Sliding Control-Non Linear Control Systems-Lecture Slides, Slides of Nonlinear Control Systems

Download Sliding Control-Non Linear Control Systems-Lecture Slides and more Slides Nonlinear Control Systems in PDF only on Docsity! Outline Sliding Control Continuous Approximations of Switching Control Laws Nonlinear Control Lecture 10: Sliding Control Department of Electrical Engineering Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 10 1/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Sliding Control Sliding Surface Integral Control Gain Margins Continuous Approximations of Switching Control Laws Farzaneh Abdollahi Nonlinear Control Lecture 10 2/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Sliding Surface I The problem of tracking the n-dimensional vector xd (the original tracking problem) can be replaced by a 1st-order stabilization problem in s. I Given initial condition xd(0) = x(0), the problem of tracking x ≡ xd is equivalent to remaining on the surface S(t) for all t > 0 (s ≡ 0 represents a linear differential equation whose unique solution is x̃ ≡ 0) I In (1),s contains x̃(n − 1) we only need to differentiate s once for the input u to appear. I Bounds on s can be directly translated into bounds on x̃ s represents a true measure of tracking performance. When x̃(0) = 0: ∀t ≥ 0, |s(t)| ≤ Φ⇒ ∀t ≥ 0, |x̃ (i)| ≤ (2λ)iε, i = 0, ..., n − 1 (3) where ε = Φ/λn−1 Farzaneh Abdollahi Nonlinear Control Lecture 10 5/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I Proof: x̃ is obtained from s through a sequence of first-order lowpass filters, shown in Fig. I Let y1 output of first filter: y1 = ∫ t 0 e−λ(t−T )s(T )dT , |s| ≤ Φ⇒ |y1| ≤ Φ ∫ t 0 e−λ(t−T )dT = (Φ/λ)(1− e−λt) ≤ Φ/λ I Repeat the same procedure all the way to yn−1 = x̃ |x̃ | ≤ Φ/λn−1 = ε I To obtain x̃ (i), see the Fig b I The output of the (n − 1− i)th filter: z1 < Φ/λn−1−i I Note that p±λp+λ = 1− λ λ+p ≤ 1 + λ λ+p I ∴|x̃ (i)| ≤ (Φ/λn−1−i )(1 + λλ ) i = (2λ)iε I If x̃(0) 6= 0, , (3) is obtained asymptotically, within a short time-constant (n − 1)/λ. Farzaneh Abdollahi Nonlinear Control Lecture 10 6/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Sliding Condition I To keep the scalar s at zero, a control law u it should be found s.t outside of S(t): 1 2 d dt s2 ≤ −η|s| (4) where η > 0 conts. I ∴ The squared ”distance” to the surface, s2 , decreases along all system trajectories. (V = 12 s 2) I (4), so-called sliding condition, makes the surface an invariant set. I By keeping the invariant set, some disturbances or dynamic uncertainties can be tolerated. I S(t) is sliding surface; behavior of the system on the surface is sliding mode Farzaneh Abdollahi Nonlinear Control Lecture 10 7/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I After defining the sliding surface s, the control is designed in two steps 1. A feedback control law u is selected so as to verify sliding condition (4) 2. The discontinuous control law u is suitably smoothed to achieve an optimal trade-off between control bandwidth and tracking precision I To cope with modeling imprecision and disturbances, the control law has to be discontinuous across S(t). I Implementing the associated control switchings is always imperfect (switching is not instantaneous, and the value of s is not known with infinite precision) yields chattering I Chattering high control activity and may excite high frequency dynamics neglected in modeling (such as unmodeled structural modes, neglected time-delays, and so on). Farzaneh Abdollahi Nonlinear Control Lecture 10 10/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Example I Consider ẍ = f + u (5) I f is unknown,but estimated by f̂ , estimation error on f assumed to be bounded by known function F = F (x , ṫ) |f̂ − f | ≤ F I To track x ≡ xd , define the sliding surface: s = ( d dt + λ)x̃ = ˙̃x + λx̃ ṡ = f + u − ẍd + λ ˙̃x I Best approximation û to achieve ṡ = 0 û = −f̂ + ẍd − λ ˙̃x I The feedback control strategy is chosen intuitive ”if the error is negative, push hard enough in the positive direction (and conversely)” I To satisfy (4), a term discontinuous across the surface s = 0: u = û − ksgn(s) where sgn(s) = 1 if s > 0 sgn(s) = −1 if s < 0 Farzaneh Abdollahi Nonlinear Control Lecture 10 11/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I Note that this strategy works only for first-order systems. I By choosing k to be large enough (4) can be guaranteed 1 2 d dt s2 = ṡ.s = (f − f̂ )s − k |s| ’ I letting k = F + η 12 d dt s 2 ≤ −η|s| I Integral Control: To minimize the reaching time and make s(t = 0) = 0, one can use integral control, i.e. ∫ t 0 x̃(r)dr as variable of interest. I The previous example is third order relative to this variable, so s: s = ( ddt + λ) 2( ∫ t 0 x̃(r)dr) = ˙̃x + 2λx̃ + λ2 ∫ t 0 x̃(r)dr I The approximation of control law will be changed to û = −f̂ + ẍd − 2λ ˙̃x − λ2x̃ I The control law, u and k will remain the same I Now if x̃(0) 6= 0 s = ˙̃x + 2λx̃ + λ2 ∫ t 0 x̃(r)dr − ˙̃x(0)− 2λx̃(0) I ∴ Although x̃(0) 6= 0, s(t = 0) = 0 Farzaneh Abdollahi Nonlinear Control Lecture 10 12/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I Outside of B(t), the control law u is like before to guarantee that the boundary layer is invariant I All trajectories starting inside B(t = 0) remain inside B(t) for all t > 0 I Inside B(t), u is interpolated I For instance,, inside B(t), in the expression of u replace sgn(s) by s/Φ, as shown in Fig I As it has been shown before, instead of perfect tracking, tracking to within a guaranteed precision εis guaranteed. I For all trajectories starting inside B(t = 0) ∀t ≥ 0|x̃ (i)| ≤ (2λ)iε i = 1, ..., n − 1 Farzaneh Abdollahi Nonlinear Control Lecture 10 15/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Example I Consider the system dynamics ẍ + a(t)ẋ2 cos 3x = u I 1 ≤ a(t) ≤ 2, for simulation a(t) = | sin t|+ 1, I λ = 20, η = 0.1 I f̂ = 1.5ẋ2 cos 3x , F = 0.5ẋ2| cos 3x | I By using the switching control law: u = û − ksgn(s) = 1.5ẋ2cos3x + ẍd − 20 ˙̃x − (0.5ẋ2| cos 3x |+ 0.1)sgn( ˙̃x + 20x̃) Farzaneh Abdollahi Nonlinear Control Lecture 10 16/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Example Cont’d I Tracking performance is excellent at the price of high control chattering Farzaneh Abdollahi Nonlinear Control Lecture 10 17/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Case 1: b = b̂ = 1 I the control signal is modified as: u = û − k̄(x)sat(s/Φ) I k̄(x) = k(x)− Φ̇ I sat(y) = { y if |y | ≤ 1 sgn(y) otherwise I So the system trajectories inside the boundary layer: ṡ = −k̄(x) sΦ −∆f (x) = −k̄(xd) s Φ + (−∆f (xd) + O(ε)) where ∆f = f̂ − f I We can consider a first order filter: I its dynamic depends on desired state xd I s : a measure of the algebraic distance to the surface S(t) is its output I the ”perturbations,” (uncertainty ∆f (xd)) is its input Farzaneh Abdollahi Nonlinear Control Lecture 10 20/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I s provides tracking error x̃ by further low pass filtering (2) I λ is break-frequency of the filter I It must be chosen to be ”small” with respect to high-frequency unmodeled dynamics (such as unmodeled structural modes or neglected time delays) I Let us define Φ based on bandwidth λ: k̄(xd )Φ = λ I and: φ̇+ λΦ = k(xd) (7) k̄(x) = k(x)− k(xd) + λΦ Farzaneh Abdollahi Nonlinear Control Lecture 10 21/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws I The boundary layer thickness Φ is defined based on the evolution of dynamic model uncertainty I Control signal depends on s I s-trajectory represents a TV measure of the validity of the assumptions on model uncertainty I tracking error x̃ is a filtered version of s Farzaneh Abdollahi Nonlinear Control Lecture 10 22/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Case 2: β 6= 1 I Define: βd = β(xd) = b(xd ) b̂(xd ) I If k(xd) ≥ λΦβd⇒Φ̇ + λΦ = βdk(xd) I If k(xd) ≤ λΦβd⇒Φ̇ + λΦ β2d = k(xd )βd I Φ(0) = βdk(xd(0))/λ I Modify λ = k̄(xd )βdΦ I And finally k̄(x) = k(x)− k(xd) + λΦβd Farzaneh Abdollahi Nonlinear Control Lecture 10 25/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Remarks 1. The desired trajectory xd must be smooth enough not to excite the high-frequency unmodeled dynamics. 2. The sliding control guarantees the best tracking performance given the desired control bandwidth and the extent of parameter uncertainty. 3. If the model or its bounds are so imprecise that F can only be chosen as a large constant, then define Φ a large constant, s.t. the term k̄sat(s/Φ) = λs/β like simple P.D. Farzaneh Abdollahi Nonlinear Control Lecture 10 26/30 Docsity.com Outline Sliding Control Continuous Approximations of Switching Control Laws Remarks 4. For exceptional disturbances which their intensity is high s.t may take the traj. out of the boundary: I If integral control is applied, the integral term in the control may become unreasonably large I once the disturbance stops, the system goes through large amplitude oscillations in order to return to the desired trajectory (integrator windup) I It is a potential cause of instability because of saturation effects and physical limits on the motion. I Solution: As long as the system is outside the boundary layer maintain the integral term constant I When the system remains in the boundary layer (returns to normal case after the exceptional disturbance) integration can resume Farzaneh Abdollahi Nonlinear Control Lecture 10 27/30 Docsity.com