ( −fa(η, ξ) + ∂φ ∂η f(η, ξ) ) cancels the nominal ξ dynamics. The term v was initially chosen to drive the system to the sliding manifold in finite time. However, to avoid problems with chatter, the choice of v was modified to v = −β(η, ξ) 1 − k sat( z ǫ ) (2) where sat( z ǫ ) :=    sat(z1 ǫ ) ... sat(zm ǫ )    . This control law drives z to the boundary layer {(η, ξ) | |zi| < ǫ, 1 ≤ i ≤ m} in finite time. To assess the behavior of the closed-loop system, note that the dynamics restricted to the sliding manifold (i.e., the dynamics with z = 0, or equivalently ξ = φ(η)) are asymptotically stable, by assumption. In fact, we make a slightly stronger assumption. Assumption. A function V (η) exists which satisfies α1(‖η‖) ≤ V (η) ≤ α2(‖η‖) and V̇ = ∂V ∂η (f(η,φ(η) + z) + δη(η,φ(η) + z)) < −α3(‖η‖) for all ‖η‖ ≥ γ(‖z‖) (3) for all (η, ξ) in some domain D containing the origin. The functions αi and γ are class K.  In words, this assumption says that, • for z 6= 0, V decreases whenever ‖η‖ is larger than γ(‖z‖) within the domain D, and • for z = 0, V is strictly decreasing for all (η,φ(η)) ∈ D. This notion should be somewhat familiar. Considering z as an input, the assumption implies local input- to-state stability. (Note that the ISS property is only local because D is not necessarily all of Rn and αi and γ are only class K.) Recall that our choice of control attempts to make z (and thus γ(‖z‖)) small. The origin of the η-subsystem is asymptotically stable on the sliding manifold. An important question is: How does the η-subsystem behave within an epsilonic boundary layer of the sliding manifold? Recall that z converges to a small neighborhood of the sliding surface in finite time. That is, our choice of feedback drives |zi| to an ǫ-neighborhood of zero in finite time. Moreover, because |zi| is non-increasing 1 z = 0 x( )t x( )t » ´ ´ z ·c ·² Figure 1: The boundary layer. outside this “boundary layer,” there exists a non-negative constant c ≥ ǫ such that |zi| ≤ c for each i ∈ {1, . . . ,m}. It follows that ‖z‖ ≤ κc for some positive constant κ which depends on the norm. (If ‖ · ‖ denotes the Euclidean norm, then κ = √ m.) By assumption (3), this implies that V̇ < −α3(‖η‖) for all ‖η‖ ≥ γ(κc) ≥ γ(‖z‖). Now, define the class K function α(c) = α2(γ(κc)). V (η) ≥ α(c) ⇐⇒ V (η) ≥ α2(γ(κc)) by definition of α, =⇒ α2(‖η‖) ≥ α2(γ(κc)) because α2(‖η‖) ≥ V (η), by assumption, =⇒ ‖η‖ ≥ γ(κc) by applying class K function α−12 to both sides, =⇒ V̇ ≤ −α3(‖η‖). Therefore, the set {(η,z(η, ξ)) ∈ D | V (η) ≤ α(c)} is positively invariant because V is strictly decreasing on the boundary of this set. It follows that the compact set Ω = {η | V (η) ≤ α(c)} × {z | |zi| ≤ c, 1 ≤ i ≤ m} is positively invariant for c ≥ ǫ. Any trajectory starting in Ω remains there for all future time. Furthermore, we know that |zi(t)| ≤ ǫ after some finite time. By the preceding series of arguments, V̇ ≤ −α3(γ(κǫ)) for all η large enough that V (η) ≥ α(ǫ). Therefore, any trajectory starting in Ω will reach the compact, positively invariant set Ωǫ = {η | V (η) ≤ α(ǫ)} × {z | |zi| ≤ ǫ, 1 ≤ i ≤ m} in finite time.1 Notice that, in the limit ǫ → 0, the set Ωǫ shrinks to the origin and we recover the result of the discontinuous controller. This makes sense because as ǫ → 0, the slope of the saturation function goes to ∞ and this function approximates the signum function in our original stabilizing controller. In practice, ǫ is chosen according to a trade-off between accuracy and undesirable effects such as high frequency excitation and actuator wear due to chattering. We have shown asymptotic convergence to the set Ωǫ. To conclude asymptotic convergence to the origin requires additional assumptions. For example, if ξ = φ(η) exponentially stabilizes the η dynamics and δξ vanishes at the origin, then the given feedback (i.e., with the saturation function in place of the signum function) exponentially stabilizes the origin of the entire system. (See Theorem 14.2 in Khalil.) 1The fact that V̇ ≤ −α3(γ(κǫ)) whenever V (η) ≥ α(ǫ) implies that V ≤ −α3(γ(κǫ))t + V (η(0)) until V (η) < α(ǫ). 2 where ∆(x1, x2, v) = Gaδξ − ∂φ ∂η δη = c̄ ( c − c̄ c̄ ) u − (−2αx1) ( (a − ā)x31 + (b − b̄)x1x2 ) = (c − c̄) ( 1 c̄ (−2αx1)(āx31 + b̄x1x2) + 1 c̄ v ) + (2αx1) ( (a − ā)x31 + (b − b̄)x1x2 ) = ( −2αā ( c − c̄ c̄ ) + 2α(a − ā) ) x41 + ( −2αb̄ ( c − c̄ c̄ ) + 2α(b − b̄) ) x21x2 + c − c̄ c̄ v Recall that, in contrast to backstepping which drives x1 and z to zero simultaneously, sliding mode control drives z to zero in finite time, after which x1 converges to zero by design of φ(x1). To find a choice of feedback which does this in spite of the uncertainty represented by ∆, we note that |∆| ≤ k1x41 + k2x21|x2| + k|v| where k1 ≥ ∣ ∣ ∣ ∣ −2αā ( c − c̄ c̄ ) + 2α(a − ā) ∣ ∣ ∣ ∣ ≥ 2α ( ā c̄ |c − c̄| + |a − ā| ) , k2 ≥ ∣ ∣ ∣ ∣ −2αb̄ ( c − c̄ c̄ ) + 2α(b − b̄) ∣ ∣ ∣ ∣ ≥ 2α ( b̄ c̄ |c − c̄| + |b − b̄| ) , and (6) k ≥ ∣ ∣ ∣ ∣ c − c̄ c̄ ∣ ∣ ∣ ∣ . (7) We assume that k can be chosen less than one, which equates to assuming that 0 < |c| < 2|c̄|, which is satisfied according to the given bounds on c and the given choice of c̄. We thus note that |∆| ≤ ρ(x1, x2) + k|v| where ρ(x1, x2) = k1x41 + k2x21|x2|. Define β(x1, x2) = ρ(x1, x2) + b0 for some choice b0 > 0 and let v = −β(x1, x2) 1 − k sign(x2 + αx 2 1). (8) The control law (4) with ueq given by (5) and v given by (8) drives z to zero (i.e., it drives x2 to −αx21) in finite time. After z = 0, x1 and x2 converge to zero as t → ∞. Figure 3 shows a simulation with model parameters a = 1.2, b = 1.3, c = 0.8. and with control parameters α = 50, k = 0.5, k1 = 150, k2 = 100, b0 = 1. As can be seen in the figure, z converges to zero in less than one-tenth of one second. Note that the magnitude of x2 grows relatively large during this transient. However, after z = 0, the state converges asymptotically to zero. Suppose that, instead of choosing v according to (8), we replace the signum function with the saturation function with slope ǫ. We may expect the control to behave identically for z ≥ ǫ. After z < ǫ, however, we expect different behavior. As can be seen in Figure 4, for ǫ = 10, the system settles quickly upon a non- zero equilibrium. (One can easily check that non-zero equilibria exist for ǫ 6= 0 and that these equilibria converge to the origin as ǫ → 0.) For smaller ǫ, the system state continues to converge toward zero. We may control the accuracy with which the closed-loop system tracks the desired state (i.e., the equilibrium at the origin) through our choice of ǫ. 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −30 −20 −10 0 10 x 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20 0 20 40 60 z t Figure 3: Simulation using the signum function. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −30 −20 −10 0 10 x 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20 0 20 40 60 z t Figure 4: Simulation using the saturation function: ǫ = 10: Solid; ǫ = 0.1: Dashed; ǫ = 0.001: Dotted. 6