Stabilization of TS fuzzy systems with time-delay and nonlinear - - PowerPoint PPT Presentation
Stabilization of TS fuzzy systems with time-delay and nonlinear consequents Zolt an Nagy, Am alia M aty as, Zs ofia Lendek Technical University of Cluj-Napoca, Romania 11-17 July 2020, IFAC World Congress Online (Berlin, Germany)
Zolt´ an Nagy, Am´ alia M´ aty´ as, Zs´
Technical University of Cluj-Napoca, Romania
11-17 July 2020, IFAC World Congress Online (Berlin, Germany)
Background Fuzzy system with nonlinear consequents Controller design Example Summary
2nd year MSc student Technical University of Cluj-Napoca ROCON research group
Background Fuzzy system with nonlinear consequents Controller design Example Summary
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Background
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Fuzzy system with nonlinear consequents
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Controller design
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Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
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Background
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Fuzzy system with nonlinear consequents
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Controller design
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Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Controller design method for Takagi-Sugeno (TS) systems with nonlinear consequents Time-delayed input The membership functions may depend on both current and delayed states Slope-bounded local nonlinearities LMI conditions for stabilization of the TS systems
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model: ˙ x(t) =
s
hi
u(t) ∈ Rnu input vector s - number of rules z(t) - premise vector, available hi(z) - membership function convex sum: hi(z) ∈ [0, 1], s
i=1 hi(z) = 1
Ai and Bi are local linear models
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model: ˙ x(t) =
s
s
hi
hi(z(t − τ(t))) - membership function that depends on the delayed states
Notation τ(t) := τ Fzzτ :=
s
s
hi
Model ˙ x(t) =Azzτ x(t) + Dzzτ x(t − τ) + Bzzτ u(t − τ)
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Disadvantage The number of fuzzy rules increases exponentially with the number of nonlinearities.
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Disadvantage The number of fuzzy rules increases exponentially with the number of nonlinearities. Idea Keep some of the nonlinearities in their
Background Fuzzy system with nonlinear consequents Controller design Example Summary
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Background
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Fuzzy system with nonlinear consequents
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Controller design
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Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model: ˙ x(t) =Azzτ x(t) + Dzzτ x(t − τ) + Bzzτ Gψ(Hx(t)) + Bzzτ u(t − τ) x(t − τ) is the time-delayed state vector u(t − τ) is the time-delayed input vector ψ(Hx(t)) is a vector function, ψ(·) = ψ1(·) ψ2(·) ... ψr(·) each entry is a function of a linear combination of x(t) G system matrix Advantages Fewer fuzzy rules Measured-state & unmeasured-state nonlinearities separated
Background Fuzzy system with nonlinear consequents Controller design Example Summary
For each ψi, i = 1, ..., r there exist bi such that 0 ≤ ψi(v) − ψi(w) v − w ≤ bi There exist a δi(t) ∈ [0, bi], so that ψi(v) − ψi(w) = δi(t)(v − w), ∀ v, w ∈ R, v = w
Background Fuzzy system with nonlinear consequents Controller design Example Summary
˙ x1(t) = −2x1(t) + 2x1(t − τ) + x2(t − τ) ˙ x2(t) = ( − 6+sin(x1(t)))x2(t)+ ( 0.9+0.1sin(x1(t)))x1(t−τ) + ( 4+sin(x1(t−τ)))x2(t−τ)+(0.75+0.25sin(x1(t)))(u(t−τ)+ α1(x1)+α2(x2)
) Rearranging the system: ˙ x(t)= −2 −6+sin(x1(t))
i=1
s
j=1 hi(z)hj(z−τ)Aij
x(t) +
1 0.9+0.1sin(x1(t)) 4+sin(x1(t − τ))
i=1
s
j=1 hi(z)hj(z−τ)Dij
x(t − τ) +
i=1
s
j=1 hi(z)hj(z−τ)Bij
Background Fuzzy system with nonlinear consequents Controller design Example Summary
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Background
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Fuzzy system with nonlinear consequents
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Controller design
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Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model ˙ x(t) =Azzτ x(t) + Dzzτ x(t − τ) + Bzzτ Gψ(Hx(t)) + Bzzτ u(t − τ) Proposed controller structure u(t) = −Kzx(t) − Gψ(Hx(t)) Delayed in time u(t − τ) = −Kzτ x(t − τ) − Gψ(Hx(t − τ))
Background Fuzzy system with nonlinear consequents Controller design Example Summary
˙ x(t) =Azzτ x(t) + (Dzzτ −Bzzτ Kzτ )x(t − τ) + Bzzτ G(ψ(Hx(t)) − ψ(Hx(t − τ))) Closed-loop system ψ(Hx(t)) − ψ(Hx(t − τ)) =δ(t)(Hx(t) − Hx(t − τ)) =δ(t)H(x(t) − x(t − τ)), Notation η := H(x(t) − x(t − τ)) Closed-loop system ˙ x(t) =Azzτ x(t) + (Dzzτ − Bzzτ Kzτ )x(t − τ) + Bzzτ Gδ(t)η
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Lyapunov functional V (t, x) =xT(t)Px(t) + t
t−τ
xT(s)Qx(s)ds P = PT > 0, Q = QT > 0 delay-independent conditions
Background Fuzzy system with nonlinear consequents Controller design Example Summary
System: ˙ x(t) =Azzτ x(t) + Dzzτ x(t − τ) + Bzzτ Gψ(Hx(t)) + Bzzτ u(t − τ) Controller: u(t) = −Kzτ x(t) − Gψ(Hx(t)) Closed-loop system: ˙ x(t) =Azzτ x(t) + (Dzzτ − Bzzτ Kzτ )x(t − τ) + Bzzτ Gδ(t)η
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Theorem (1) Assume that τ is differentiable, ˙ τ ≤ d and d ∈ [0, 1) is a given constant. If there exist P = PT > 0, Q = QT > 0, M = diag(m1, ..., mr) > 0, Ni, for i = 1, ..., s and constant ǫ > 0 so that Fijj ≤ 0 and 2 s − 1Fijj + Fijk + Fikj ≤0 ∀i, j, k = 1, ..., s, j = k. where Fijk = PAT
ij + ∗ + Q
DijP − BijNk BijGM + PHT P ∗ −(1 − d)Q −PHT ∗ ∗ ν(M) ∗ ∗ ∗ −ǫI , and ν(M) = −2Mdiag( 1
b1 , ..., 1 b1 ) then the closed-loop system is asymptotically stable.
Controller gains are recovered: Ni = KiP−1.
Background Fuzzy system with nonlinear consequents Controller design Example Summary
1
Background
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Fuzzy system with nonlinear consequents
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Controller design
4
Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
˙ x1(t) = −2x1(t) + 2x1(t − τ) + x2(t − τ) ˙ x2(t) = ( − 6+sin(x1(t)))x2(t)+ ( 0.9+0.1sin(x1(t)))x1(t−τ) + ( 4+sin(x1(t−τ)))x2(t−τ)+(0.75+0.25sin(x1(t)))(u(t−τ)+ α1(x1)+α2(x2)
) Rearranging the system: ˙ x(t)= −2 −6+sin(x1(t))
i=1
s
j=1 hi(z)hj(z−τ)Aij
x(t) +
1 0.9+0.1sin(x1(t)) 4+sin(x1(t − τ))
i=1
s
j=1 hi(z)hj(z−τ)Dij
x(t − τ) +
i=1
s
j=1 hi(z)hj(z−τ)Bij
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model: ˙ x(t) =Azzτ x(t) + Dzzτ x(t − τ) + Bzzτ Gψ(Hx(t)) + Bzzτ u(t − τ) Matrices A11 = A12 = −2 −5
−2 −7
D11 = 2 1 0.8 3
2 1 0.8 5
2 1 1 3
2 1 1 5
G =
1
1 1
0.5
1
h1(z) = 1 − sin(z) 2 , h2(z) = 1 − h1(z), z = x1.
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Slowly varying time-delay function: ˙ τ ≤ d = 0.01 τ(t) = 1 + 0.5cos(0.01t) Controller gains: K1 =
6.99
5.06
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Open-loop system Closed-loop system
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Model: ˙ x(t) =Ax(t) + Dx(t − τ) + Bu(t) + BGψ(H(x(t)) where A = −2 −5
2 a1 3 4
B = 1
Background Fuzzy system with nonlinear consequents Controller design Example Summary
’x’- Corollary 1 (Moodi and Kazemy, 2019), ’o’- Theorem 1
Background Fuzzy system with nonlinear consequents Controller design Example Summary
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Background
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Fuzzy system with nonlinear consequents
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Controller design
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Example
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Summary
Background Fuzzy system with nonlinear consequents Controller design Example Summary
Summary To handle unmeasured-state nonlinearities Reduce computational complexity by keeping some of the nonlinearities in their
Time-delay in the input LMI design conditions Ongoing work More complex Lyapunov functional
Background Fuzzy system with nonlinear consequents Controller design Example Summary
This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project number PN-III-P1-1.1-TE-2016-1265, contract number 11/2018.