1. A Deeper Look at LPV Stephan Bohacek USC

2. General Form of Linear Parametrically Varying (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  R n u  R m  - 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)) + f u (  (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))  g x (0,  (k),0) x(k) + g u (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,  )} nominal type 1 failure type n failure nominal type 1 failure type n failure ball of radius  centered at  n

6. SS (A  S  + B  E  ) T (C  S  ) T (D  S  ) T A  S + B  E  SS C  S  I D  S  I Stabilization of LPV Systems Packard and Becker, ASME Winter Meeting, 1992. Find S  R n  n and E  R m  n such that 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. > 0 for all 

7. x T 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 Cost For LPV systems, you only get an upper bound on the 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 = A T XA - A T XB(D T D + B T XB) -1 B T XA + CC } depends on  For LTI systems, you get the exact cost. If the LMI is not solvable, then the inequality is too conservative, or the system is unstabilizable.

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

9. S  +  i  [1,n]  i S i (A  S  + B  E  ) T (C  S  ) T (D  S  ) T A  S + B  E  SS (C  S  ) I (D  S  ) I Note that X(t)=(  i  [1,n] b i (t)S i ) -1 So X(t+d)=X(t)+X’(z)d by the mean value thm. Hence X(t+d)=X(t) + d/dx (  i  [1,n] b i (t)S i ) -1 =X(t) + (  i  [1,n] b i (t)S i ) -1  i  [1,n] (db i /dx)d i (  i  [1,n] b i (t)S i ) -1 Therefore, post and premultiplying the 1-1 element of the above matrix by X(t) Yeilds X(t+d)

10. 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] b i (  ) S i ) -1 and F  (k) = E  (k) X  (k) You still only get an upper bound on the cost Might the solution to the LMI be discontinuous? 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.

11. |x(k+j)|    (0)   (0) |x(k)| Linear Dynamically Varying (LDV) Systems Bohacek and Jonckheere, IEEE Trans. AC Assume that f is known. Def: The LDV system defined by (f,A,B) is stabilizable if there exists F :   Z  R m  n such that, if x(k+1) = (A  (k) + B  (k) F(  (0),k)) x(k)  (k+1) = f(  (k)) then j for some   (0) <  and   (0) < 1. 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.

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

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

14. LDV Controller for the Henon Map

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

16. 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  X f(  ) A  - L  (R  ) -1 L  In this case, X is continuous. TTT

17. X May Become Discontinuous as  is Reduced

18. LPV with Rate Limited Parameter Variation If the LMI is not solvable, then the set {b i } is too small (or  is too small), the inequality is too conservative, or the system is unstabilizable. S  +  i  {1,n}  i S i (A  S  + B  E  ) T (C  S  ) T (D  S  ) T A  S + B  E  SS C  S  I D  S  I Suppose that | f(  )-  |  <  and S  =  i  {1,N} b i (  ) S i E  =  i  {1,N} b i (  ) E i > 0 for all  and |  i |<   where S i  R n  n, E i  R m  n and {b i } is a set of orthogonal functions such that |b i (  ) - b i (  +  )| <  .

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

20. nominal type 1 failure type 2 failure LSVDV systems

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

22. -2-1.5-0.500.511.5 2 -2 -1.5 -0.5 0 0.5 1 1.5 2 Cost if Alternative 1 Occurs where Q = A  X 1 A  + C  C  TT

23. -2-1.5-0.500.511.52 -2 -1.5 -0.5 0 0.5 1 1.5 2 Cost if Alternative 2 Occurs where Q = A  X 2 A  + C  C 

24. Worst Case Cost -2-1.5-0.500.511.52 -2 -1.5 -0.5 0 0.5 1 1.5 2

25. The LMI Approach is Conservative -2-1.5-0.500.511.52 -2 -1.5 -0.5 0 0.5 1 1.5 2 conservative

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

27. quadratic Piecewise Quadratic Approximation of the Cost Define X(x,  ) := max i  N x T X i (  )x

28. Piecewise Quadratic Approximation of the Cost not an LMI

29. -3-20123 -3 -2 0 1 2 3 Piecewise Quadratic Approximation of the Cost

30. -3-20123 -3 -2 0 1 2 3

31. -0.8-0.6-0.4-0.200.20.40.60.81 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Piecewise Quadratic Approximation of the Cost Allowing non-positive definite X i permits good approximation.

32. -2.5-2-1.5-0.500.51 1.5 2 2.5 -2.5 -2 -1.5 -0.5 0 0.5 1 1.5 2 2.5 Piecewise Quadratic Approximation of the Cost

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

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

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

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

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

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