Tecniche di ottimizzazione convessa per il controllo robusto di sistemi incerti

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1 Università di Siena Tecniche di ottimizzazione convessa per il controllo robusto di sistemi incerti Andrea Garulli Dipartimento di Ingegneria dell Informazione, Università di Siena With: Graziano Chesi (University of Hong Kong) Alberto Tesi (Università di Firenze) Antonio Vicino (Università di Siena) Scuola Avanzata Giovanni Zappa Parma, novembre 2006

2 Università di Siena 1 Outline Some basic facts: convexity, LMIs, positivity of polynomial forms Homogeneous polynomial forms Case Study I: Quadratic distance problems Case Study II: Homogeneous polynomial Lyapunov functions for robustness analysis of uncertain systems

3 Università di Siena 2 Convexity The great watershed in optimization isn t between linearity and nonlinearity, but convexity and nonconvexity. (R. Rockafellar, SIAM Review 1993) Convex optimization is becoming a technology. (S. Boyd, CDC Plenary, 2002) What does this mean? efficient numerical techniques (and stable software) for solving large-size problems lots of applications in different fields

4 Università di Siena 3 Semidefinite programs A special class of optimization problems: semidefinite programs (SDPs) min c T x s.t. F 0 + x 1 F 1 + x 2 F 2 + x n F n 0 where F i R m m are given matrices, and x R n is the variable. The constraint is a Linear Matrix Inequality (LMI) ( means negative semidefinite ) SDPs are convex optimization problems that can be solved by efficient numerical techniques (interior-point methods) SDPs with 1000 variables and constraints solvable on PC (far larger if sparsity and/or structure are exploited) Lots of applications in control [Boyd et al., 1994]

5 Università di Siena 4 Positive polynomials in control Many relevant control problems can be cast in terms of positivity of polynomial forms Examples: Robust stability margin for systems affected by structured uncertainty Construction of Lyapunov functions for uncertain systems Synthesis of H robust controllers of fixed complexity Estimation of the domain of attraction of nonlinear systems Robust disturbance analysis for nonlinear systems Key point (late 90s): Sufficient conditions for testing positivity of polynomial forms can be formulated in terms of SDPs

6 Università di Siena 5 Homogeneous polynomial forms A special case is represented by homogeneous polynomial forms (all monomials have same degree) More parsimonious representation Structural properties Next slides: formulation of LMI-based positivity test for homogeneous polynomial forms

7 Università di Siena 6 Homogeneous polynomial forms: the Complete Square Matricial Representation A function f m (x) is a (real n-variate) Homogeneous Form (HF) of degree m in x R n if f m (x) = i 1 +i i n =m c i1,i 2,...,i n x i 1 1 x i x i n n where i 1, i 2,...,i n are nonnegative integers, and c i1,i 2,...,i n R are weighting coefficients. The HF f m (x) is - positive if f m (x) > 0 x 0 - nonnegative if f m (x) 0 x

8 Università di Siena 7 Complete Square Matricial Representation (1/3) Let x {m} R d be a vector containing all monomials of degree m in x. The number of these monomials is d = σ(n,m) n + m 1 m = (n + m 1)! (n 1)!m! Definition [SMR] Let f 2m (x) be any HF of degree 2m. The Square Matricial Representation (SMR) of f 2m (x) is given by f 2m (x) = x {m} F x {m} where F = F R d d is a suitable coefficient matrix. The matrix F is called SMR matrix of f 2m (x).

9 Università di Siena 8 Complete Square Matricial Representation (2/3) Fact: the matrix F is not unique! Indeed, f 2m (x) = x {m} (F + L)x {m} L L where L = L = L : x {m} Lx {m} = 0 x The set L is a linear space of dimension d L = 1 σ(n, m)(σ(n,m) + 1) σ(n, 2m) 2

10 Università di Siena 9 Complete Square Matricial Representation (3/3) Definition [CSMR] Let L(α) be a linear parameterization of the set L with α R d L. The Complete SMR (CSMR) of f 2m (x) is given by f 2m (x) = x {m} F(α) x {m} F(α) = F + L(α) where F = F is any SMR matrix of f 2m (x). The matrix F(α) is denoted as CSMR matrix of f 2m (x).

11 Università di Siena 10 Key problem: how to ensure that a HF is positive A sufficient condition: If there exists α R d L such that the matrix F + L(α) is positive definite, then f 2m (x) is a positive HF Facts: The converse is not true in general (counterexamples) The converse is true in some special cases (n = 2 for any m; n = 3 and m = 2) Main benefit: finding α R d L such that F + L(α) > 0 is an LMI feasibility problem

12 Università di Siena 11 Example (1/2) f 6 (x) = x 6 1 2x 5 1x 2 3x 4 1x x 3 1x x 2 1x x 6 2 n = 2 (x R 2 ) m = 3 (degree = 2m = 6) x {3} = (x 3 1 x 2 1x 2 x 1 x 2 2 x 3 2) A SMR matrix is F =

13 Università di Siena 12 Example (2/2) d L = 3 (dimension of subset L) The CSMR matrix is F + L(α) with α R 3 and L(α) = 0 0 α 1 α 2 0 2α 1 α 2 α 3 α 1 α 2 2α 3 0 α 2 α F is not positive definite α 1, α 2, α 3 are the degrees of freedom that can be exploited in order to prove positivity of f 6 (x) Choosing α 1 = 1.8, α 2 = 0.4, α 3 = 3.6, it turns out that F + L(α) > 0 and hence f 6 (x) is positive

14 Università di Siena 13 Basic idea dates back to: G. Chesi, A. Tesi, A. Vicino, R. Genesio, A convex approach to a class of minimum norm problems, in: Robustness in Identification and Control, Springer, Many people have worked on this in recent years (Parrilo, Reznick, Lasserre, Henrion, Scherer,... many others!) Several problems have been tackled within this framework. In this talk, two case studies: I) solution of nonconvex distance problems II) construction of homogeneous Lyapunov functions for robustness analysis

15 Università di Siena 14 Case Study I: Quadratic distance problems

16 Università di Siena 15 Motivation Several problems in system analysis and control can be cast as minimum distance problems: optimal estimate of the domain of attraction of equilibria parametric stability margin of a control system D-stability of real matrices region of validity of linear H controllers for nonlinear systems characterization of frequency plots of ellipsoidal plant families... Moreover... problems from different fields Example: estimation of the fundamental matrix in dynamic vision Issues: non-convex optimization, local minima, high complexity,...

17 Università di Siena 16 Canonical Quadratic Distance Problem (CQDP) Let w(x) be a (real n-variate) even polynomial of degree 2m, i.e., w(x) = m i=0 w 2i (x) where w 2i (x), 1 i m, are homogeneous forms of degree 2i. Then, compute s.t. c min = min x R n w(x) = 0 x 2 Assumptions (w.l.o.g.): 1. {x : w(x) = 0} 2. δ > 0 such that x < δ w(x) > 0

18 Università di Siena 17 Generality of CQDPs Consider a general quadratic distance problem s.t. c i = min ξ R n p(ξ) = 0 ξ x 2 U where x R n, U R n n is a positive definite matrix and p(ξ) is a generic polynomial of degree m. There exists a CQDP such that c min = c i Hint: set x = T(ξ) = U f (ξ x) where U f R n n is a non-singular matrix such that U = U fu f and w(x) = p T 1 (x) p T 1 ( x).

19 Università di Siena 18 QDP Example (1/6) s.t. c i = min ξ R 2 ξ2 1 2ξ 1 ξ 2 + 2ξ 2 2 ξ ξ ξ 1 1 = x x 1

20 Università di Siena 19 QDP Example (2/6) CQDP with x R 2 and w(x) = x x7 1 x x 6 1 x x5 1 x x4 1 x x3 1 x x2 1 x x 1x x 8 2 2x4 1 8x3 1 x 2 12x 2 1 x2 2 8x 1x 3 2 4x4 2 4x2 1 8x 1x 2 4x x x 1

21 Università di Siena 20 Basic idea for solving CQDPs (1/2) s.t. c min = min x 1,x 2 x x 2 2 w(x) = 1 x x 2 1x 2 2 x 4 4 = X X1 Constraint set: w(x) = 0; Level sets: B c = {x 1, x 2 : x x 2 2 = c}, c R

22 Università di Siena 21 Basic idea for solving CQDPs (2/2) Find the smallest c for which the level set B c is tangent to the constraint set w(x) = 0 Key observation: make w(x) homogeneous w(x) = 1 x x 2 1x 2 2 x 4 4 = w(x; c) = (x2 1 + x 2 2) 2 x 4 c x 2 1x 2 2 x 4 4 Important consequence: w(x) > 0 x B c w(x; c) > 0 x B c w(x; c) > 0 x 0

23 Università di Siena 22 Solution of CQDP: one-parameter family of positivity test Let B c = {x : x 2 = c}. The solution c min of CQDP is given by c min = sup { c R : w(x) > 0, x B c c (0, c] } Lemma 1 Let w(x; c) = Then, m i=0 x 2(m i) w 2i (x) c m i. w(x) > 0 x B c w(x; c) > 0 x 0 c min can be found by solving a one-parameter family of positivity tests on the homogeneous form w(x; c) (of degree 2m in x for all c 0)

24 Università di Siena 23 QDP Example (3/6) w(x) = x x7 1 x x 6 1 x x5 1 x x4 1 x x3 1 x x2 1 x x 1x x 8 2 2x4 1 8x3 1 x 2 12x 2 1 x2 2 8x 1x 3 2 4x4 2 4x2 1 8x 1x 2 4x Apply Lemma 1: w(x; c) = x x7 1 x x 6 1 x x5 1 x x4 1 x x3 1 x x2 1 x x 1x x8 2 + (x2 1 + x2 2 )2 c 2 + (x2 1 + x2 2 )3 c 3 2x 4 1 8x3 1 x 2 12x 2 1 x2 2 8x 1x 3 2 4x4 2 4x 2 1 8x 1x 2 4x (x2 1 + x2 2 )4 c 4

25 Università di Siena 24 QDP Example (4/6) Introduce the CSMR of w(x; c): x {4} = (x 4 1 x 3 1x 2 x 2 1x 2 2 x 1 x 3 2 x 4 2) CSMR matrix: W(c; α) = W(c) + L(α) W(c) = c c c

26 Università di Siena 25 QDP Example (5/6) and L(α) = 0 0 α 1 α 2 α 3 α 4 0 2α 1 α 2 α 3 α 5 α 1 α 2 2α 4 α 5 α 6 α 2 α 3 α 5 2α 6 0 α 3 α 4 α 5 α Objective: Exploit the 6 degrees of freedom to make W(c; α) positive definite

27 Università di Siena 26 Lower bound for the solution of CQDP Complete SMR of w(x; c) : w(x; c) = x {m} W(c; α) x {m} α R d L where W(c; α) = W(c) + L(α), with L(α) L. Theorem 1 Let and Then, η(c) = max α R d L λ min [W(c; α)] ĉ min = sup { ĉ : η(c) > 0 c (0, ĉ] } ĉ min c min.

28 Università di Siena 27 Computation of the lower bound Computation of η(c) for fixed c can be rewritten as an LMI optimization problem s.t. η(c) = max t R,α R d L t W(c; α) ti d 0 The lower bound ĉ min can be computed by solving a one-parameter family of LMIs (e.g.: bisection search over c)

29 Università di Siena 28 Size of the LMIs for computing the lower bound n=2 3 4 m=1 d = 2, d L = 0 3, 0 4, 0 2 3, 1 6, 6 10, , 3 10, 27 20, , 6 15, 75 35, 465 Table 1: Values of d and d L for some n and m. c min = min x R n x 2 η(c) = max t R,α R d L s.t. s.t. w(x) = m i=0 w 2i(x) = 0 W(c; α) ti d 0 t

30 Università di Siena 29 Optimality: a posteriori test Is the lower bound tight? How is it possible to check if ĉ min = c min? Theorem 2 Let α min R d L be such that λ min [W(ĉ min ; α min )] = 0 and K = Ker[W(ĉ min ; α min )]. Then, ĉ min = c min if and only if there exists x min R n such that x {m} min K and x min 2 = ĉ min From Theorem 2 a simple algorithmic procedure for computing x min and testing tightness of ĉ min

31 Università di Siena 30 Optimality: a priori conditions Are there CQDPs for which it can be ensured a priori that ĉ min = c min? Theorem 3 A necessary and sufficient condition for which ĉ min = c min is that for all c < c min w(x; c) = h i=1 f 2 m,i(x) where f m,i (x) are suitable homogeneous forms of degree m. There exist families of homogeneous forms which satisfy Theorem 3 Theorem 4 Let C 2, = {CQDPs such that n = 2}, C 3,2 = {CQDPs such that n = 3 and m = 2}. Then, the lower bound ĉ min for any CQDP belonging to C 2, or C 3,2 is tight.

32 Università di Siena 31 QDP Example (6/6) The sequence of LMI optimizations returns ĉ min = Two ways to check bound tightness: a posteriori LMI sequence returns α min = 10 3 (3.550, 3.295, 6.923,6.856, 3.275,3.376) and K = Ker[W(ĉ min ; α min )] = span{(0.4272,0.4369,0.4468,0.4569,0.4672) } Optimality test provides candidate minimum ˆx = (0.2340, ) which satisfies ˆx {4} K and ˆx 2 = ĉ min (see Theorem 2) a priori This CQDP has n = 2 and hence Theorem 4 applies!

33 Università di Siena 32 Extensions Same framework allows to tackle other problems Maximum distance problems upper bound (with similar tightness conditions) Seminorm optimization s s.t. c min = min x R n i=1 x 2 i s < n w(x) = 0 Two-forms case w(x) = w 2k (x) + w 2m (x), 0 k < m lower bound can be obtained from one single LMI optimization

34 Università di Siena 33 Numerical examples: randomly generated CQDPs Class of problems 1. C 3,3 = {CQDP s.t. n = 3 and m = 3} d = 10, d L = C 4,2 = {CQDP s.t. n = 4 and m = 2} d = 10, d L = randomly generated CQDPs for each class Coefficients of the forms w(x) randomly generated from normal distribution Feasibility ensured by setting w(x) = 0 for some x For every CQDP: LMI optimization + a posteriori optimality test dimension of kernel K always equal to 1 algorithm always returned a candidate minimum ˆx K

35 Università di Siena 34 Performance indexes η g = c g ĉ min ĉ min η g = c g ĉ min ĉ min c g returned by the function CONSTR of MATLAB, initialized at the candidate minimum point ˆx c g returned by the function CONSTR of MATLAB, initialized at a fixed point x (η g > 0.01 local minimum) e t = w(ˆx) v w q(ˆx) measure of non-feasibility for the candidate minimum point ˆx, with w(x) = v w q(x) and q(x) = 1 x {2} x {4}... x {2m} max(ηg) max(e t ) average(e t ) rate(η g > 0.01) C 3, % C 4, % Statistics for randomly generated CQDPs

36 Università di Siena log 10 (e t ) log (η ) 10 g e t for class C 3,3 η g for class C 3, log 10 (e t ) log 10 (η g ) e t for class C 4,2 η g for class C 4,2

37 Università di Siena 36 An example of non-tight lower bound CQDP with n = 3, m = 3 and w(x) = f ns (x) (x x x 2 3) 3 with f ns (x) = x 4 1x x 2 1x 4 2 3x 2 1x 2 2x x 6 3 f ns (x) is a non negative homogeneous form x w(x; c) = 1 c 3 1 x f ns (x) c min 1 w(x; 1) = 10f ns (x) can t be written as a sum of squares ĉ min < 1 Indeed, procedure returns ĉ min = , while c min = 1 (for x 1 = x 2 = x 3 = 1/ 3) Dimension of K greater than 1 (... generalize?...)

38 Università di Siena 37 Estimation of fundamental matrix in stereo vision systems (1/2) Fundamental matrix F: u T F u = 0 u,u rank(f) = 2 Estimation of F requires minimization of rational functions in two variables (with possible local minima) Problem can be cast as a seminorm optimization s.t. min x R 3 x2 3 w(x) = 0 and solved in the proposed framework (avoiding local mimima)

39 Università di Siena 38 Estimation of fundamental matrix in stereo vision systems (2/2) g(z) z 2 z 1 Rational function to be minimized In this case: d = 10, d L = 27 it turns out ĉ min = c min

40 Università di Siena 39 Extensions: Positivstellensatz & co. Positivstellensatz (simplified version) Let f(x) and h(x) be given polynomials. The set {x : f(x) 0, h(x) = 0} is empty if and only if there exist polynomials s 1 (x), s 2 (x) and p(x) satisfying s 1 (x) + s 2 (x) f(x) + p(x) h(x) < 0 and s 1 (x), s 2 (x) are sum of squares (SOS). x Main fact: testing if a polynomial is a SOS can be cast in terms of an LMI feasibility problem (analog to positivity of homogeneous forms)

41 Università di Siena 40 Application to distance problems min s.t. c(x) w i (x) = 0, i = 1,...,n (1) where c(x), w i (x) are given polynomials If there exist polynomials p i (x) such that c(x) γ + n p i (x)w i (x) i=1 is a SOS then γ is a lower bound for problem (1). Good news: the above test is an LMI in γ and the parameters of p i (x) Bad news: may need high degrees of p i (x) to get a tight lower bound

42 Università di Siena 41 SOS relaxations Powerful tool to treat fairly general optimization problems (in control and other fields) Families of nested relaxations to trade-off accuracy and complexity (Scherer, Henrion, etc.) Possibility to exploit structure, sparsity, etc. (Parrilo and others) Software packages available (SOSTOOLS, GloptiPoly)

43 Università di Siena 42 Case Study II: Homogeneous polynomial Lyapunov functions for robustness analysis of uncertain systems

44 Università di Siena 43 Basic facts about Lyapunov Functions (LFs) An asymptotically stable linear system ẋ(t) = A x(t) always admits a quadratic LF, V (x) = x T Px When the system is affected by parametric uncertainty, ẋ(t) = A(w) x(t), with w W (given polytope), A(w) = i w ia i if w is (arbitrarily) time-varying, robust asymptotic stability is equivalent to the existence of a common Lyapunov function for all matrices A(w), w W if w is time-invariant (or slowly time-varying), it is worth using parameter-dependent Lyapunov functions In both cases, quadratic LFs provide only sufficient conditions

45 Università di Siena 44 Homogeneous polynomial LFs Main Idea: use (homogeneous) polynomial Lyapunov functions New classes of Lyapunov functions for robust stability analysis of systems affected by structured uncertainties Stability tests formulated in terms of LMIs and GEVPs Useful properties: necessity, monotonicity,... Problems: Pb. I: systems with time-varying uncertainty Pb. II: systems with time-invariant uncertainty Pb. III: time-varying uncertainty with bounded variation rate

46 Università di Siena 45 Problem I: Polytopic systems with time-varying uncertainty Σ : ẋ(t) = A(w(t)) x(t) A(w(t)) = A 0 + q i=1 w i (t)a i A 0, A 1,...,A q R n n are given matrices The uncertain time-varying parameter vector w(t) belongs to the polytope W = co{w 1,...,w r } for all t 0, with w i R q, i = 1,...,r Problem: Construct a Lyapunov function proving robust asymptotic stability of Σ

47 Università di Siena 46 Homogeneous Polynomial Lyapunov Functions Basic Idea: Search among homogeneous polynomial forms of degree 2m Homogeneous Polynomial Lyapunov Functions (HPLFs) HPLF of degree 2m: v 2m (x) Find v 2m (x) such that: (i) v 2m (x) > 0 for all x 0; (ii) v 2m (x) < 0 for all x 0 and for all w(t) W.

48 Università di Siena 47 Extended matrix Use the SMR to write v 2m (x) = x {m} V x {m}. Then, v 2m (x) = d dt x{m} V x {m} + x {m} V We have that d dt x{m} = A {m} x {m} where A {m} R d d is the extended matrix given by: d dt x{m} A {m} = (K mk m ) 1 K m m 1 i=0 I n m 1 i A I n i K m with K m R nm d satisfying x [m] = K m x {m} and x [m] = x x x Known in the literature as power transformation [Brockett, Barkin,...] Key result: A is Hurwitz iff A {m} is Hurwitz m

49 Università di Siena 48 Sufficient condition for the existence of a HPLF Theorem 5 Let Āj = A(w j ), j = 1,...,r and Āj,{m} denote the extended matrix of Āj. If the system of LMIs V > 0 V Āj,{m} Ā j,{m}v L(α j ) > 0 j = 1,...,r admits a feasible solution V = V R d d α j R d L, j = 1,...,r then v 2m (x) = x {m} V x {m} is a HPLF for system Σ. v 2m (x) is a common quadratic LF for the extended polytopic system!

50 Università di Siena 49 Corollaries The computation of lower bounds of parametric stability margins requires the solution of a Generalized EigenValue Problem (GEVP), which is a quasi-convex optimization problem and can be tackled by efficient optimization tools [Boyd, Nemirovski,...] Also the construction of the Lyapunov function that achieves the best transient performance (optimal performance HPLF) can be cast as a GEVP

51 Università di Siena 50 Other results (necessity, conservativeness) For some values of n and m the LMI condition is also necessary (a) n = 2, m N (b) n = 3, 2m = 4 (c) n N, 2m = 2 (quadratic LF... well known!) positive homogeneous forms that are also SOS Conservativity of LMI sufficient condition does not increase with m (m = 1 HPLFs encompass QLFs)

52 Università di Siena 51 Related work The proposed parameterization of HPLFs encompasses other families of polynomial Lyapunov functions, for which sufficient conditions have been derived using the S-procedure [Zelentsovsky] For this problem, it is useless to consider the lower degree terms: polynomial LFs cannot do better than HPLFs of the same degree [Packard] HPLFs are a universal class of LFs: robust stability of Σ implies the existence of a HPLF (which is also a sum-of-squares) [Blanchini & Miani; Dayawansa & Martin]

53 Università di Siena 52 HPLF Example 1 ẋ(t) = (A 0 + w(t)a 1 )x(t) A 0 = A 1 = Objective: compute κ 2m = sup {κ : v 2m (x) for the system, with 0 w(t) κ} Quadratic LF : κ < 3.82 HPLF with S-procedure [Zelentsovsky] : κ < 5.73 (m = 2) Polyhedral LF [Blanchini & Miani] : κ = 6 Piecewise quadratic LF [Xie et al.] : κ = 6.2 HPLFs of increasing degree 2m κ 2m Solution of one GEVP, with (3m 2 + m + 2)/2 free variables

54 Università di Siena 53 HPLF Example 2 (1/2) ξ(t) + ξ(t) + k(t)ξ(t) = 0 Objective: compute the maximum κ such that the solution remains bounded, for all 0 k(t) κ The maximum value must satisfy k exp 1 4κ 1 π arctan( 4κ 1) 1 = 0 and is equal to κ [Brockett] Problem can be tackled in the framework of HPLF by computing κ 2m with A 0 = , A 1 =

55 Università di Siena 54 HPLF Example 2 (2/2) Values of κ 2m obtained from solution of the GEVP 2m κ 2m Trajectory of the system corresponding to the worst-case sequence k(t) Compare with the level curves of the HPLFs! limit trajectory x 2 0 x m=4 m=3 m=2 0.5 m=12 1 m= (a) level curves of v 2m (x) x (b) v 24 (x) and limit trajectory x 1

56 Università di Siena 55 HPLF Example 3 Helicopter model with robust controller [Narendra, Y.H. Chen] Resulting closed-loop uncertain system ẋ(t) = w 3 (t) w 3 (t) w 3 (t) w 3 (t) w 1 (t) w 2 (t) x(t) with w 1 (t) , w 2 (t) , w 3 (t) Objective: compute λ 2m = inf v 2m sup x R n \0 sup q(t) v(x) v(x) System Σ with n = 4. Uncertainty set W with r = 8 vertices. Quadratic LF : λ = [Olas] HPLF of degree 4: ˆλ 4 = (upper bound returned by GEVP)

57 Università di Siena 56 Extension to systems with rational time-varying uncertainty Σ rat : ẋ(t) = A rat (w(t)) x(t) A rat (w(t)) real-valued rational function of w(t) w(t) W = co{w 1,...,w l } for all t 0, with w i R r, i = 1,...,l Problem: Construct a Lyapunov function proving robust asymptotic stability of Σ rat

58 Università di Siena 57 Linear Fractional Representation (LFR) Exploiting Linear Fractional Representation (LFR), system Σ rat can be rewritten as: Σ rat : ẋ = Ax + Bq p = Cx + Dq q = E(w)p, E(w) = diag(w 1 I s1,...,w r I sr ) where q,p R h are auxiliary vectors, A R n n, B R n h, C R h n, D R h h are suitable matrices, and h = s s r. Use HPLFs! Find a HPLF v 2m (x) = x {m} V x {m} which ensures robust asymptotic stability of Σ rat for all w(t) W.

59 Università di Siena 58 Derivative representation (1/2) We have v 2m (x) = d dt x{m} V x {m} + x {m} V d dt x{m} From the LFR one has d dt x{m} = x{m} x (Ax + BE(w)p) = A {m}x {m} + B(w) p x {m 1} where B(w) is a suitable extended matrix. Hence, v 2m (x) = y m (x; p) U(V, w)y m (x; p) where y m (x; p) = x {m} p x {m 1}, U(V, w) = V A {m} + A {m}v V B(w) 0

60 Università di Siena 59 Derivative representation (2/2) Degrees of freedom in the representation of v 2m (x)? 1. Look at the set N = N = N : y m(x; p)ny m (x; p) = 0 x p There exists a linear parameterization N(β) of N 2. Let C, D(w) be the extended matrices satisfying x p Cx x {m 1} = Cx {m}, DE(w)p x {m 1} = D(w) p x {m 1} Then, for all matrices G,H of suitable dimension, y m(x; p)w(g,h, w)y m (x; p) = 0 where W(G, H, w) = G C + C G G D(w) G + C H H D(w) + D(w) H H H

61 Università di Siena 60 Existence of a HPLF for LFR systems Theorem 6 If there exist matrices V = V, G, H and vectors β i, i = 1,...,l, satisfying V > 0 U(V,w i ) + W(G, H, w i ) + N m (β i ) < 0 i = 1,...,l then v 2m (x) = x {m} V x {m} is a HPLF for the uncertain system Σ rat. LMI feasibility problem! If d LFR = max{s 1,...,s r } = 1, a less conservative LMI condition can be formulated

62 Università di Siena 61 HPLF Example 4 (1/3) ẋ(t) = A rat (θ(t)) x(t) where A rat (θ(t)) = θ 2 (t) 2 + θ 1(t) + 2θ 1 (t)θ 2 (t) 2 θ 1 (t) and Θ = co{[0, 0], [1,3], [1, 1] }. The LFR is A = , B = , C = ,. D = , E(θ) = n = 2, d = 2, d LFR = 1 θ θ 2

63 Università di Siena 62 HPLF Example 4 (2/3) m ˆγ 2m Size N m Free var x m= x 1 Left: 2m-HPLF stability margin ˆγ 2m = sup {γ : HPLF of degree 2m over ηθ for all η [0,γ]}, size of matrix N m and free variables in the LMI test Right: level curves of v 2m (x) corresponding to ˆγ 2m, m = 1,..., 5

64 Università di Siena 63 HPLF Example 4 (3/3) Alternative way to establish robust stability over ˆγ 10 Θ: define A rat (a) = {A rat (θ) : θ aθ} C rat (a) = {[c 1, c 2 ] : c 1 = 3 4θ 2, c 2 = (2 + θ 1 + 2θ 1 θ 2 )/(2 θ 1 ), θ aθ}. The polytope A(ˆγ 10 ) = {[0, 1; c 1, c 2 ] : [c 1,c 2 ] C(ˆγ 10 )} bounds the set A rat (ˆγ 10 ), where C(ˆγ 10 ) is the polytope bounding set C rat (ˆγ 10 ) (dashed) c c 1 However: robust stability of ẋ(t) = A(t)x(t) for A(t) A(ˆγ 10 ) cannot be proved even by a HPLF of degree 20 (robust stability margin )

65 Università di Siena 64 Problem II: Polytopic systems with time-invariant uncertainty ẋ(t) = A(w) x(t) w P = w R q : q i=1 w i = 1, w i 0, i = 1,2,...,q A(w) = q i=1 w i A i, with A i R n n, i = 1,...,q, given real matrices Problem: Consider the polytope of matrices A = A(w) R n n : w P Establish if A contains only Hurwitz matrices..

66 Università di Siena 65 HPD-QLFs Fact: parameter-dependent quadratic Lyapunov functions usually provide less conservative results for polytopic systems Idea: use Homogeneously Parameter-Dependent Quadratic Lyapunov Functions (HPD-QLFs) v s (x, w) = x P s (w)x, where P s (w) R n n is a homogeneous matricial form of degree s (a matrix whose entries are real q-variate homogeneous polynomial forms of degree s)

67 Università di Siena 66 A basic result on HPD-QLFs The set A is Hurwitz if and only if there exists a HPD-QLF v s (x, w) such that P s (w) > 0 A (w)p s (w) + P s (w)a(w) < 0 w P Moreover, un upper bound on the degree can be established: s 1 n(n + 1) 1. 2 Aim: provide sufficient conditions for the existence of a HPD-QLF for the polytope of matrices A, in terms of LMIs

68 Università di Siena 67 Homogeneous matricial forms There exists a CSMR also for homogeneous polynomial matricial forms Let C 2s (w) R n n be a homogeneous matricial form of degree 2s C 2s (w) = w {s} I n [Cs + U s (α)] w {s} I n α : vector of free parameters U s (α) is the linear parameterization of U s = U s = U s : w {s} I n Us w {s} I n = 0 n n

69 Università di Siena 68 Computing HPD-QLFs: v s (x, w) > 0 Lemma 2 Let sq([w 1, w 2,...,w q ] ). = [w 2 1, w 2 2,...,w 2 q]. Then, P s (w) > 0 w P P s (sq(w)) > 0 w R q 0. Notice that for some S s S s where P s (sq(w)) = w {s} I n Ss w {s} I n S s = S s = S s : w {s} I n contain entries w i 1 1 w i w i q q Ss w {s} I n with any odd i j does not. Also S s admits a linear parameterization S s (β) S s (β) > 0 v s (x, w) > 0 This induces a corresponding linear parameterization P s (w, β)

70 Università di Siena 69 Computing HPD-QLFs: v s (x, w) < 0 Let Q s+1 (w, β) = A (w)p s (w, β) P s (w, β)a(w) and consider the corresponding SMR: Q s+1 (sq(w), β) = w {s+1} I n Rs+1 (β) w {s+1} I n If there exists α such that then, v s (x, w) < 0, w P. R s+1 (β) + U s+1 (α) > 0

71 Università di Siena 70 Sufficient condition for the existence of a HPD-QLF Theorem 7 The polytope A is Hurwitz if there exists a nonnegative integer s, and parameter vectors α, β such that S s (β) > 0 R s+1 (β) + U s+1 (α) > 0 where S s (β) S s and U s+1 (α) U s+1. LMI feasibility problem!

72 Università di Siena 71 Conservativeness of HPD-QLFs Is it possible to reduce conservativeness by increasing the degree s? Result: if there exists a HPD-QLF of degree s for the polytope of matrices A, then there exists also a HPD-QLF of degree s + 1 Why restricting to homogeneous forms? Result: if there exists a QLF with polynomial dependence on the parameters of degree s, for the polytope of matrices A, then there is also a HPD-QLF with homogeneous dependence of the same degree for fixed s the only conservativeness comes from the SOS gap

73 Università di Siena 72 HPD-QLF Example 1 (1/2) Compute the robust parametric margin ρ = sup { η R : A(w, η) is Hurwitz for all w P for all η [0, η]} where A(w, η) = q w i A i (η), A i (η) = Ā0 + ηāi, i = 1,...,q i=1 Consider Ā 0 = , Ā 1 = , Ā 2 = q = 2, n = 3

74 Università di Siena 73 HPD-QLF Example 1 (2/2) s ˆρ s Free var ˆρ s : lower bound to ρ, obtained via HPD-QLF of degree s ρ = (true robust parametric margin achieved by s = 2) linearly PD-QLFs are conservative [Leite & Peres, 2003] returns ρ L = (same as s = 1)

75 Università di Siena 74 HPD-QLF Example 2 (1/2) Same problem, with Ā 0 = Ā 1 = Ā 2 = Ā 3 = q = 3, n = 4

76 Università di Siena 75 HPD-QLF Example 2 (2/2) s ˆρ s Free var ˆρ s : lower bound to ρ, obtained via HPD-QLF of degree s ρ = (true robust parametric margin achieved by s = 2) linearly PD-QLFs are conservative [Leite & Peres, 2003] returns ρ L = < ˆρ {s=1} (!)

77 Università di Siena 76 Problem III: Polytopic systems with bounded variation rate uncertainty Σ : ẋ(t) = A(w(t)) x(t) (2) w P = w R q : q i=1 w i = 1, w i 0, i = 1,2,...,q ẇ D = co{d 1,...,d h } ( A(w) = q i=1 q i=1 dj i = 0, 0 D) w i A i, with A i R n n, i = 1,...,q, given real matrices

78 Università di Siena 77 Problem formulation [P1] Robust stability: Establish if system Σ is robustly asymptotically stable [P2] Maximum variation rate: Compute the maximum scaling factor γ of polytope D for which robust asymptotic stability is still guaranteed, i.e. γ = sup {a R : the origin is an asymptotically stable equilibrium point under the constraints w(t) P and ẇ(t) ad}

79 Università di Siena 78 HPD-HLFs Key step: construction of a Homogeneously Parameter-Dependent Homogeneous Lyapunov Function (HPD-HLF) v(x, w) = a i,j w i x j i N q, j N n q k=1 i k = s n k=1 j k = 2m HPD-HLFs generalize previous classes of LFs: s = 0 m = 1 HPLFs HPD-QLFs s = 1 & m = 1 [Gahinet et al. 1996, Montagner & Peres 2003]

80 Università di Siena 79 Stability analysis via HPD-HLFs (1/3) Aim: find a HPD-HLF v(x, w) such that v(x, w) > 0 v(x, w) < 0 (x, w) R n 0 P (x, w, ẇ) R n 0 P D One has v(x, w) = v(x, w) x A(w)x + v(x, w) w ẇ = v(x, w) x A(w)x + q i=1 w i 2 v(x, w) ẇ w v(x, w) can be written as a homogeneous form in x and w

81 Università di Siena 80 Stability analysis via HPD-HLFs (2/3) Getting rid of P via Lemma 2, the stability condition can be rewritten as v(x, sq(w)) > 0 (x, w) R n 0 R q 0 v(x, sq(w)) < 0 (x, w, ẇ) R n 0 R q 0 D How to represent v(x, sq(w)) and v(x, sq(w))? SMR of HPD-HLFs: v(x, sq(w)) = w {s} x {m} = ξ V, w {s} V w {s} x {m} Matrix V must belong to the linear set V = V = V : ξ V, w {s} does not contain elements of w with odd power

82 Università di Siena 81 Stability analysis via HPD-HLFs (3/3) Let D 1 (V ) and D 2 (V, ẇ) be symmetric matrices such that ξ D 1 (V ), w {s+1} ξ D 2 (V, ẇ), w {s+1} = = v(x, sq(w)) x v(x, w) w A(sq(w)) x w=sq(w) q i=1 w 2 i 2 ẇ R.H.S. terms in v(x, sq(w)) Hence, v(x, sq(w)) = ξ D 1 (V ),w {s+1} + ξ D 2 (V, ẇ), w {s+1} = ξ D 1 (V ) + D 2 (V, ẇ) + N, w {s+1} where N is any symmetric matrix belonging to the linear set N = N = N : ξ N, w {s+1} = 0 x w

83 Università di Siena 82 Sufficient condition for the existence of a HPD-HLF Theorem 8 (Sufficient condition for [P1]) Let V (α) and N(β) be linear parameterizations of the sets V and N where α and β are free parameter vectors. If there exist α and β j, j = 1,...,h, such that the following set of LMIs is satisfied 0 < V (α) 0 > D 1 (V (α)) + D 2 (V (α),d j ) + N(β j ) j = 1,...,h then v(x,w) is a HPD-HLF for system Σ.

84 Università di Siena 83 Lower bound on maximum variation rate Corollary 1 (Lower bound for [P2]) Consider the generalized eigenvalue problem (GEVP) 0 < V (α) b = inf b b,α,β 0,...,β h s.t. 0 > D 1 (V (α)) + N(β 0 ) 0 > b D 1 (V (α)) + N(β 0 ) + D 2 (V (α), d j ) +N(β j ) j = 1,...,h Then, ˆγ = 1 b γ.

85 Università di Siena 84 Conservativeness of HPD-HLFs If there exist m and s such that the LMI condition for HPD-HLF is satisfied, then it is also satisfied for m and s + 1. If there exist m and s such that the LMI condition for HPD-HLF is satisfied, then it is also satisfied for am and as where a is any positive integer. There is no guarantee that conservativeness does not increase, when m grows for a constant s (different from HPLFs!)

86 Università di Siena 85 HPD-HLF Example 1 ẋ(t) = w(t) 1 x(t), 0 w(t) k, ẇ(t) γ Problem: for a fixed k, compute the maximum variation rate γ such that the system is asymptotically stable

87 Università di Siena 86 m = ˆγ s=3 15 s=2 10 s=1 s= Linearly parameter-dependent QLFs: - dash-dotted blue line: [Gahinet et al. 1996] - dashed red line: [Montagner & Peres 2003] - solid black line with s = 1: HPD-HLF and [Geromel & Colaneri 2005] k

88 Università di Siena 87 m = 2 m = s=3 20 s=3 ˆγ ˆγ 15 s=2 15 s=2 10 s=1 10 s=1 s=0 s= k k

89 Università di Siena 88 HPD-HLF Example ẋ(t) = x(t), 0 w(t) k, ẇ(t) γ 1 3 w(t) 2 Problem: for a fixed k, compute the maximum variation rate γ such that the system is asymptotically stable

90 Università di Siena m = s=3 65 ˆγ s= s=0 s= Linearly parameter-dependent QLFs: - dashed red line: [Montagner & Peres 2003] - solid black line with s = 1: HPD-HLF and [Geromel & Colaneri 2005] k

91 Università di Siena 90 HPD-HLF Example 3 Consider the system [Boyd et al 1994, Montagner and Peres 2003] ẋ(t) = w(t) x(t), 0 w(t) 1, ẇ(t) γ Problem: compute the maximum variation rate γ such that the system is asymptotically stable By using Corollary 1, for m = 2 and s = 0 one finds ˆγ = + (lower bound provided by linearly parameter-dependent QLFs 63.25).

92 Università di Siena 91 HPD-HLF Example 4 ẋ(t) = A(w(t))x(t) with w(t) P ẇ(t) D = co{d 1, d 2 } d 1 = γ[1, 0, 1], d 2 = γ[ 1,0, 1] A(w) = where 3 i=1 w i A i A i = Ā0 + kāi and

93 Università di Siena 92 Ā 0 = Ā 1 = Ā 2 = Ā 3 = Problem: for a fixed k, compute the maximum variation rate γ such that the system is asymptotically stable

94 Università di Siena 93 ˆγ Linearly parameter-dependent QLFs: - solid line: HPD-HLF with s = 1 and m = 1 - dashed red line: [Montagner & Peres 2003] - dash-dotted green line: [Geromel & Colaneri 2005] k

95 Università di Siena 94 Structure of homogeneous Lyapunov functions Which class of homogeneous Lyapunov function should be chosen? Time-varying uncertainty with no bound on the variation rate: parameter dependence is useless HPLFs (s = 0) Time-invariant uncertainty: quadratic parameter-dependent Lyapunov functions are sufficient HPD-QLFs (m = 1) Time-varying uncertainty with bounds on the variation rate: there is benefit from homogeneous dependence both in the state and in the parameters HPD-HLFs (increase both m and s)

96 Università di Siena 95 Open problems Extension to other uncertainty structures Example: polynomial dependence on the uncertain parameters (can be done with the same machinery, but size of LMIs can become very high) Establish relationships between different sufficient conditions (hierarchy?) Comparisons with similar approaches: SOS, Positivstellensatz, moments [Parrilo, Scherer, Henrion & Lasserre, Prajna...] Different uncertainty parameterizations: trade-off performance (e.g. stability margin) and LMIs size Use polynomial LFs for synthesis of robust controllers

97 Università di Siena 96 References Books 1. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge (UK), Y. Nesterov and A. Nemirovskii, Interior Point Polynomial Methods in Convex Programming, SIAM, D. Henrion and A. Garulli, Eds, Positive Polynomials in Control, Lecture Notes in Control and Information Sciences, Vol. 312, Springer, Papers 1. P. A. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Math. Prog., Series B, vol. 96, pp , G. Chesi, A. Garulli, A. Tesi and A. Vicino, Solving quadratic distance problems: an LMI based approach, IEEE Trans. on Automatic Control, vol. 48, no. 2, pp , 2003.

98 Università di Siena G. Chesi, A. Garulli, A. Tesi and A. Vicino, Homogeneous Lyapunov functions for systems with structured uncertainties, Automatica, vol. 39, no. 6, pp , G. Chesi, A. Garulli, A. Tesi, and A. Vicino, Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach, IEEE Trans. on Automatic Control, vol. 50, no. 3, pp , D. Henrion and J.-B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback, IEEE Trans. on Automatic Control, vol. 51, no. 2, pp , C. W. Scherer, LMI relaxations in robust control, European Journal of Control, Vol. 12, no.1, 2006.