Non-Conservatism of LMI Conditions for Graziano Chesi Time-Varying - - PowerPoint PPT Presentation

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 1\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems

Graziano Chesi Department of Electrical and Electronic Engineering University of Hong Kong IFAC Workshop on Uncertain Dynamical Systems Udine, Italy, August 23-26, 2011

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

◮ LMIs are a standard tool for uncertain systems

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear,

polytopic/non-polytopic, time-invariant/time-varying uncertainty

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear,

polytopic/non-polytopic, time-invariant/time-varying uncertainty

◮ ... obtains conservative bounds on worst-case

performances

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear,

polytopic/non-polytopic, time-invariant/time-varying uncertainty

◮ ... obtains conservative bounds on worst-case

performances

◮ ... can (possibly) reduce the bounds conservatism

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

Basic problem: establishing robust stability

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

Basic problem: establishing robust stability

◮ Time-invariant (TI): there exist non-conservative

LMI conditions, e.g. [Bliman SICON 2004], [Scherer EJC 2006], [Chesi, Garulli, Tesi, Vicino SPRINGER 2009] (in general asymptotically, possibly dependent on the uncertainty set)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems

Basic problem: establishing robust stability

◮ Time-invariant (TI): there exist non-conservative

LMI conditions, e.g. [Bliman SICON 2004], [Scherer EJC 2006], [Chesi, Garulli, Tesi, Vicino SPRINGER 2009] (in general asymptotically, possibly dependent on the uncertainty set)

◮ Time-varying (TV): what do we know?

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Polytopic system with linear dependence: ˙ x(t) = A(u(t))x(t), u(t) ∈ U A(u) = A0 +

i uiAi,

U = conv{u(1), u(2), . . .}

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Polytopic system with linear dependence: ˙ x(t) = A(u(t))x(t), u(t) ∈ U A(u) = A0 +

i uiAi,

U = conv{u(1), u(2), . . .}

◮ TI case: stable iff

A(u) is Hurwitz ∀u ∈ U

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Polytopic system with linear dependence: ˙ x(t) = A(u(t))x(t), u(t) ∈ U A(u) = A0 +

i uiAi,

U = conv{u(1), u(2), . . .}

◮ TI case: stable iff

A(u) is Hurwitz ∀u ∈ U

◮ TV case: stable iff

∀ε > 0 ∃δ > 0 : x(0) < δ ⇓ x(t) < ε ∀t ≥ 0 and limt→∞ x(t) = 0 ∀u(·) ∈ U

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Basic stability condition:

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Basic stability condition:

◮ Time-invariant (TI) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Basic stability condition:

◮ Time-invariant (TI) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ Time-varying (TV) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Basic stability condition:

◮ Time-invariant (TI) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ Time-varying (TV) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ In both cases the (same) condition is only sufficient

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem

Basic stability condition:

◮ Time-invariant (TI) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ Time-varying (TV) case: stable if

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ In both cases the (same) condition is only sufficient ◮ This condition is based on a common quadratic LF

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 6\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples

◮ TI case: consider A(u) = A0 + u1A1 with

A0 = −2 1 −3 1

• , A1 =

2 −2

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 6\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples

◮ TI case: consider A(u) = A0 + u1A1 with

A0 = −2 1 −3 1

• , A1 =

2 −2

• ◮ A(u) is Hurwitz for all u ∈ U = [0, 1], however

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 6\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples

◮ TI case: consider A(u) = A0 + u1A1 with

A0 = −2 1 −3 1

• , A1 =

2 −2

• ◮ A(u) is Hurwitz for all u ∈ U = [0, 1], however

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ TV case: consider A(u(t)) = A0 + u1(t)A1 with

A0 =

• 1

−2 −1

• , A1 =
• −1

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 6\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples

◮ TI case: consider A(u) = A0 + u1A1 with

A0 = −2 1 −3 1

• , A1 =

2 −2

• ◮ A(u) is Hurwitz for all u ∈ U = [0, 1], however

∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

◮ TV case: consider A(u(t)) = A0 + u1(t)A1 with

A0 =

• 1

−2 −1

• , A1 =
• −1
• ◮ the origin is asymptotically stable for all

u(t) ∈ U = [0, 4], however ∃P > 0 : PA(u) + A(u)′P < 0 ∀u ∈ ver(U)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 7\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Why Conservative?

◮ TI case: stable iff [Chesi et al. 2003]

∃P(u) > 0 : P(u)A(u) + A(u)′P(u) < 0 ∀u ∈ U deg P(u) ≤ b

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 7\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Why Conservative?

◮ TI case: stable iff [Chesi et al. 2003]

∃P(u) > 0 : P(u)A(u) + A(u)′P(u) < 0 ∀u ∈ U deg P(u) ≤ b

◮ We need a polynomially parameter-dependent

quadratic LF (of known degree)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 7\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Why Conservative?

◮ TI case: stable iff [Chesi et al. 2003]

∃P(u) > 0 : P(u)A(u) + A(u)′P(u) < 0 ∀u ∈ U deg P(u) ≤ b

◮ We need a polynomially parameter-dependent

quadratic LF (of known degree)

◮ TV case: stable iff [Blanchini and Miani 1999]

∃v(x) = Vx2p : ˙ v(x, u) < 0 ∀x = 0 ∀u ∈ ver(U)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 7\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Why Conservative?

◮ TI case: stable iff [Chesi et al. 2003]

∃P(u) > 0 : P(u)A(u) + A(u)′P(u) < 0 ∀u ∈ U deg P(u) ≤ b

◮ We need a polynomially parameter-dependent

quadratic LF (of known degree)

◮ TV case: stable iff [Blanchini and Miani 1999]

∃v(x) = Vx2p : ˙ v(x, u) < 0 ∀x = 0 ∀u ∈ ver(U)

◮ We need a polynomial LF (of unknown degree)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 8\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Problem

◮ How to search for such LFs?

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 8\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Problem

◮ How to search for such LFs? ◮ Establishing positivity of a polynomial is an NP-hard

problem

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 8\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Problem

◮ How to search for such LFs? ◮ Establishing positivity of a polynomial is an NP-hard

problem

◮ Searching for a positive polynomial that satisfies

some desired properties is even more difficult

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 9\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SMR

Square matrix representation (SMR) of a polynomial h : Rn → R of degree 2m: h(x) = x{m}′ (H + L(α)) x{m}

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 9\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SMR

Square matrix representation (SMR) of a polynomial h : Rn → R of degree 2m: h(x) = x{m}′ (H + L(α)) x{m}

◮ x{m} contains all monomials in x of degree m

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 9\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SMR

Square matrix representation (SMR) of a polynomial h : Rn → R of degree 2m: h(x) = x{m}′ (H + L(α)) x{m}

◮ x{m} contains all monomials in x of degree m ◮ H is a symmetric matrix

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 9\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SMR

Square matrix representation (SMR) of a polynomial h : Rn → R of degree 2m: h(x) = x{m}′ (H + L(α)) x{m}

◮ x{m} contains all monomials in x of degree m ◮ H is a symmetric matrix ◮ L(α) is a linear parametrization of

L =

• L = L′ : x{m}′Lx{m} = 0 ∀x

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 9\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SMR

Square matrix representation (SMR) of a polynomial h : Rn → R of degree 2m: h(x) = x{m}′ (H + L(α)) x{m}

◮ x{m} contains all monomials in x of degree m ◮ H is a symmetric matrix ◮ L(α) is a linear parametrization of

L =

• L = L′ : x{m}′Lx{m} = 0 ∀x
• ◮ SMR also known as Gram matrix method

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 10\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SOS Polynomial

◮ A polynomial h(x) is sum of squares of polynomials

(SOS) iff h(x) =

• i

hi(x)2 with hi(x) polynomial

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 10\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SOS Polynomial

◮ A polynomial h(x) is sum of squares of polynomials

(SOS) iff h(x) =

• i

hi(x)2 with hi(x) polynomial

◮ SOS polynomials are clearly nonnegative

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 10\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

SOS Polynomial

◮ A polynomial h(x) is sum of squares of polynomials

(SOS) iff h(x) =

• i

hi(x)2 with hi(x) polynomial

◮ SOS polynomials are clearly nonnegative ◮ A polynomial h(x) is SOS iff the following LMI holds

[Chesi, Genesio, Tesi, Vicino ECC 1999]: ∃α : H + L(α) ≥ 0

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 11\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Extension to Matrix Polynomials

SMR of a matrix polynomial H : Rn → Rr×r of degree 2m: H(x) = (x{m} ⊗ Ir)′ ¯ H + L(α)

• (x{m} ⊗ Ir)

◮ A matrix polynomial H(x) is SOS iff

H(x) =

• i

Hi(x)′Hi(x) with Hi(x) matrix polynomial

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 11\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Extension to Matrix Polynomials

SMR of a matrix polynomial H : Rn → Rr×r of degree 2m: H(x) = (x{m} ⊗ Ir)′ ¯ H + L(α)

• (x{m} ⊗ Ir)

◮ A matrix polynomial H(x) is SOS iff

H(x) =

• i

Hi(x)′Hi(x) with Hi(x) matrix polynomial

◮ SOS matrix polynomials are clearly positive

semidefinite for all x

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 11\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Extension to Matrix Polynomials

SMR of a matrix polynomial H : Rn → Rr×r of degree 2m: H(x) = (x{m} ⊗ Ir)′ ¯ H + L(α)

• (x{m} ⊗ Ir)

◮ A matrix polynomial H(x) is SOS iff

H(x) =

• i

Hi(x)′Hi(x) with Hi(x) matrix polynomial

◮ SOS matrix polynomials are clearly positive

semidefinite for all x

◮ A matrix polynomial H(x) is SOS iff the following

LMI holds [Chesi, Garulli, Tesi, Vicino CDC 2003]: ∃α : ¯ H + L(α) ≥ 0

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 12\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TI

◮ We look for a polynomially parameter-dependent

quadratic LF with dependence of degree m

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 12\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TI

◮ We look for a polynomially parameter-dependent

quadratic LF with dependence of degree m

◮ With w = (u2 1, u2 2, . . .), we express P(w) and

P(w)A(w) + A(w)′P(w) with the SMR of matrix polynomials

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 12\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TI

◮ We look for a polynomially parameter-dependent

quadratic LF with dependence of degree m

◮ With w = (u2 1, u2 2, . . .), we express P(w) and

P(w)A(w) + A(w)′P(w) with the SMR of matrix polynomials

◮ The origin is robustly stable if there exist α and β

satisfying the LMIs [Chesi, Garulli, Tesi, Vicino CDC 2003] < S(β) > R(β) + L(α) (1)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 13\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TV

◮ We look for a polynomial LF of degree 2m

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 13\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TV

◮ We look for a polynomial LF of degree 2m ◮ Extended matrix A# of A:

∇x{m}Ax = A#x{m}

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 13\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMI/SMR Condition for TV

◮ We look for a polynomial LF of degree 2m ◮ Extended matrix A# of A:

∇x{m}Ax = A#x{m}

◮ The origin is robustly stable if there exist V and α(i)

satisfying the LMIs [Chesi, Garulli, Tesi, Vicino CDC 2002]

• <

V > VB#

i

+ B#

i ′V + L(α(i)) ∀i

(2) where Bi = A(u(i))

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 14\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conservative or Optimal?

◮ The previous LMI/SMR conditions impose that a

(matrix) polynomial is SOS

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 14\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conservative or Optimal?

◮ The previous LMI/SMR conditions impose that a

(matrix) polynomial is SOS

◮ Unfortunately there exist nonnegative polynomials

that are not SOS (called PNS polynomials)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 14\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conservative or Optimal?

◮ The previous LMI/SMR conditions impose that a

(matrix) polynomial is SOS

◮ Unfortunately there exist nonnegative polynomials

that are not SOS (called PNS polynomials)

◮ An example [Motzkin 1967]:

hMotzkin(x) = x4

1x2 2 + x2 1x4 2 + x6 3 − 3x2 1x2 2x2 3

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 14\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conservative or Optimal?

◮ The previous LMI/SMR conditions impose that a

(matrix) polynomial is SOS

◮ Unfortunately there exist nonnegative polynomials

that are not SOS (called PNS polynomials)

◮ An example [Motzkin 1967]:

hMotzkin(x) = x4

1x2 2 + x2 1x4 2 + x6 3 − 3x2 1x2 2x2 3 ◮ Are we lost?

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 15\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Hilbert’s 17th Problem and Other

◮ Any nonnegative polynomial can be expressed as

ratio of two SOS polynomials [Artin 1927]

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 15\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Hilbert’s 17th Problem and Other

◮ Any nonnegative polynomial can be expressed as

ratio of two SOS polynomials [Artin 1927]

◮ A homogeneous polynomial h(x) is positive on the

simplex if and only if there exists an integer k such that all coefficients of h(x)(x1 + . . . + xn)k are positive [Polya 1974]

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 15\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Hilbert’s 17th Problem and Other

◮ Any nonnegative polynomial can be expressed as

ratio of two SOS polynomials [Artin 1927]

◮ A homogeneous polynomial h(x) is positive on the

simplex if and only if there exists an integer k such that all coefficients of h(x)(x1 + . . . + xn)k are positive [Polya 1974]

◮ Any PNS polynomial is the vertex of a cone of PNS

polynomials [Chesi TAC 2007]

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 16\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TI case

◮ Let m be the degree of the parameter-dependence of

the quadratic LF

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 16\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TI case

◮ Let m be the degree of the parameter-dependence of

the quadratic LF

◮ The origin is robustly stable if and only if there exists

a sufficiently large m such that the LMI/SMR condition (1) holds [Chesi AUT 2008]

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 17\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

More difficult!

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 17\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

More difficult!

◮ In fact, let v(x) be a SOS LF (which exists from

[Blanchini and Miani 1999])

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 17\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

More difficult!

◮ In fact, let v(x) be a SOS LF (which exists from

[Blanchini and Miani 1999])

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with w(x)v(x) where w(x) is SOS

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 17\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

More difficult!

◮ In fact, let v(x) be a SOS LF (which exists from

[Blanchini and Miani 1999])

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with w(x)v(x) where w(x) is SOS

◮ however, this provides −w(x)˙

v(x) − v(x) ˙ w(x), which could not even be pd

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 17\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

More difficult!

◮ In fact, let v(x) be a SOS LF (which exists from

[Blanchini and Miani 1999])

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with w(x)v(x) where w(x) is SOS

◮ however, this provides −w(x)˙

v(x) − v(x) ˙ w(x), which could not even be pd

◮ in fact, w(x)v(x) could not be a LF!

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 18\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Other possibilities?

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 18\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Other possibilities?

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with v(x)k where k is a positive integer

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 18\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Other possibilities?

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with v(x)k where k is a positive integer

◮ this provides −kv(x)k−1 ˙

v(x) which is pd (clearly, v(x)k is a LF)

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 18\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Other possibilities?

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with v(x)k where k is a positive integer

◮ this provides −kv(x)k−1 ˙

v(x) which is pd (clearly, v(x)k is a LF)

◮ is −kv(x)k−1 ˙

v(x) also SOS?

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 18\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Other possibilities?

◮ if −˙

v(x) is not SOS, then one can think of replacing v(x) with v(x)k where k is a positive integer

◮ this provides −kv(x)k−1 ˙

v(x) which is pd (clearly, v(x)k is a LF)

◮ is −kv(x)k−1 ˙

v(x) also SOS?

◮ we also need that v(x)k and −kv(x)k−1 ˙

v(x) have pd SMR matrices...

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result: Sketch of the proof:

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result:

◮ Let 2m be the degree of the LF

Sketch of the proof:

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result:

◮ Let 2m be the degree of the LF ◮ The origin is robustly stable if and only if there exists

a sufficiently large m such that the LMI/SMR condition (2) holds [Chesi AUT 2011] Sketch of the proof:

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result:

◮ Let 2m be the degree of the LF ◮ The origin is robustly stable if and only if there exists

a sufficiently large m such that the LMI/SMR condition (2) holds [Chesi AUT 2011] Sketch of the proof:

◮ if f (x) is pd, then for all SOS pd g(x) with deg(g)

dividing deg(f ) there exists k such that g(x)kf (x) is SOS [Scheiderer 2009]

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result:

◮ Let 2m be the degree of the LF ◮ The origin is robustly stable if and only if there exists

a sufficiently large m such that the LMI/SMR condition (2) holds [Chesi AUT 2011] Sketch of the proof:

◮ if f (x) is pd, then for all SOS pd g(x) with deg(g)

dividing deg(f ) there exists k such that g(x)kf (x) is SOS [Scheiderer 2009]

◮ v(x) = Vx2p 2p admits a pd SMR matrix

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 19\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Non-conservatism: TV case

Final result:

◮ Let 2m be the degree of the LF ◮ The origin is robustly stable if and only if there exists

a sufficiently large m such that the LMI/SMR condition (2) holds [Chesi AUT 2011] Sketch of the proof:

◮ if f (x) is pd, then for all SOS pd g(x) with deg(g)

dividing deg(f ) there exists k such that g(x)kf (x) is SOS [Scheiderer 2009]

◮ v(x) = Vx2p 2p admits a pd SMR matrix ◮ −kv(x)k−1 ˙

v(x) admits a pd SMR matrix

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 20\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conclusion

◮ LMIs are a standard tool for uncertain systems

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 20\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conclusion

◮ LMIs are a standard tool for uncertain systems ◮ By exploiting the SMR, LMIs impose that (matrix)

polynomials are SOS

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 20\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conclusion

◮ LMIs are a standard tool for uncertain systems ◮ By exploiting the SMR, LMIs impose that (matrix)

polynomials are SOS

◮ However, there exists a gap between nonnegative

polynomials and SOS

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 20\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conclusion

◮ LMIs are a standard tool for uncertain systems ◮ By exploiting the SMR, LMIs impose that (matrix)

polynomials are SOS

◮ However, there exists a gap between nonnegative

polynomials and SOS

◮ This means that LMI/SMR conditions may fail to

recognize a LF

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 20\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Conclusion

◮ LMIs are a standard tool for uncertain systems ◮ By exploiting the SMR, LMIs impose that (matrix)

polynomials are SOS

◮ However, there exists a gap between nonnegative

polynomials and SOS

◮ This means that LMI/SMR conditions may fail to

recognize a LF

◮ Nevertheless, non-conservatism with LMI/SMR

conditions is obtained by using LFs with larger degree than the minimum required to prove stability

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 21\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Key References

• G. Chesi, A. Garulli, A. Tesi, A. Vicino

Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems Springer, 2009

• G. Chesi, “LMI Techniques for Optimization Over

Polynomials in Control: a Survey,” IEEE Trans. on Automatic Control, 2010

• G. Chesi

Domain of Attraction: Analysis and Control via SOS Programming Springer, 2011

Thank you!

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 22\22 Graziano Chesi www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Software: SMRSOFT

• G. Chesi

SMRSOFT: a Matlab toolbox for solving basic

• ptimization problems over polynomials and studying

dynamical systems via SOS programming Available from 1 Sep 2011 at http://www.eee.hku.hk/˜chesi Currently SMRSOFT allows one to:

◮ investigate positivity/non-positivity of

polynomials/matrix polynomials;

◮ determine the minimum of rational functions; ◮ solve systems of polynomial equations.

More functions coming soon!