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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

A Symbolic Approach for Solving Algebraic Riccati Equations

• G. Rance, Y. Bouzidi, Al. Quadrat, Ar. Quadrat

Journées Nationales de Calcul Formel Monday, January 22th 2018

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

The linear optimal control problem

Input : a linear dynamical system

• ˙

x(t) = A x(t) + B u(t) x(0) = x0 where x(t) ∈ Rn the state vector, u(t) ∈ Rm the control vector A (resp. B) is an n × n (resp. n × m) real matrix Output : a control u that stabilizes the system and minimizes a quadratic cost functional 1 2

+∞

[x(t)T Q x(t) + u(t)T R u(t)]d t where Q (resp. R) is a positive semi-definite (resp. positive definite) symmetric real matrix. Goal : Achieve a control reference using the minimum energy

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Optimal control : mathematical simplifications

Let introduce the Lagrange multiplier λ = (λ1, . . . , λn) and the following functional 1 2

• +∞

[x(t)T Q x(t) + u(t)T R u(t)−λ(t)(˙ x(t) − A x(t) − B u(t))]d t By a variation computation, the problem is reduced to solving the following OD systems

• ˙

λ(t)T + AT λ(t)T + Q x(t) = 0, ˙ x(t) − A x(t) − B u(t) = 0, R u(t) + BT λ(t)T = 0.

u = −R−1 BT λ(t)T

− − − − − − − − − − →

• ˙

x(t) = A x(t) − B R−1 BT λ(t)T , ˙ λ(t)T = −Q x(t) − AT λ(t)T . If we seek for a solution of the form λ(t)T = P(t) x(t), P(t) must satisfy the differential equation ˙ P = A P + AT P + P B R−1 BT PT + Q If we consider a constant matrix P, this yields the following algebraic equation A P + AT P + P B R−1 BT PT + Q = 0 The optimal control is then given as u(t) = −R−1 BT P x(t)

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Algebraic Riccati Equations

An Algebraic Riccati Equation is the following quadratic matrix equation AT X + X A + X B R−1 BT X + Q = 0 (1) where A is a real n × n matrix and Q, R are real symmetric n × n matrices Solving Algebraic Riccati Equations

• Computing all the solutions X of (1)
• Computing specific solutions of (1) : real, hermitian, positive definite...
• A positive definite solution is stabilizing

Algebraic Riccati Equations are fundamental in many linear control theory problems (Estimation, Filtering, Robust control,...)

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Riccati Equations and invariant subspaces

Solutions of (1) can be constructed in term of the invariant subspaces of the following 2n × 2n Hamiltonian matrix H :=

A

−B R−1 BT −Q −AT

• Theorem [Zhou et al. (1996)]

Let V ⊂ C2n be an n-dimensional invariant subspace of H and let X1, X2 ∈ Cn×n be two complex matrices such that V = Im

X1

X2

• If X1 is invertible, then X := X2 X −1

1

is a solution of the Riccati Equation (1). Invariant subspaces can be obtained via eigenvalues and eigenvectors computation

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

The spectral factorization problem

The spectrum of H is symmetric with respect to the real and imaginary axis If we consider the characteristic polynomial of H f (λ) = det(H − λ I2n) Then f (λ) = f (−λ) Invariant subspaces can be obtained by computing factorizations of the form f (λ) = g(λ) g(−λ) where g(λ) ∈ C[λ] This problem is known as the spectral factorization problem

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

The problem under consideration

nth order Single Input (u) Single Output (y) systems

• ˙

x = A x + B u y = C x

A :=

      

1 . . . 1 ... . . . . . . . . . ... ... . . . 1 −a0 −a1 . . . −an−2 −an−1

      

B := (0 . . . 0 1)T C := (c0 . . . cn−1)

where a := (a0, · · · , an−1), c := (c0, · · · , cn−1) are unknown parameters. Goal : Compute a closed loop control u that stabilizes y and minimizes 1 2

+∞

[y(t)2 + u(t)2]dt This control will depend on the parameters a,c observe the effect of parameters on the optimization problem !

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

The problem under consideration

This yields the following Algebraic Riccati Equation R := X A + AT X − X B BT X + CT C = 0 (2) where X is a symmetric matrix Theorem [Zhou et al. (1996)] If the pair (A, C) is observable, then The positive definite solution X of (2) is unique The positive definite solution X of (2) is a stabilizing solution Goal : Compute the positive definite solution of (2)

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Algebraic description

R = 0 ⇔ n(n+1)

2

polynomial equations of n(n+1)

2

unknowns Noting X = (xi,j) then n(n−1)

2

elements of R yields xi,j = xi−1,j+1 + f (ak, ck, xk,n|k = 1 · · · n) Recursion → xi,j = f (ak, ck, xk,n|k = 1 · · · n) Two halting conditions :

• Strictly above the anti-diagonal → First row
• Below the anti-diagonal → Last column

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Algebraic description

For k = 1 . . . n, we set xk,0 = x0,k := 0, and for (i, j) ∈ N2, we define :

N(i, j) := i − 1,

2 ≤ i + j ≤ n + 1 (stly. above anti-diag.) N(i, j) := n − j + 1, n + 1 < i + j ≤ 2 n + 1 (below anti-diag.) The elements of X solution of R = 0 are determined only by the bk’s xk,n = bk−1 − ak−1 (last column of X) xi,j−1 =

N(i,j)

• k=0

(−1)k bi−1−k bj−1+k − θN(i,j) where 1 ≤ k ≤ n, 1 ≤ i < j ≤ n, and θm is defined by : θm :=

m

• k=0

(−1)k ai−1−k aj−1+k + ci−1−k cj−1+k

• The number of variables is now equal to n

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

A new polynomial system

Polynomial system of n equations in bk B :=

    

B0 := b2

0 − d0 = 0,

Bk := b2

k + 2 M(k)

• m=1

(−1)mbk−m bk+m − d2k = 0, 1 ≤ k ≤ n − 1 where the constants d2k are defined by

        

d0 := a2

0 + c2

d2k := 2

M(k)

• m=1

(−1)m (ak−m ak+m + ck−m ck+m) + a2

k + c2 k ,

d2n := 1 Theorem - [Rance et al. (2016)] The polynomial system B = {B0, · · · , Bn−1}

• is a reduced Gröbner basis of the ideal B w.r.t. the DRL order bn−1 ≻ . . . ≻ b0
• has generically 2n distinct complex solutions

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Relation with the spectral factorization

Find the solutions via invariant spaces of the Hamiltonian H :=

• A

−B BT −CT C −AT

• ∈ Q(a0, . . . , an−1, c0, . . . , cn−1)2n×2n.

f (λ) is the characteristic polynomial of H f (λ) = (−1)n

n

• k=0

n

• l=0

(−1)k(cl ck + al ak)λl+k. Theorem -[Rance et al. (2016)] Let f (λ) = g(λ) g(−λ) be a factorization of f , where g(λ) :=

n

• k=0

bk λk. The equations that stems from the equality are those in B Theorem - [Kanno et al. (2009)] X > 0 ⇔ σ := max{bn−1 ∈ R | solution of B}

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Some interesting properties

Theorem - [Rance et al. (2016)] The polynomial ideal generated by B is in shape position with respect to any variable bn−k where k is odd. Our proof is based on the spectral factorization formulation Moreover, the system has certain symmetries that should be identified (Ongoing work)

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Parametrization of the solutions of B

Next step : parametrize the solutions of B

• Generically, the system B can be written in the following form :

      

P(bn−1) = 0, bn−2 = fn−2(bn−1), · · · b0 = f0(bn−1), Solving computing the roots of P(bn−1) + substitution in the fi Study the maximum real root of P(bn−1) with respect to the parameters Some theoretical and practical barriers

• Size of expressions grows exponentially

→ limited to low order systems

• Degrees of polynomials grows exponentially

→ No closed-form solutions for high order systems ⇒ Interest in small order systems

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Small order systems - n = 3

B :=

  

B0 := b2

0 − d0 = 0

⇒ b0 completely determined B1 := b2

1 − 2 b0 b2−d2 = 0

B2 := b2

2 − 2 b1−d4 = 0

⇒ b1 = 1

2 (b2 2 − d4)

B1 ⇒ univariate polynomial P in b2 P(b2) := b4

2 − 2 d4 b2 2 − 8 b0 b2 + d2 4 − 4 d2 = 0

Its roots are b2(ε) = ε1 1

2

√ 2 u + ε2 1

2

√∆2 with

  

ε1 := ±1, ε2 := ±1, ε := (ε1, ε2), p2 := 4 d2 − 4

3 d2 4 ,

q2 := 8

3 d2 d4 − 16 27 d3 4 − 8 b2 0,

            

α :=

• −27 q2+

27(4 p3

2+27 q2 2)

2

1/3

u := 1

3

• α − 3 p2

α + 2 d4

• ,

∆2 := 2

• 2 d4 + ε1

8 b0 √ 2 u − u

• .

Determine X > 0 : which root is the greatest ?

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Discriminants of univariate polynomials

Choosing b2,max ⇔ Computing the discriminant of P Discriminant of a quadratic polynomial Let P(x) = x2 + a x + b, x ∈ R, (a, b) ∈ R2. The discriminant of P is defined by : ∆(a, b) = a2 − 4 b. Roots of P : x1,2 = −a ±

• ∆(a, b).

⇒ When ∆(a, b) = 0, x1 and x2 are crossing !

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Using the Discriminants of univariate polynomials

Exemple of a second order polynomial P(x) = x2 + a x + b = 0 discx(P) = a2 − 4 b x1,2(a, b) = −a ±

• discx(P)

Red cell : x1,2(0, 1) = ±2 i ∈ C Blue cell : x1,2(0, −1) = ±2 ∈ R Discriminant of P(b2) := b4

2 − 2 d4 b2 2 − 8 b0 b2 + d2 4 − 4 d2

We apply the same reasoning to P(b2) to prove that σ = 1 2 √ 2 u + 1 2

• ∆2

is the maximal real root of P for any values of the parameters.

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

A practical example

Two-mass-spring system : c := k m1 m2 a2 := m1 + m2 m1 m2 k G := y1 e1 = c0 s2(s2 + a2) X :=

   

b0 b1 b0 b2 b0 b3 b0 b0 b2 b1 b2 − b0 b3 b1 b3 − b0 b1 b0 b3 b1 b3 − b0 b2 b3 − b1 b2 − a2 b0 b1 b2 − a2 b3

   

B ⇒

      

B0 := b2

0 − c2 0 = 0

B1 := b2

1 − 2 b0 b2 = 0

B2 := b2

2 − 2 b1 b3 + 2 b0 − a2 2 = 0

B3 := b2

3 − 2 b2 + 2 a2 = 0 JNCF 2018 Yacine Bouzidi A Symbolic Approach for Solving Algebraic Riccati Equations 23 / 27

Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

A practical example

• A parametrization of B is easily found :

          

b0 = c0 b1 = b4

3 + 4 a2 b2 3 + 8 c0

8 b3 b2 = 1

2 b2 3 + a2

P(b3) := b8

3 + 8 a2 b6 3 + 16 (a2 2 − 3 c0) b4 3 − 64 a2 c0 b2 3 + 64 c2 0 = 0

• P is of degree 4 ⇒ symbolic
• Positive definite solution is given by X(σ) where :

σ := √ 2

• 2 c0 − a2
• +
• 2 c0 − a2

2

+ 2 c0

• In this case : X(c0, a2)

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Overview

1 Algebraic Riccati Equations for the optimal control problem 2 A new algebraic description 3 The case of 3 order systems 4 A practical example 5 Conclusion and perspectives

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Conclusion and perspectives

• Contributions :
• Symbolic techniques in automatic control problems to handle parameters
• Closed form control with respect to the parameters
• Also used for H∞ control
• Ongoing work :
• Study the symmetries of the systems that stem from the Riccati Equations
• Extension to higher order systems work with implicit equations
• Extension to MIMO systems

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

Thank you for your attention

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

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Algebraic Riccati Equations for the optimal control problem A new algebraic description The case of 3 order systems A practical example Conclusion and persp

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