Chapter 7 Sliding Control

1. Chapter 7 Sliding Control

2. To Cope with Model Imprecision Model imprecision –actual uncertainty about the plant (e. g., unknown plant parameters) –the purposeful choice of a simplified representation of the system's dynamics (e. g., modeling friction as linear, or neglecting structural modes in a reasonably rigid mechanical system).

3. Modeling inaccuracies • structured (or parametric) uncertainties • –corresponds to inaccuracies on the terms actually included in the model • unstructured uncertainties (or unmodeled dynamics) • –corresponds to inaccuracies on (i. e., underestimation of) the system order

4. Sliding control A simple approach to robust control

5. 7.1 Sliding Surfaces Consider the single-input dynamic system x :output of interest, ex.: the position of a mechanical system, u : control input, ex: a motor torque :state vector. (7.1)

6. –function f (x) (in general nonlinear) is not exactly known, but the extent of the imprecision on f (x) is upper bounded by a known continuous function of x; –Control gain b(x) is not exactly known, but is of known sign and is bounded by known, continuous function of x.

7. Control Problem make the state x to track a specific time-varying state in the presence of model imprecision on f (x) and b(x). The initial desired state must be such that (7.2)

8. 7.1.1 A Notational Simplification Let be the tracking error and let be the tracking error vector.

9. Define a time-varying surface S(t) in the state-space by the scalar equation s(x; t) = 0, whereand  is a strictly positive constant. For instance, if n = 2, i. e., s is simply a weighted sum of the position error and the velocity error; if n = 3, (7.3)

10. Given initial condition (7.2), the problem of tracking x  xdis equivalent to that of remaining on the surface S(t) for all t > 0; indeed s  0 represents a linear differential equation whose unique solution is given initial conditions (7.2). Thus, the problem of tracking the n-dimensional vector can be reduced to that of keeping the scalar quantity s at zero.

11. Tracking problem 1st-order stabilization problem in s. (why?) Bounds on s can be directly translated into bounds on the tracking error vector , and therefore the scalar s represents a true measure of tracking performance.

12. 1st-order problem of keeping the scalar s at zero: Choosing u such that outside of S(t) (7.5) where  >0 –The squared "distance" to the surface decreases along all system trajectories. Thus, it constrains trajectories to point towards the surface S(t) , as illustrated in Figure 7.2.

13. –Once on the surface the system trajectories remain on the surface. (ideal situation) –Satisfying condition (7.5), or sliding condition, makes the surface an invariant set. –Sliding surface: S(t) verifying (7.5) -Sliding regime or sliding mode: the system's behavior on the surface.

14. Figure 7.2: The sliding condition

15. –Satisfying (7.5) if condition (7.2) is not exactly verified, i. e., if x(t = 0) is actually off xd(t = 0), the surface S(t) will nonetheless be reached in a finite time smaller than |s(t = 0)|/. -Assume s(t = 0) > 0, and let treach be the time required to hit the surface s = 0. Integrating (7.5) between t = 0 and t = treach :

16. Similar result for s(t = 0)<0, that Definition (7.3) on the surface, the tracking error tends exponentially to zero, with a time constant (n - 1)/ (from the sequence of (n - 1) filters of time constants equal to 1/.

17. Figure 7.3: Graphical interpretation of equations (7.3) and (7.5) (n = 2)

18. -Control Switching:

19. –Since the implementation of the associated control switchings is necessarily imperfect (for instance, switching, in practice, is not instantaneous, and the value of s is not known with infinite precision), this leads to chattering (Figure 7.4).

20. Figure 7.4:chattering as a result of imperfect control switchings

21. –Chattering is undesirable in practice, since it involves high control activity and further may excite high-frequency dynamics neglected in the course of modeling (such as unmodeled structural modes, neglected time-delays, and so on)

22. 7.1.2 Filippov's Construction of the Equivalent Dynamics The dynamics in sliding mode :Equivalent control, ueq: continuous control law to maintain if the dynamics were exactly known. (7.6)

23. Example: for a system of the formwe have and the system dynamics in sliding mode is Geometrically, the equivalent control can be constructed asi. e., as a convex combination of the values of u on both sides of the surface S(t). (7.7)

24. 7.1.3 Perfect Performance - At a Price Given the bounds on uncertainties of f (x) and b(x), constructing a control law to satisfy the sliding condition (7.5) : A basic example consider the second-order systemwhere f (possibly nonlinear or time-varying) is not exactly known, but estimated as (7.8)

25. The estimation error on f is assumed to be bounded by some known function F = FFor instance, given the systemwhere a(t) is unknown but satisfiesone has (7.9) (7.10)

26. In order to have the system track we define a sliding surface s = 0 according to (7.3), namely: We then have:The best approximation of a continuous control law that would achieve is thus (7.11) (7.12) (7.13)

27. Add to a term discontinuous across the surface s = 0:where sgn is sign function: k to be dtermined (7.14)

28. By choosing in (7.14) to be large enough: letting we get from (7.9) (7.15)

29. –Note from (7.15) the control discontinuity k across the surface s = 0 increases with the extent of parametric uncertainty. –Note that and F need not depend only on x or They may more generally be functions of any measured variables external to system (7.8), and may also depend explicitly on time.

30. Integral Control A similar result would be obtained by using integral control, i. e., formally letting to be the variable of interest. The system (7.8) is now third-order relative to this variable, and (7.3) gives: We then obtain, instead of (7.13),

31. with (7.14) and (7.15) formally unchanged. Note that can be replaced by i. e., the integral can be defined to within a constant. The constant can be chosen to obtain s (t = 0) = 0 regardless of xd(0), by letting

32. Gain Margins Assume now that (7.8) is replaced by where the (possibly time-varying or state-dependent) control gain b is unknown but of known bounds (themselves possibly time-varying or state-dependent) (7.16) (7.17)

33. Choose the estimate of gain b as the geometric mean of the above bounds: Bounds (7.17) can then be written in the form where :gain margin and (7.18)

34. With s and defined as before, the control law with Satisfies the sliding condition . (7.19) (7.20)

35. Indeed, using (7.19) in the expression of leads to so that k must verify Since where this in turn leads to and thus to (7.20). Note that the control discontinuity has been increased in order to account for the uncertainty on the control gain b.

36. 7.2 Continuous Approximations of Switching control Laws • Boundary Layer • where  is the boundary layer thickness, and is the boundary layer width. (7.25)

37. Figure 7.6.a: The boundary layer for n=2

38. –Choose u as before (i. e., satisfying sliding condition (7.5)), outside B(t), the boundary layer is attractive, hence invariant –All trajectories starting inside B(t=0) remain inside B(t) for all t  0; and we then interpolate u inside B(t)- for instance, replacing in the expression of u the term sgn(s) by s/, inside B(t).

39. Figure 7.6.b: Control interpolation in the boundary layer

40. Example 7.2: Consider the system (7.10). Assume the desired trajectory is xd = sin (t / 2). Figure 7.7 shows that tracking error and control law using the switched control law (with  = 20,  =0.1) at a sampling rate of 1 kHz. Actual value of a(t) used in the simulations is a(t) = | sint | + 1

41. The estimation error on f is assumed to be bounded by some known function F = FFor instance, given the systemwhere a(t) is unknown but satisfiesone has (7.9) (7.10)

42. Figure 7.7:Switched control input and resulting tracking performance

43. –Tracking performance is excellent, but is obtained at the price of high control chattering. –Assume now that we interpolate the above control input in a thin boundary layer of thickness 0.1

44. As shown in figure 7.8, the tracking performance, while not as “perfect” as above, is still very good, and is now achieved using a smooth control law. Note that the bounds on tracking error are consistent with (7.25).

45. Figure 7.8: Smooth control input and resulting tracking performance

46. How to improve the performance? Instead of the fixed boundary layer, using the varying thickness

47. Consider again the system (7.1) with –To guarantee the distance to the boundary layer always decreases we require in (7.26) reflects that the boundary layer attraction condition is more stringent during boundary layer contraction and less stringent during boundary layer expansion (7.26)

48. –To satisfy (7.26), the quantity is added to control discontinuity gain k(x), i. e., the term k(x)sgn(s) obtained from switched control law u is replaced by where and sat is the saturation function, which can be formally defined as Accordingly, control law u becomes: (7.27)

49. Let us now consider the system trajectories inside the boundary layer, where they lie by construction: they can be expressed directly in terms of the variable s as where (7.28)

50. Now since and are continuous in x, we can exploit (7.4) to rewrite (7.28) in the form We see from (7.29) that the variable s (which is a measure of the algebraic distance to the surface S(t)) can be viewed as the output of a first-order filter, whose dynamics only depends on the desired state xd(t), and whose inputs are, to the first order, “perturbations,” i. e., uncertainty f(xd). (7.29)