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A Unified View of Sampled-Data Control Systems
Jose C. Geromel
School of Electrical and Computer EngineeringUniversity of Campinas - Brazil
Kyoto UniversityKyoto, December 9th, 2015
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Contents
Main notation
Tutorial
MotivationPlantControlNetworkPerformance optimization
Sampled-data controlHybrid systemsSampled-data controlMarkov jump linear systems
Practical applicationsInverted pendulumMass-spring-damper system
Conclusion
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Main notation
◮ Real (R), nonnegative real (R+) and natural numbers (N)
◮ K = {1, · · · ,N}
◮ The trace function is tr(·)
◮ For ξ(t) defined for all t ≥ 0 the value ξ(τ−) indicates thelimit of ξ(t) as t goes to τ ≥ 0 from the left
◮ Sampling of a real signal s[k] = s(tk) for all k ∈ N
◮ Sets of bounded signals in continuous-time (t ∈ R+) anddiscrete-time (k ∈ N) domains:
◮ L2 equipped with ‖w‖22 =∫∞
0‖w(t)‖22dt < ∞
◮ ℓ2 equipped with ‖w‖22 =∑
k∈N‖w [k ]‖22 < ∞
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Tutorial
◮ LTI systems expressed through a state space realization
x(t) = Ax(t) + Ew(t), x(0) = x0
z(t) = Cx(t)
◮ x0 ∈ Rn is the initial condition
◮ x(t) : R+ → Rn is the state
◮ z(t) : R+ → Rq is the controlled output
◮ w(t) : R+ → Rp is the exogenous perturbation input
x(t) = eAtx0 +
∫ t
0eA(t−τ)Ew(τ)dτ, ∀t ≥ 0
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Tutorial
◮ Stability: Matrix A ∈ Rn×n is said to be
◮ Hurwitz (asymptotically) stable if
Re{λi(A)} < 0, ∀i = 1, · · · , n
eigenvalues inside the left open half plane!◮ Schur (asymptotically) stable if
|λi (A)| < 1, ∀i = 1, · · · , n
eigenvalues inside the unit circle!
Simple and useful relationship holds for a given h > 0
A Hurwitz stable ↔ eAh Schur stable
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Tutorial
◮ The celebrated Lyapunov Lemma states that:◮ A ∈ R
n×n is Hurwitz stable if and only if there exists P > 0satisfying the linear matrix inequality (LMI)
A′P + PA < 0
◮ A ∈ Rn×n is Schur stable if and only if there exists S > 0
satisfying the linear matrix inequality (LMI)
A′SA− S < 0
Simple and important fact that holds for a given h > 0
∃P > 0 s.t. A′P + PA < 0 → eA′hPeAh − P < 0
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Tutorial
◮ Performance: To evaluate performance we need to be ableto calculate the quantity
‖z‖22 =
∫ ∞
0z(t)′z(t)dt
where z(t) is the output corresponding to w(t) = 0 andarbitrary initial condition x(0) = x0 ∈ R
n. Hence
‖z‖22 = x ′0
(∫ ∞
0eA
′tC ′CeAtdt
)
︸ ︷︷ ︸
A′P+PA+C ′C=0
x0
= infP>0
{x ′0Px0 : A
′P + PA+ C ′C < 0}
requires A Hurwitz stable!
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Tutorial
◮ Let h > 0 be given, define the matrices
eAh = Ah ,
∫ h
0eA
′τC ′CeAτdτ = C ′hCh
Adopting the uniform sequence {tk = kh}k∈N it follows that
∫ ∞
0eA
′tC ′CeAtdt =∑
k∈N
∫ tk+1
tk
eA′tC ′CeAtdt
=∑
k∈N
(eA′h)k
∫ h
0eA
′τC ′CeAτdτ (eAh)k
⇓∫ ∞
0eA
′tC ′CeAtdt
︸ ︷︷ ︸
A′P+PA+C ′C=0
=∑
k∈N
A′kh C
′hCh Ak
h
︸ ︷︷ ︸
A′
hPAh−P+C ′
hCh=0
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Tutorial
◮ Evolving from the same initial condition x(0) = x [0] = x0
x(t) = Ax(t)z(t) = Cx(t)
→x [k + 1] = Ahx [k]
z [k] = Chx [k]
⇓∫ ∞
0z(t)′z(t)dt =
∑
k∈N
z [k]′z [k]
◮ No kind of approximations involved!◮ Both systems are said to be equivalent!
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Tutorial
◮ Performance: Let us now calculate the quantity
‖z‖22 =
∫ ∞
0z(t)′z(t)dt
where z(t) is the output corresponding to w ∈ L2 andarbitrary initial condition x(0) = x0 ∈ R
n. Hence
x(t) = eA(t−tk )x [k] +
∫ t
tk
eA(t−τ)Ew(τ)dτ, t ∈ [tk , tk+1)
depends exclusively on (x [k],w [k]) for all t ∈ [tk , tk+1) if
w ∈ L2h ⊂ L2
a set composed by all signals w(t) = w [k], ∀t ∈ [tk , tk+1)such that w ∈ ℓ2.
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Tutorial
◮ Defining the matrices of appropriate dimensions
F =
[A E
0 0
]
−→ eFh =
[Ah Eh
0 I
]
the fundamental formula[
x(t)w(t)
]
= eF (t−tk )
[x [k]w [k]
]
, ∀t ∈ [tk , tk+1)
holds and provides z(t) in the same time interval. As before,with matrices
G =[C 0
]−→
∫ h
0eF
′τG ′GeFτdτ =
[C ′h
D ′h
] [C ′h
D ′h
]′
the equivalent system follows.
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Tutorial
◮ Evolving from the same initial condition x(0) = x [0] = x0
x(t) = Ax(t) + Ew(t)z(t) = Cx(t)
→x [k + 1] = Ahx [k] + Ehw [k]
z [k] = Chx [k] + Dhw [k]
⇓∫ ∞
0z(t)′z(t)dt =
∑
k∈N
z [k]′z [k]
◮ No kind of approximations involved!◮ This is restricted to the class of inputs w ∈ L2h
◮ For w ∈ L2 it is not possible to obtain an equivalent system offinite dimension!
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Tutorial
◮ Claim: Hybrid systems are well adapted to modelsampled-data control systems!
◮ Consider the LTI continuous-time system
x(t) = Ax(t) + Bu(t) + Ew(t), x(0) = x0
z(t) = Cx(t) + Du(t)
◮ u(t) : R+ → Rm is the control variable such that
u ∈ U ↔ u(t) = u(tk), t ∈ [tk , tk+1), ∀k ∈ N
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Tutorial
◮ For instance, in the simplest case of state feedback
u ∈ U ↔ u(t) = Lx(tk), t ∈ [tk , tk+1), ∀k ∈ N
defining the state variable ξ(t)′ = [x(t)′ u(t)′] ∈ Rn+m, the
closed-loop sampled-data system can be rewritten as
ξ(t) =
[A B
0 0
]
︸ ︷︷ ︸
F
ξ(t) +
[E
0
]
︸ ︷︷ ︸
J
w(t)
z(t) =[C D
]
︸ ︷︷ ︸
G
ξ(t)
ξ(tk) =
[I 0L 0
]
︸ ︷︷ ︸
H
ξ(t−k ), k ∈ N
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Tutorial
◮ This a special class of hybrid systems of the form
ξ(t) = F ξ(t) + Jw(t)
z(t) = Gξ(t)
ξ(tk) = Hξ(t−k )
◮ Necessary and sufficient condition for asymptotic stabilityfollows from the discrete-time process
ξ(tk ) → ξ(t−k+1) → ξ(tk+1), ∀k ∈ N
◮ Performance indexes of interest must be calculated (analysis)◮ In general, matrix H depends on the matrix variables to be
determined (synthesis)
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Tutorial
◮ The celebrated Bellman’s principle of optimality◮ HJB equation (continuous-time) and dynamic programming
(discrete-time) are the most important theoretical devices fordynamic systems optimization
◮ Hard to solve due to generality!
Consider the general problem
supw∈W
∫ ∞
0f (x(t),w(t))dt
subject tox(t) = F (x(t),w(t)), x(0) = x0
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Tutorial
◮ Consider the sequence {tk}k∈N. Bellman’s principle ofoptimality states that
V (x(tk), tk) = supw∈W
{∫ tk+1
tk
f (x ,w)dt + V (x(tk+1), tk+1)
}
where◮ The function V (x , t) : Rn × R+ → R is called cost-to-go!◮ Assuming (x∗,w∗ ∈ W ) is optimal, adding terms
supw∈W
∫∞
0
f (x(t),w(t))dt =∑
k∈N
∫ tk+1
tk
f (x(t)∗,w(t)∗)dt
= V (x0, 0)
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Tutorial
◮ Upper bounds to the optimal cost are easily generated!
◮ If f (x ,w) > 0, ∀(x ,w) 6= 0 and f (0, 0) = 0 then◮ V (x , t) is positive definite◮ V (x(tk ), tk) > V (x(tk+1), tk+1) for all k ∈ N
◮ The discrete-time sequence {x(tk)}k∈N converges!
◮ The cost-to-go function satisfies the HJB equation
∂V
∂t+
∂V ′
∂xF (x ,w)
︸ ︷︷ ︸dVdt
≤ −f (x ,w), w ∈ W
for t ∈ [tk , tk+1), ∀k ∈ N.
◮ For time invariant systems and tk+1 − tk = h, ∀k ∈ N asolution satisfying some adequate initial and final boundaryconditions on the time interval t ∈ [0, h) suffices!
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Motivation
◮ Sampled-data control in a general framework
◮ Digital implementation
◮ Remote control supported by internet facilities
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Motivation
◮ From the very basic control structure ...
◮ Plant P → Modeling◮ Controller C → Control design
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Motivation
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◮ ... to networked control!
◮ Plant P → Modeling◮ Controller C → Control design◮ Network → Signal quality and limitations
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Plant
◮ The plant P is linear time invariant - LTI
x(t) = Ax(t) + Bu(t) + Ew(t)
z(t) = Cx(t) + Du(t)
◮ x0 ∈ Rn is the initial condition
◮ x(t) : R+ → Rn is the state
◮ u(t) : R+ → Rm is the control
◮ z(t) : R+ → Rq is the controlled output
◮ w(t) : R+ → Rp is the exogenous perturbation input
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Control
◮ Communication channel modeling! – Operator R
f (t)︸︷︷︸
trasmitted signal
→ Rf (t)︸ ︷︷ ︸
received signal
◮ Effective remote control applied to the plant
u(t) = RC(Rx)(t)
◮ The state x(t) is measured and the controller receives Rx(t)◮ The transmitted controller output C(Rx(t)) provides u(t)
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Network
◮ Bandwidth limitation: Data transmission with maximumfrequency 1/h. Successive sampling instants satisfyhk = tk+1 − tk ≥ h,∀k ∈ N, (Matveev and Savkin, 2009)
Rf (t) = f (tk), ∀t ∈ [tk , tk+1), ∀k ∈ N
◮ Sampled-data control operator, (Chen and Francis, 1995),(Ichicawa and Katayama, 2001)
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Network
◮ Packet dropout: Network information loss imposes
Rf (t) = Γθ(t)f (t)
where θ(t) ∈ K = {1, · · · ,N} are the states of a Markovchain, (Costa, Fragoso and Todorov, 2013)
◮ Control synthesis is merged in a stochastic framework!
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Performance optimization
◮ H2 performance: From zero initial condition
J2(u) =
p∑
l=1
‖ zl︸︷︷︸
w(t)=el δ(t)
‖22
◮ H∞ performance: From zero initial condition
J∞(u) = infγ
{γ2 : ‖z‖22 ≤ γ2‖w‖22 , ∀w ∈ L2
}
worst case perturbation. It passes through the network?
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Performance optimization
◮ The main goal is to solve
infu∈U
Jα(u)
where:◮ α ∈ {2,∞}◮ u ∈ U define feasible signals subject to the network limitations◮ Solution strategy → remove the constraint u ∈ U and
redefine an equivalent design problem
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Sampled-data control
◮ Considering tk+1 − tk = h > 0, ∀k ∈ N and preparing data asbefore, that is
F =
[A B
0 0
]
−→ eFh =
[Ah Bh
0 I
]
and
G =[C D
]−→
∫ h
0eF
′τG ′GeFτdτ =
[C ′h
D ′h
] [C ′h
D ′h
]′
the associated equivalent discrete-time system is determined.
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Sampled-data control
◮ Optimal solution of the H2 control problem
infu∈U
J2(u) = minL
∥∥∥(Ch + DhL) (zI − (Ah + BhL))
−1E∥∥∥
2
2
◮ The optimal matrix gain follows from the stabilizing solution ofa discrete time algebraic Riccati equation
◮ The optimal control is of the form
u(t) = Lx(tk), ∀t ∈ [tk , tk+1)
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Sampled-data control
◮ Consider the class of not too severe external perturbations
w ∈ L2h ⊂ L2
and a new performance index
Jh(u) = infγ
{γ2 : ‖z‖22 ≤ γ2‖w‖22 , ∀w ∈ L2h
}
which satisfiesinfu∈U
J∞(u) ≥ infu∈U
Jh(u)
⇓
Is this lower bound useful in the context of H∞ control?
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Sampled-data control
◮ (Sub)Optimal solution of the H∞ control problem
infu∈U
Jh(u) = minL
∥∥∥(Ch + DhL) (zI − (Ah + BhL))
−1 Eh
∥∥∥
2
∞
◮ Matrix Eh is obtained as before by replacing B → [B E ]◮ The optimal matrix gain follows from the stabilizing solution of
a discrete time algebraic Riccati equation◮ The optimal control is of the form
u(t) = Lx(tk), ∀t ∈ [tk , tk+1)
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Sampled-data control
◮ Sampled-data optimal control in a general framework
⇓
Hybrid Systems
+
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Sampled-data control
◮ Sampled-data optimal control in a general framework
⇓
Hybrid Systems
+
Bellman’s Principle of Optimality
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Hybrid systems
◮ Remember that in the case of state feedback
u ∈ U ↔ u(t) = Lx(tk), t ∈ [tk , tk+1), ∀k ∈ N
defining the state variable ξ(t)′ = [x(t)′ u(t)′] ∈ Rn+m, the
closed-loop sampled-data system can be rewritten as
ξ(t) =
[A B
0 0
]
︸ ︷︷ ︸
F
ξ(t) +
[E
0
]
︸ ︷︷ ︸
J
w(t)
z(t) =[C D
]
︸ ︷︷ ︸
G
ξ(t)
ξ(tk) =
[I 0L 0
]
︸ ︷︷ ︸
H
ξ(t−k ), k ∈ N
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Hybrid systems
◮ Let us consider a generic hybrid system of the form
ξ(t) = F ξ(t) + Jw(t)
z(t) = Gξ(t)
ξ(tk) = Hξ(t−k )
◮ Necessary and sufficient condition for asymptotic stabilityfollows from the discrete-time process
ξ(tk) → ξ(t−k+1) → ξ(tk+1), ∀k ∈ N
◮ Performance indexes J2(u) and J∞(u) calculation
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H2 performance
◮ Theorem 1: If the matrix differential equation
P(t) + F ′P(t) + P(t)F = −G ′G
admits a solution in the time interval t ∈ [0, h] satisfying theboundary conditions
P(0) < S−1 , P(h) > H ′S−1H
for some symmetric matrix S > 0, then the hybrid system isasymptotically stable and satisfies
J2(u) < tr(J ′H ′S−1HJ)
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H2 performance
◮ Matrix S > 0 satisfies the Lyapunov inequality:
eF′hH ′S−1HeFh < S−1 −
∫ h
0eF
′tG ′GeFtdt
︸ ︷︷ ︸
≥0
which admits a solution if and only if HeFh is Schur stable.
◮ The optimal sampled-data H2 control follows from
infL,S>0
tr(J ′H ′S−1HJ) → CONVEX?
Notice that H depends on the state feedback gain L
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H∞ performance
◮ Theorem 2: If the matrix differential equation
P(t) + F ′P(t) + P(t)F + γ−2P(t)JJ ′P(t) = −G ′G
admits a solution in the time interval t ∈ [0, h] satisfying theboundary conditions
P(0) < S−1 , P(h) > H ′S−1H
for some symmetric matrix S > 0, then the hybrid system isasymptotically stable and satisfies
J∞(u) < γ2
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H∞ performance
◮ We have to solve a nonlinear differential equation!
◮ There exists a solution P(t),∀t ∈ [0, h] if and only if thealgebraic Riccati equation
FQ + QF ′ + γ−2JJ ′ + QG ′GQ = 0
admits a solution. Defining F = F + QG ′G◮ In general Q is sign indefinite◮ The following matrices can be readily calculated
Fh = e F h,
∫ h
0
e F′τG ′Ge Fτdτ = G ′
hGh
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H∞ performance
◮ Solution of the Riccati differential equation, (Geromel, 1978)
◮ Lemma 1: The H∞ two boundary value problem admits asolution if and only if there exist W , S such that S > Q and
[W WH ′
HW S
]
> 0
[W − Q 0
0 I
]
>
[FhGh
]
(S − Q)
[FhGh
]′
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H∞ performance
◮ Jointly convex problem in the matrix variables (W ,S)
◮ The limit caseγ → +∞ , Q = 0
provides a solution to the H2 two boundary value problem
◮ Including the matrix gain L, in the set of variables, theproblem remains CONVEX?
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H∞ performance
◮ Important: Continuous-time systems characterized by H = I◮ From the solution of the H∞ two boundary value problem
P(t) = P = S−1, ∀t ∈ [0, h]
where P > 0 is the stabilizing solution of the algebraic Riccatiequation
F ′P + PF + γ−2PJJ ′P = −G ′G
the classical results are recovered
J2 = ‖G(sI − F )−1J‖22
J∞ = ‖G(sI − F )−1J‖2∞
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Sampled-data control
◮ Main property of the Riccati equation solution Q
F = F︸︷︷︸
A B
0 0
+ Q︸︷︷︸
Q 00 0
G ′︸︷︷︸
C ′
D ′
G =
[A B
0 0
]
matrices F and F have the same structure!
⇓
Fh =
[Ah Bh
0 I
]
, Gh =[Ch Dh
]
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Sampled-data control
◮ The determination of the state feedback gain L requires theblock structure
S =
[X Y
Y ′ Z
]
◮ Lemma 2: The H∞ two boundary value problem admits asolution if and only if there exists S > 0 such that S > Q and
[X − Q 0
0 I
]
>
[Ah Bh
Ch Dh
] [X − Q Y
Y ′ Z
] [Ah Bh
Ch Dh
]′
(∗)
In the affirmative case L = Y ′X−1.
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Sampled-data control
◮ H∞ optimal control
infS>0,S>Q,γ
{γ2 : (∗)}
◮ H2 optimal control
infS>0
{tr(E ′X−1E ) : (∗)}
adopting the limit case solution γ → +∞ , Q = 0
Both are convex problems expressed by LMIs
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Example
◮ A second order system with state space realization
A =
[0 1
−6 1
]
, B =
[01
]
, E =
[11
]
C =
[1 00 0
]
, D =
[01
]
has been treated. Three problems have been solved:◮ H∞ optimal control◮ H∞ optimal control with w ∈ L2h ⊂ L2
◮ H∞ suboptimal control
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Example
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
h [s]
log10
(
γ)
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Nonuniform sampling
◮ Time distance between successive samplings is uncertain
tk+1 − tk = hk ∈ T , ∀k ∈ N
◮ H2 optimal control
infS>0
tr(E ′X−1E ) : (∗)︸︷︷︸
Ψ(S,h)>0
,∀h ∈ T
LMI =⇒ convex set whenever h ∈ T is fixed!
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Nonuniform sampling
◮ Globally convergent algorithm!
Jℓ+1 ≥ Jℓ , ℓ ∈ {0, 1, · · · }
◮ Example
A =
[0 1
−16 4.8
]
, B =
[016
]
, E =
[016
]
C =
[1 00 0
]
, D =
[00.1
]
, T = (0, π/25]
◮ L = [ 0.8430 − 0.4781] → γ = 24.407, (Suplin et al., 2007)◮ L = [−2.0620 − 0.8793] → γ = 1.0401
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Markov jump linear systems
◮ Consider a MJLS with state space realization
x(t) = Aθ(t)x(t) + Bθ(t)u(t) + Eθ(t)w(t)
z(t) = Cθ(t)x(t) + Dθ(t)u(t)
◮ θ(t) ∈ K is a Markovian process defined by
P(θ(t + h) = j |θ(t) = i) = δi−j + λijh + o(h)
◮ x(0) = 0◮ θ(0) = θ0 with probability P(θ0 = i) = πi0, ∀i ∈ K
◮ Performance indices are defined accordingly to
‖η‖22 =
∫∞
0
E{η(t)′η(t)}dt
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Hybrid Markov jump linear systems
◮ Hybrid Markov jump linear system
ξ(t) = Fθ(t)ξ(t) + Jθ(t)w(t)
z(t) = Gθ(t)ξ(t)
ξ(tk) = Hθ(tk )ξ(t−k )
◮ Sampling instants h = tk+1 − tk > 0, ∀k ∈ N are independentof the Markov process
◮ Jumps may occur inside the sampling interval◮ State feedback control design of the form
u(t) = u(tk ) = Lθ(tk)x(tk), ∀t ∈ [tk , tk+1)
◮ Impossible to determine an equivalent discrete-time system!
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H2 performance
◮ Theorem 3: If the coupled matrix differential equations
Pi (t) + F ′i Pi(t) + Pi(t)Fi +
∑
j∈K
λijPj(t) = −G ′iGi
for i ∈ K admit a solution in the time interval t ∈ [0, h]satisfying the boundary conditions
Pi(0) < Si−1 , Pi(h) > H ′
iSi−1Hi , i ∈ K
for some symmetric matrices Si > 0, then the hybrid Markovjump linear system is mean square stable and satisfies
J2(u) <∑
i∈K
πi0tr(J′iH
′iSi
−1HiJi)
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H2 performance
◮ The previous result characterizes the optimal solution
◮ As before, the following matrices are defined
Fi = Fi + (λii/2)I =⇒ Fhi = eFih, i ∈ K
G ′di Gdi=
∫ h
0eF
′
iτ
G ′iGi +
∑
j 6=i∈K
λijPj(τ)
︸ ︷︷ ︸
≥0
eFi τdτ, i ∈ K
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H2 performance
◮ The determination of the state feedback gains Li , i ∈ K
requires the block structure
Si =
[Xi Yi
Y ′i Zi
]
> 0, i ∈ K
and to solveinfSi>0
∑
i∈K
πi0tr(E′i X
−1i Ei )
[Xi 00 I
]
>
[Ahi Bhi
Chi Dhi
]
Si
[Ahi Bhi
Chi Dhi
]′
, i ∈ K
⇓
In the affirmative case Li = Y ′i X
−1i , i ∈ K
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H2 performance
◮ Important facts:
◮ N uncoupled subproblems◮ Matrices Ghi , i ∈ K are coupled through the dependence of
Pj , j 6= i ∈ K. For this reason a globally convergent algorithmhas been developed
Si(ℓ+1) ≥ Si(ℓ) ≥ 0 , ℓ ∈ {0, 1, · · · }
◮ There is no difficulty to treat H∞ control design problems
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Inverted pendulum
◮ Inverted pendulum without friction
u
mg
ftr
φ
xh
whose vertical displacement θv = φ− π/2 follows from thelinear model. Data are given in (Geromel and Korogui, 2011)
(M +m)xh −mℓθv = u
ℓθv − xh − gθv = 0
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Inverted pendulum
◮ Sampled-data H2 optimal - uniform sampling
h = 500 [ms]
⇓
L = [1.3023 3.3221 −159.1264 −46.8910] =⇒ J2(u) = 156.6714
◮ Sampled-data H2 guaranteed - nonuniform sampling
hk ∈ T = [300, 700] ms ∀k ∈ N
⇓
L = [0.6539 1.9751 −146.6085 −43.0491] =⇒ J2(u) = 370.4710
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Inverted pendulum
◮ Time simulation - hk uniformly distributed inside T
0 2 4 6 8 10−50
0
50
0 2 4 6 8 10−100
0
100
200
t [s]
t [s]
θv[o]
u[N
]
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Mass-spring-damper system
◮ Two masses without friction connected with a damper andtwo springs, (Lutz, 2014)
◮ The control signal is transmitted through a network:◮ Bandwidth limitation: 1/h = 2 [Hz]◮ Packet dropout: Two Markov modes K = {1, 2}
corresponding to package loss and transmission success definedby a transition rate matrix Λ such that
eΛh0 =
[0.85 0.150.10 0.90
]
, h0 = 20 [ms]
◮ Initial probability π0 = [0.4 0.6]′
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Mass-spring-damper system
◮ Sampled-data H∞ optimal control - uniform sampling
h = 500 [ms]
⇓
J∞(u) = γ2opt , γopt = 1.5351
Considering γ = 1.6 we have calculated state feedback gains
[L1L2
]
=
[0.8653 1.0451 −0.6580 −3.12970.7204 0.7531 −0.5783 −2.7536
]
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Mass-spring-damper system
◮ Algorithm evolution for γ = 1.6
0 10 20 300
20
40
60
80
ℓ
‖Sℓ‖2
‖∆Sℓ‖2
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Mass-spring-damper system
◮ Monte Carlo simulation of 500 runs provides the trajectoriesof the closed-loop system excited by the exogenousperturbation w(t) = sin(6.38t), for t ∈ [0, 2] [s] and w(t) = 0elsewhere, (Leon-Garcia, 2008)
0 2 4 6 8 10 12−0,1
0,3
0,7
1,0
t [s]
z(t)′z(t)
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Conclusion
◮ Linear filtering: Optimal filtering in the context ofnetworked systems
◮ Dynamic output feedback control: Optimal control underpartial information taking into account bandwidth limitationand packet dropout occurrence
◮ Switched control: Optimal decision rule for thedetermination of hk ∈ T at each k ∈ N towards performanceoptimization.
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Conclusion
Thank you for your attention!!!
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