A Deeper Look at LPV

1. A Deeper Look at LPV Stephan Bohacek USC

2. General Form of Linear ParametricallyVarying (LPV) Systems x(k+1) = A(k)x(k) + B(k)u(k) z(k) = C(k)x(k) + D(k) u(k) (k+1) = f((k)) linear parts nonlinear part xRn u Rm  - compact A, B, C, D, and f are continuous functions.

3. How do LPV Systems Arise? Nonlinear tracking (k+1)=f((k),0) – desired trajectory (k+1)=f((k),u(k)) – trajectory of the system under control Objective: find u such that | (k)-(k) | 0 as k  . (k+1)= f((k),0) + f((k),0) ((k)- (k)) + fu((k),0) u(k) Define x(k) = (k) - (k) x(k+1) = A(k)x(k) + B(k)u(k)   A(k) B(k)

4. How do LPV Systems Arise ? Gain Scheduling x(k+1) = g(x(k), (k), u(k))  gx(0,(k),0) x(k) + gu(0,(k),0) u(k) (k+1) = f(x(k), (k), u(k)) – models variation in the parameters Objective: find u such that |x(k)| 0 as k    A(k) B(k)

5. Types of LPV Systems Different amounts of knowledge about f lead to a different types of LPV systems. f()  - know almost nothing about f (LPV) |f()- |< - know a bound on rate at which  varies (LPV with rate limited parameter variation) f() - know f exactly (LDV)  is a Markov Chain with known transition probabilities (Jump Linear) f() where f() is some known subset of  (LSVDV) f()={0, 1, 2,…, n} f()={B(0,), B(1,), B(2,),…, B(n ,)} type 1 failure ball of radius  centered at n type n failure nominal type n failure type 1 failure nominal

6. Stabilization of LPV SystemsPackard and Becker, ASME Winter Meeting, 1992. Find SRnn and ERmn such that for all  > 0 x(k+1) = (A+B (ES-1)) x(k) (k+1) = f((k)) In this case, is stable. If  is a polytope, then solving the LMI for all  is easy.

7. Cost For LTI systems, you get the exact cost. x(0) X x(0) = k[0,] |Cj[0,k](A+BF)x(0)|2 + |DF(j[0,k](A+BF))x(0)|2 where X = ATXA - ATXB(DTD + BTXB)-1BTXA + CC For LPV systems, you only get an upper bound on the cost. } xT X x k[0,] |C(k)j[0,k](A(j)+B(j)F)x|2 + |D(k)F(j[0,k](A(j)+B(j)F))x|2 where X=S-1 depends on  • If the LMI is not solvable, then • the inequality is too conservative, • or the system is unstabilizable.

8. LPV with Rate Limited Parameter VariationWu, Yang, Packard, Berker, Int. J. Robust and NL Cntrl, 1996Gahinet, Apkarian, Chilali, CDC 1994 Suppose that | f()-  | <  and where Si Rnn, Ei Rmnand {bi} is a set of orthogonal functions such that |bi() - bi(+)| < . S = i{1,N}bi() Si E = i{1,N}bi() Ei We have assumed solutions to the LMI have a particular structure. for all  and |i|<  > 0 x(k+1) = (A(k) + B(k)E(k) X(k))x(k) then is stable. where X = (S)-1

9. Cost You still only get an upper bound on the cost x(0) X(0) x(0)k{0,} |C(k)j{0,k}(A(j)+B(j)F(k))x(0)|2 + |D(k)F(k)(j{0,k}(A(j)+B(j)F(k)))x(0)|2 where X = (i[1,N]bi() Si)-1 and F(k) = E(k) X(k) • If the LMI is not solvable, then • the assumptions made on S are too strong, • the inequality is too conservative, • or the system is unstabilizable. Might the solution to the LMI be discontinuous?

10. Linear Dynamically Varying (LDV) SystemsBohacek and Jonckheere, IEEE Trans. AC Assume that f is known. x(k+1) = A(k)x(k) + B(k)u(k) z(k) = C(k)x(k) + D(k) u(k) (k+1) = f((k)) A, B, C, D and f are continuous functions. Def: The LDV system defined by (f,A,B) is stabilizable if there exists F :   Z  Rmn x(k+1) = (A(k) + B(k)F((0),k)) x(k) (k+1) = f((k)) such that, if |x(k+j)|  (0)(0)|x(k)| then j for some (0) <  and (0) < 1.

11. Continuity of LDV Controllers X = AXA + CC - AXB(DD + BXB)-1BXA T T T T T T u(k) = - (D (k) D (k) + B (k) X (k) B (k))-1B (k) X (k) A(k) x(k) T T T Theorem: LDV system (f,A,B) is stabilizable if and only if there exists a bounded solution X :  Rnn to the functional algebraic Riccati equation In this case, the optimal control is and X is continuous. Since X is continuous, X can be estimated by determining X on a grid of .

12. Continuity of LDV Controllers Continuity of X implies that if |1- 2| is small, then is small. Which is true if which only happened when f is stable, where  and  are independent of , which is more than stabilizability provides. or

13. LDV Controller for the Henon Map

14. HControl for LDV Systems Bohacek and Jonckheere SIAM J. Cntrl & Opt. Objective:

15. Continuity of the H Controller Theorem: There exists a controller such that if and only if there exists a bounded solution to X = CC + AXf()A - L(R)-1L T T T In this case, X is continuous.

16. X May Become Discontinuous as  is Reduced

17. LPV with Rate Limited Parameter Variation Suppose that | f()-  | <  and where Si Rnn, Ei Rmnand {bi} is a set of orthogonal functions such that |bi() - bi(+)| < . S = i{1,N}bi() Si E = i{1,N}bi() Ei for all  and |i|<  > 0 • If the LMI is not solvable, then • the set {bi} is too small (or  is too small), • the inequality is too conservative, • or the system is unstabilizable.

18. Linear Set Valued Dynamically Varying (LSVDV) Systems Bohacek and Jonckheere, ACC 2000 set valued dynamical system A, B, C, D and f are continuous functions.  is compact.

19. LSVDV systems type 1 failure nominal type 2 failure

20. 1 - Step Cost For example, let f()={1, 2} alternative 1 alternative 2

21. Cost if Alternative 1 Occurs 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 where Q = AX1A + CC T T

22. Cost if Alternative 2 Occurs 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 where Q = AX2A + CC

23. Worst Case Cost 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

24. The LMI Approach is Conservative 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 conservative

25. Worst Case Cost piece 2 piece 1 • non-quadratic cost • piece-wise quadratic

26. Piecewise Quadratic Approximation of the Cost Define X(x,) := maxiN xTXi()x quadratic

27. Piecewise Quadratic Approximation of the Cost not an LMI

28. Piecewise Quadratic Approximation of the Cost 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3

29. Piecewise Quadratic Approximation of the Cost 3 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3

30. Piecewise Quadratic Approximation of the Cost 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Allowing non-positive definite Xi permits good approximation.

31. Piecewise Quadratic Approximation of the Cost 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 2 1.5

32. The Cost is Continuous Theorem: If 1. the system is uniformly exponentially stable, 2. X : Rn    R solves 3. X(x, )  0, thenX is uniformly continuous. • Hence, X can be approximated: • partition Rn into N cones, and • grid  with M points.

33. Piecewise Quadratic Approximation of the Cost X(x,,T,N,M)  maxf()X(Ax,,T-1,N,M) + xTCCx T Define X(x,,T,N,M) := maxiNxTXi(,T,N,M)x such that X(x,,0,N,M) = xTx. X(x,,0,N,M)  X(x,) as N,M,T   Would like time horizon number of cones number of grid points in 

34. X can be Found via Convex Optimization The cone centered around first coordinate axis C1 := {x :  > 0, x = e1 + y, y1=0, |y|=1} depends N, the number is cones convex optimization:

35. X can be Found via Convex Optimization The cone centered around first coordinate axis C1 := {x :  > 0, x = e1 + y, y1=0, |y|=1} depends N, the number is cones convex optimization: X(x,,0,N,M,K)  X(x,) as N,M,T,K   Theorem: In fact, related to the continuity of X

36. Optimal Control of LSVDV Systems only the direction is important the optimal control is homogeneous but not additive

37. Summary LPV increasing knowledge about f increasing computational complexity increasing conservativeness LPV with rate limited parameter variation optimal in the limit LSVDV might not be that bad optimal LDV