Advances and Challenges in Linear Control Systems, Rome, Italy

Periodic control challenges Computational detours New computational paradigms Algorithms & Open problems 

Computational issues for linear periodic 

systems - paradigms, algorithms, open 

problems 

Andreas Varga 

German Aerospace Center (DLR) 

Symposium on Advances and Challenges in Linear Control 

Systems – Rome, Italy, March 19, 2012 

On the occasion of 70th birthday of Osvaldo Maria Grasselli


Outline 

periodic control challenges 

computational detours 

new computational paradigms 

algorithms & open computational problems


Satellite attitude control 

Linearized LEO satellite dynamics with magnetic actuation 

ẋ(t) = Ax(t) + B(t)u(t) 

y(t) = Cx(t) 

x(t) ∈ R 4 , u(t) ∈ R 2 

y(t) ∈ R 2 

• A has only imaginary eigenvalues, B(t) is T -periodic matrix




ẋ(t) = Ax(t) + B(t)u(t) 

y(t) = Cx(t) 

x(t) ∈ R 4 , u(t) ∈ R 2 

y(t) ∈ R 2 

• A has only imaginary eigenvalues, B(t) is T -periodic matrix 

Discrete-time periodic output feedback stabilization 

For a sampling period ∆, stabilize the discretized 

N = T /∆–periodic system 

x((k + 1)∆) = A k x(k∆) + B k u(k∆) 

y(k∆) = Cx(k∆) 

with A k = exp(A∆), B k = ∫ (k+1)∆ 

k∆ 

e [A(k+1)∆−τ] B(τ)dτ, using 

the periodic output feedback u(k∆) = F k y(k∆) .




ẋ(t) = Ax(t) + B(t)u(t) 

y(t) = Cx(t) 

x(t) ∈ R 4 , u(t) ∈ R 2 

y(t) ∈ R 2 

• A has only imaginary eigenvalues, B(t) is T -periodic matrix 

Discrete-time periodic output feedback stabilization 

For a sampling period ∆, stabilize the discretized 

N = T /∆–periodic system 

x((k + 1)∆) = A k x(k∆) + B k u(k∆) 

y(k∆) = Cx(k∆) 

with A k = exp(A∆), B k = ∫ (k+1)∆ 

k∆ 

e [A(k+1)∆−τ] B(τ)dτ, using 

the periodic output feedback u(k∆) = F k y(k∆) . 

Challenge 

T =5400 s, ∆=10 s ⇒ very large period N =540!


Wind turbine disturbance attenuation 

Closed-loop linear FE wind turbine blade control system 

ẋ(t) = A(t)x(t) + E(t)d(t) 

x(t) ∈ R n , 

y(t) = C(t)x(t) + F(t)d(t) 

• A(t) is T -periodic; only A(i∆), i = 0, . . . , K available 

Typical values: n = 1200, T = 6 s, ∆=0.2 s, K = 30




ẋ(t) = A(t)x(t) + E(t)d(t) 

x(t) ∈ R n , 

y(t) = C(t)x(t) + F(t)d(t) 


Typical values: n = 1200, T = 6 s, ∆=0.2 s, K = 30 

Floquet stability analysis 

Compute the eigenvalues of the state-transition matrix Φ A (T , 0) 

and check their stability.




ẋ(t) = A(t)x(t) + E(t)d(t) 

x(t) ∈ R n , 

y(t) = C(t)x(t) + F(t)d(t) 


Typical values: n = 1200, T = 6 s, ∆=0.2 s, K = 30 

Floquet stability analysis 

Compute the eigenvalues of the state-transition matrix Φ A (T , 0) 

and check their stability. 

Challenge 

Many eigenvalues have nearly unit magnitudes ⇒ accurate 

eigenvalue computation difficult because of very large state 

dimension n =1200!


Vibration and noise reduction in helicopter forward flight 

Linearized helicopter model in forward flight 

ẋ(t) = A(t)x(t) + B(t)u(t) + E(t)d(t) 

x(t) ∈ R n , 

y(t) = C(t)x(t) + D(t)u(t) + F(t)d(t) 

• A(t), B(t), E(t), C(t), D(t), F(t) are T -periodic; 

• only A(i∆), B(i∆), C(i∆), D(i∆), . . . i = 0, . . . , K available 

Typical values: n = 56, T = 0.15 s, ∆=0.003 s, K = 50





x(t) ∈ R n , 




Typical values: n = 56, T = 0.15 s, ∆=0.003 s, K = 50 

Individual blade control for noise and vibration reduction 

Use a periodic controller based on the internal model principle 

parameterized by constant gains





x(t) ∈ R n , 




Typical values: n = 56, T = 0.15 s, ∆=0.003 s, K = 50 

Individual blade control for noise and vibration reduction 

Use a periodic controller based on the internal model principle 

parameterized by constant gains 

Challenge 

High computational effort required when employing an 

optimization-based tuning procedure for closed-loop 

stabilization: each function evaluation involves solving a pair of 

periodic Lyapunov/Sylvester equations


Multi-rate sampled system analysis and design 

Multirate sampling of continuous-time systems 

ẋ(t) = Ax(t) + Bu(t) 

y(t) = Cx(t) + Du(t) 

For given ∆, sample input u j with sampling period k j ∆ and 

output y i with sampling period l i ∆ (k j , l i integers).




ẋ(t) = Ax(t) + Bu(t) 

y(t) = Cx(t) + Du(t) 


output y i with sampling period l i ∆ (k j , l i integers). 

Minimal discretized periodic system 

x k+1 = A k x k + B k u k 

y k = C k x k + D k u k 

A k ∈ IR n k+1×n k 

, B k ∈ IR n } 

k+1×m 

C k ∈ IR p×n k 

, D k ∈ IR p×m – N-periodic matrices




ẋ(t) = Ax(t) + Bu(t) 

y(t) = Cx(t) + Du(t) 


output y i with sampling period l i ∆ (k j , l i integers). 

Minimal discretized periodic system 

x k+1 = A k x k + B k u k 


A k ∈ IR n k+1×n k 

, B k ∈ IR n } 

k+1×m 

C k ∈ IR p×n k 

, D k ∈ IR p×m – N-periodic matrices 

Challenge 

Time-varying state dimensions!


Computational detours 

Solving LTP computational problems using LTI techniques 

1 Form a LTI lifted system representation 

2 Apply computational methods for LTI systems 

3 Recover the LTP representation of the solution 

LTI – linear time-invariant 

LTP – linear time-periodic


Computational detours 

Solving LTP computational problems using LTI techniques 

1 Form a LTI lifted system representation 

2 Apply computational methods for LTI systems 

3 Recover the LTP representation of the solution 

LTI – linear time-invariant 

LTP – linear time-periodic 

Caveat 

Using lifting based computational techniques is a bad practice: 

worsened conditioning, large dimensions, numerical instability 

⇒ Working on original data must be always preferred!


D1. Standard discrete-time lifting (1) 

Standard periodic system: 

x k+1 = A k x k + B k u k 


Lifted signals: (Meyer & Burrus, 1975) 

Lifted system: 

uk L(h) = [uT (k + hN) · · · u T (k + hN + N − 1)] T , 

yk L(h) = [y T (k + hN) · · · y T (k + hN + N − 1)] T , 

xk L (h) = x(k + hN) 

x L k (h + 1) = F L k x L k (h) + GL k uL k (h) 

y L k (h) = HL k x L k (h) + LL k uL k (h)



Transition matrix: Φ A (j, i) := A j−1 · · · A i+1 A i , 

Lifted system matrices: 

Fk L = Φ A (k + N, k) 

Gk L = [ ⎡Φ A (k + N, k + 1)B k · · · B k+N−1 ⎤ ] 

H L k 

= 

L L k 

= 

⎢ 

⎣ 

⎡ 

⎢ 

⎣ 

C k 

. 

C k+N−1 Φ A (k + N − 1, k) 

⎤ 

D k 0 · · · 0 

∗ D k · · · 0 

. 

. . .. 

⎥ 

. ⎦ 

∗ ∗ · · · D k 

Φ A (i, i) := I ni 

⎥ 

⎦



Transition matrix: Φ A (j, i) := A j−1 · · · A i+1 A i , 


Fk L = Φ A (k + N, k) 

Gk L = [ ⎡Φ A (k + N, k + 1)B k · · · B k+N−1 ⎤ ] 

H L k 

= 

L L k 

= 

Disadvantages 

⎢ 

⎣ 

⎡ 

⎢ 

⎣ 

C k 

. 

C k+N−1 Φ A (k + N − 1, k) 

⎤ 

D k 0 · · · 0 

∗ D k · · · 0 

. 

. . .. 

⎥ 

. ⎦ 

∗ ∗ · · · D k 

– worsened problem conditioning 

– reliable computations not possible 

Φ A (i, i) := I ni 

⎥ 

⎦


D2. Stacked discrete-time lifting (1) 

Descriptor periodic system: 

E k x k+1 = A k x k + B k u k 


E k ∈ R ν k ×n k+1 

, A k ∈ R ν k ×n k 

, B k ∈ R ν k ×m k 

, C k ∈ R p k ×n k 

, 

D k ∈ R p k ×m k 

– N-periodic and ∑ N 

k=1 ν k = ∑ N 

k=1 n k 

Lifted signals: (Grasselli, 1991) 

Lifted system: 

u L k (h) = [uT (k + hN) · · · u T (k + hN + N − 1)] T , 

y L k (h) = [y T (k + hN) · · · y T (k + hN + N − 1)] T , 

x S k (h) = [x T (k + hN) · · · x T (k + hN + N − 1)] T 

Ek Sx k S(h + 1) = F k Sx k S(h) + GS k uL k (h) 

yk L(h) = HS k x k S(h) + LS k uL k (h)




⎡ 

⎤ 

A k −E k O · · · O 

. O .. . .. . .. . 

Fk 

S − λE k S = . . .. . .. . .. O 

⎢ 

. 

⎣ O .. ⎥ 

Ak+N−2 −E k+N−2 

⎦ 

−zE k+N−1 O · · · O A k+N−1 

G S k = diag {B k, . . . , B k+N−1 } 

H S k = diag {C k, . . . , C k+N−1 } 

L S k = diag {D k, . . . , D k+N−1 }




⎡ 

⎤ 

A k −E k O · · · O 

. O .. . .. . .. . 

Fk 

S − λE k S = . . .. . .. . .. O 

⎢ 

. 

⎣ O .. ⎥ 

Ak+N−2 −E k+N−2 

⎦ 

−zE k+N−1 O · · · O A k+N−1 

Disadvantages 

G S k = diag {B k, . . . , B k+N−1 } 

H S k = diag {C k, . . . , C k+N−1 } 

L S k = diag {D k, . . . , D k+N−1 } 

– excessive computational effort 

– structure not exploited and not preserved 

– delicate final step (periodic descriptor realization)


D3. Computing periodic solutions of matrix differential equations 

Periodic Riccati differential equation 

Ẋ(t) = A(t)X(t)+X(t)A T (t)+R(t)−X(t)Q(t)X(t) 

Periodic Lyapunov differential equation 

Ẋ(t) = A(t)X(t) + X(t)A T (t) + C(t) 

Periodic Sylvester differential equation 

Ẋ(t) = A(t)X(t) + X(t)B(t) + C(t) 

A(t), B(t), C(t), R(t), Q(t) – T -periodic matrices (given) 

X(t) – T -periodic (to be computed)


D3. Periodic generator method 

Fact: For the T -periodic solution X(t), X(0) fulfills appropriate 

standard algebraic Riccati, Lyapunov or Sylvester equations! 

Approach: 

1 Compute X(0) by solving the algebraic matrix equation 

using standard techniques 

2 Integrate the differential equation to compute the solution 

on [0, T ].


D3. Periodic generator method 

Fact: For the T -periodic solution X(t), X(0) fulfills appropriate 

standard algebraic Riccati, Lyapunov or Sylvester equations! 

Approach: 

1 Compute X(0) by solving the algebraic matrix equation 

using standard techniques 

2 Integrate the differential equation to compute the solution 

on [0, T ]. 

Expected difficulties 

– unstable A(t) leads to unstable integration of ODEs 

– severe accuracy losses expected for large T


D3. A challenging example (Colaneri) 

Computation of characteristic exponents 

[ 

] 

0 1 

A(t) = 

, with α(t) = 15 + 5 sin t and T = 2π. 

−2 ˙α(t) 6 − 2α(t) 

[ ] 

1 0 

With P(t)= 

, perform the Lyapunov transformation 

6−2 α (t) 1 

[ ] 

Ã(t) := P −1 (t)A(t)P(t) − P −1 (t)Ṗ(t) = 6 − 2α(t) 1 

0 0 

⇒ A(t) has characteristic exponents: µ 1 = 0 and µ 2 = −24. 

Using numerical integration with reltol = 10 −10 to determine 

Φ A (T , 0), we get for µ = 1 T log λ(Φ A(T , 0)): 

µ 1 = −3.9 · 10 −7 , µ 2 = −2.05... ⇒ No accurate digits in µ 2 !!!


Satisfactory algorithms 

Main requirements 

– general (e.g., time-varying dimensions) 

– numerically stable 

– efficient ⇔ complexity O(Nn 3 )






– efficient ⇔ complexity O(Nn 3 ) 

Structure exploiting "fast" algorithms 

– exploit zero structure of lifted representations 

– kind of (weak) numerical stability can be often proven 

– lowest computational complexity






– efficient ⇔ complexity O(Nn 3 ) 

Structure exploiting "fast" algorithms 

– exploit zero structure of lifted representations 

– kind of (weak) numerical stability can be often proven 

– lowest computational complexity 

Structure preserving algorithms 

– preserve cyclic structure of lifted representations 

– structurally (strong) backward stable algorithms 

– less efficient than "fast" methods


New computational paradigms 

P1. Manipulation of matrix products without forming them 

P2. Reduction of large structured pencils without forming them 

and exploiting structure ⇒ structure exploiting (fast) 

algorithms 

P3. Reduction of large structured pencils without forming them 

and preserving their structure ⇒ structure preserving 

(strong numerically stable) algorithms 

P4. Employing multiple shooting methods instead periodic 

generator based approaches


P1. Computation of eigenvalues of periodic systems 

Problem: Given an N-periodic matrix A k ∈ R n×n , compute the 

eigenvalues of 

Φ A (N + 1, 1) := A N · · · A 2 A 1 

Periodic real Schur form (PRSF): For given N-periodic A k 

there exists orthogonal N-periodic Q k s.t. 

Ã k = Q T k+1 A kQ k , 

k = 1, . . . , N 

is in PRSF (Ã1 – quasi-upper triangular, Ã2, . . ., ÃN upper 

triangular) ⇒ 

Q T 1 Φ A(N + 1, 1)Q 1 = ΦÃ(N + 1, 1) = ÃN · · · Ã2Ã1 

is in a real Schur form!


P1. Example (cont.) 

Alternative approach 

For N > 1, define t i = iT /N for i = 1, . . . , N. Compute the 

eigenvalues of Φ(T , 0) via the periodic real Schur form of 

Φ(T , 0) = Φ(t N , t N−1 ) · · · Φ(t 2 , t 1 )Φ(t 1 , 0) 

Results: N |µ 1 | µ 2 

1 3.9 · 10 −7 -2.05... 

2 2.6 · 10 −7 complex !! 

5 4.5 · 10 −7 -11.2653 

10 2.1 · 10 −9 -19.7407 

25 7.7 · 10 −12 -23.9921 

50 2.4 · 10 −15 -23.9982 

100 2.7 · 10 −15 -23.99995 

200 3.2 · 10 −15 -23.9999993 

500 1.9 · 10 −14 -23.999999998 

10-digits accuracy possible by exploiting structure!


P2. Zeros-based analysis of periodic descriptor systems 

N-periodic descriptor system: 

E k x k+1 = A k x k + B k u k 


, x k ∈ R n k 

Poles: zeros of S(z) := Fk 

S − zE k 

S 

[ F 

S 

Zeros: zeros of S(z) := k 

− zEk S Gk 

S 

Reachability/Stabilizability: 

input decoupling zeros ⇔ zeros of S(z) := [ Fk S − zE k S GS k ] 

Observability/Detectability: 

[ F 

S 

output decoupling zeros ⇔ zeros of S(z) := k 

− zEk 

S ] 

H S k 

L S k 

] 

H S k


P2. Fast structure exploiting algorithm for zeros computation 

Computational approach: Determine orthogonal Q and Z 

such that 

[ ] 

R ∗ 

QS(z)Z = 

0 ˜F − zẼ 

where: 

R has full row rank and size O (N max{n k }) 

˜F − zẼ has the same finite/infinite zeros as S(z) and size O (n k) 

Properties: 

orthogonal block row compressions ⇒ certain kind of 

weak numerical stability can be proven 

efficient computations ⇔ fast algorithm 

zero-nonzero structure of S(z) destroyed


P3. Structure preserving analysis 

N-periodic pairs: (S k , T k ) := 

([ ] 

Ak B k 

, 

C k D k 

[ 

Ek O 

]) 

O O 

Alternative system pencil: 

⎡ 

⎤ 

S k −T k O · · · O 

O S k+1 −T k+1 · · · O 

˜S k (z) := 

. 

⎢ . .. . .. . .. . 

⎥ 

⎣ O S k+N−2 −T k + N −2 ⎦ 

−zT k+N−1 O · · · O S k+N−1 

[ F 

S 

Fact: k 

− zEk S Gk 

S 

structure. 

H S k 

L S k 

] 

and ˜S k (z) have the same Kronecker


P3. Structure preserving reduction to periodic Kronecker-like form 

For the N-periodic pair (S k , T k ), determine orthogonal 

N-periodic Q k and Z k such that 

⎡ 

B r ⎤ 

⎡ 

k Ar k 

∗ ∗ ∗ 

O E O O A ∞ k r ∗ ∗ ∗ 

k 

∗ ∗ 

Q k S k Z k = 

⎢ O O O A f k 

∗ 

⎥ 

⎣ O O O O A l ⎦ , Q O O Ek ∞ ∗ ∗ 

kT k Z k+1 = 

⎢ O O O Ek f ∗ 

⎣ 

k 

O O O O E l 

O O O O Ck 

l k 

O O O O O 

(a) E r k invertible and ( (E r k )−1 A r k , (E r k )−1 B r k) 

completely 

reachable ⇒ right Kronecker structure 

(b) E l k invertible and ( C l k , (E l k )−1 A l k) 

completely observable 

⇒ left Kronecker structure 

(c) A ∞ k 

invertible and (A ∞ k )−1 Ek ∞ . . . (A∞ k+N−1 )−1 Ek+N−1 

∞ 

nilpotent ⇒ (A ∞ k , E k 

∞ ) contains the infinite structure 

(d) Ek f invertible ⇒ (Af k , E k f ) contains the finite structure 

⎤ 

⎥ 

⎦


P4. Multiple shooting methods 

Basic approach 

1 Discretize the continuous-time problem: for ∆ = T /N, define 

X k := X((k − 1)∆), for k = 1, . . . , N. 

2 Compute X 1 , X 2 , ..., X N , s.t. X 1 = X N+1 (i.e., X(0)=X(T )) 

3 Integrate the appropriate differential equation to compute the 

solution on [(k − 1)∆, k∆], for k = 1, . . . , N.


P4. Multiple shooting methods 

Basic approach 

1 Discretize the continuous-time problem: for ∆ = T /N, define 

X k := X((k − 1)∆), for k = 1, . . . , N. 

2 Compute X 1 , X 2 , ..., X N , s.t. X 1 = X N+1 (i.e., X(0)=X(T )) 

3 Integrate the appropriate differential equation to compute the 

solution on [(k − 1)∆, k∆], for k = 1, . . . , N. 

Computational aspects 

computation of suitable transition matrices over small intervals 

[(k − 1)∆, k∆] ⇒ unstable or fast dynamics can be handled 

solving appropriate discrete-time periodic Lyapunov, Sylvester or 

Riccati equations ⇒ numerically reliable algorithms available 

using structure preserving integration (e.g., symplectic, positivity 

preserving ) ⇒ enhanced accuracy for large T 

steps 1 & 3 are "embarrassingly" parallelizable ⇒ highly efficient 

computations


P4. Solution of periodic Lyapunov differential equation 

1. Discretize the continuous-time equation: For ∆ := T /N, 

X(t) and X(t + ∆) are related as 

X(t + ∆) = Φ A (t + ∆, t)X(t)Φ T A 

(t + ∆, t)+ 

Φ A (t + ∆, τ)C(τ)Φ T A 

(t + ∆, τ)dτ 

∫ t+∆ 

t 

2. Solve the discrete-time periodic Lyapunov equation: 

X k+1 = F k X k Fk T + W k, k = 1, . . . , N; X N+1 = X 1 

where X k = X((k − 1)∆), F k := Φ A (k∆, (k − 1)∆) and 

W k := 

∫ k∆ 

(k−1)∆ 

Φ A (k∆, τ) C(τ)Φ T A (k∆, τ) dτ 

Parallelization: Step 1 is "embarrassingly" parallelizable!


Algorithms for discrete-time periodic systems 

Basic analysis techniques: poles, zeros, structural 

properties, minimal realization, frequency 

response, system norms, model reduction. 

Basic synthesis techniques: periodic matrix equations 

(Lyapunov, Sylvester, Riccati), pole assignment, 

stabilization, inverses, left/right annihilators, 

minimal dynamic covers, proper/stable coprime 

factorizations, normalized coprime factorizations. 

Synthesis methods: exact fault detection and isolation (FDI), 

H 2 /H ∞ synthesis techniques, optimal periodic 

output feedback


Contributions – discrete-time 

Theory of discrete-time periodic systems: Bittanti, Bolzern, 

Colaneri, Grasselli, Longhi, De Nicolao, 

Tornambè, Verriest 

Product eigenvalue problems: Benner, Bojanczik, Byers, 

Hench, Granat, Kågström, Kressner, Sreedhar, 

Van Dooren, Varga 

Condensed forms, system analysis: Benner, Van Dooren, 

Varga 

Matrix equations: Benner, Hench, Laub, Kressner, Van 

Dooren, Varga 

Synthesis problems: Bittanti, Colaneri, Longhi, Varga


Algorithms for continuous-time periodic systems 

Basic analysis techniques: Floquet-analysis using multiple 

shooting, system norms, frequency lifting based 

methods (e.g., frequency response), discretization 

Basic synthesis techniques: multiple shooting methods for 

solving periodic matrix differential equations 

(Lyapunov, Sylvester, Riccati) 

Synthesis methods: optimal periodic output feedback, 

H 2 /H ∞ synthesis


Contributions – continuous-time 

Theory of PDEs: Bittanti, Bolzern, Brocket, Colaneri 

Periodic systems norms: Bittanti, Colaneri, Hagiwara, 

Lampe, Rosenwasser, Zhou 

Multiple shooting based methods: Varga 

Relevant discrete-time techniques: Benner, Bojanczik, 

Byers, Granat, Guo, Johansson, Hench, Laub, Van 

Dooren, Varga 

Geometric integration: Dieci, Hairer, Hench, Kenney, Laub, 

Lubich, Warnner


Open problems – discrete-time 

descriptor periodic realization 

inner-outer factorization (non-standard case) 

Hankel-norm approximation 

solution of exact/approximate model matching problems 

solution of approximate FDI synthesis problems 

efficient solution of periodic LMIs


Open problems – continuous-time 

efficient computation of frequency-responses 

periodic stabilization and pole assignment 

solution of periodic differential LMIs 

structure exploiting/preserving algorithms for 

frequency-lifted representations 

application of continuous matrix decompositions to solve 

various structural problems


Additional information 

A. Varga and P. Van Dooren 

Computational methods for periodic systems - an overview, 

Prepr. IFAC Workshop on Periodic Control Systems, Como, 

Italy, pp. 177–176, 2001. 

A. Varga. 

A PERIODIC SYSTEMS Toolbox for MATLAB. 

Proc. of IFAC 2005 World Congress, Prague, Czech 

Republic, 2005. 

(available only within cooperation projects) 

A. Varga. 

An overview of recent developments in computational 

methods for periodic systems. 

Proc. of IFAC Workshop on Periodic Control Systems, St. 

Petersburg, Russia, 2007.

Advances and Challenges in Linear Control Systems, Rome, Italy

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