Introduction to Time-Delay Systems -...

Introduction to Time-Delay Systems

Emilia FRIDMAN

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

I The simplest time-delay equation with constant delay h > 0 is

x(t) = −x(t− h).

I Time-Delay Systems (TDS) are also called systems with aftereffect ordead-time, hereditary systems, or differential-difference equations.They belong to the class of functional differential equations which areinfinite-dimensional.

I Time-delays appear in many engineering systems - aircraft, chemical controlsystems, in lasers models, in internet, biology, medicine ...There can be transport, communication or measurement delay.

Figure 1: Showering person.

I A person wishes to achieve the desired water tempreture Td by rotating themixer.

I Let T (t) denote the water temperature in the mixer output.I Let h be the time needed by the water to go from the mixer output to the

person’s head.I Assume that the change of the temperature is proportional to the angle of

rotation of the handle, whereas the rate of rotation of the handle is proportionalto T (t)− Td.

I At time t the person feels the water temperature leaving the mixer at timet− h, which results in the following equation with the constant delay h:

T (t) = −k[T (t− h)− Td], k ∈ R.

Effects of delay on stability

I Delay may be destabilizing:

x(t) = −x(t− h)

is stable for h < π/2 and unstable for h > π/2.I Delay may be stabilizing:

x(t) = u(t), y(t) = x(t).

The system is not stabilizable by the non-delayed u(t) = K0y(t)

The system is stabilizable by delayed [Niculescu, Michiels, Kharitonov 04]

u(t) = K0y(t) + K1y(t− r), r > 0. (1)

since it is stabilizable by u(t) = K0y(t) +K1y(t) and

y(t) ≈ y(t)− y(t− r)r

, r > 0.

A time-delay approach to:I DANCES [Fridman, Tutorial on TDS, Israel, 2011]

I Destabilizing delay in Venice WaltzI Stabilizing delay in Tango

The system is stabilizable by delayed [Niculescu, Michiels, Kharitonov 04]

u(t) = K0y(t) + K1y(t− r), r > 0. (1)

since it is stabilizable by u(t) = K0y(t) +K1y(t) and

y(t) ≈ y(t)− y(t− r)r

, r > 0.

A time-delay approach to:I DANCES [Fridman, Tutorial on TDS, Israel, 2011]

I Destabilizing delay in Venice WaltzI Stabilizing delay in Tango

A time-delay approach to:I Sampled-data control [Mikheev, Sobolev & Fridman, 1988]

x(t) = Ax(t) +BKx(tk), t ∈ [tk, tk+1), limk→∞

tk =∞ (2)

This system can be represented as a continuous system with time-varying delayτ (t) = t− tk (note that τ = 1 for t 6= tk)

x(t) = Ax(t) +BKx(t− τ (t)), t ∈ [tk, tk+1). (3)

A time-delay approach became popular inI Networked Control System is feedback control loop closed through

communication network.

Physical plantx

?Sampler

?u(tk)=Kx(tk−ηk)

Delay

¾Controller

6Delay

6ZOH

-u(t)

Network Medium

1

Figure 2: Networked state-feedback control system.

Packet dropouts result in variable sampling. The closed-loop

x(t) = Ax(t) +B2Kx(tk − ηk), t ∈ [tk, tk+1).

Input delay approach

x(t) = Ax(t) +B2Kx(t− τ (t)), (4)

where0 ≤ τ (t) = t− tk + ηk ≤ tk+1 − tk + ηk ≤ τM , t ∈ [tk, tk+1)

andτ (t) = 1, t 6= tk

.

Figure 3: Drilling pipe model

The drilling pipe can be modeled by the wave equation

GJ

L2∂2z

∂σ2 (σ, t)− IB∂2z

∂t2(σ, t)− β ∂z

∂t(σ, t) = 0, σ ∈ [0, 1],

under the boundary conditions

z(0, t) = 0, GJL

∂z∂σ (1, t) + IB

∂2z∂t2 (1, t) = −T ∂z

∂t (1, t).

z(σ, t) is the deviation of the angle of rotation from its steady state value,T ∂z∂t (1, t) is the (linearized) torque on the bit,

IB is a lumped inertia (the assembly at the bottom hole),β ≥ 0 is the damping (viscous and structural),I is the inertia, G is the shear modulus, J is the geometrical moment of inertia.

The main variable of interest is the angular velocity at the drill bottom zt(1, t).

Assuming β = IB = 0, the model reduces to 1-D wave equation

∂2z∂t2 (σ, t) = a ∂

2z∂σ2 (σ, t), σ ∈ (0, 1), t > 0,

z(0, t) = 0, ∂z∂σ (1, t) = −k ∂z∂t (1, t),

(5)

where a = GJIBL2 , k = LT

GJ , r = LGJ ∈ R.

By D’Alembert method, a general solution of the 1-D wave equation is given by

z(σ, t) = φ(t+ sσ) + ψ(t− sσ), t > 0, (6)

where φ, ψ are C1 and s =√

1a .

This leads to time-delay system

ψ(t+ s) = −c0ψ(t− s), t > 0, (7)

with c0 = s−ks+k and c1 = r

s+k .

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

I 18 cent.: 1-st eqs with delay by Bernoulli, Euler , ConcordetI 1940-... Systematical study by A. Myshkis, R. BellmanI 1956-... Lyapunov method for stability by Krasovskii and by Razumikhin.

Smith predictor.I 1960-... 50+ monographs in English

N. Krasovskii (1963), R. Bellman & K. Cooke (1963), Elsgol’z & Norkin(1972), J. Hale (1977), V. Kolmanovskii & R. Nosov (1986), M. Malek-Zavarei& M. Jamshidi (1987), Kolmanovskii & Myshkis (1992)

I 1995-... Robust control of systems with uncertain delay τ (t)2000-... Delay boom

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

Solution concept. Initial value problem:

x(t) = −x(t− h), t ≥ 0,x(s) = φ(s), s ∈ [−h, 0]. (8)

Step method for solution:

h 0 th 2h

t ∈ [0,h], x(t) = −φ(t− h), x(0) = φ(0),t ∈ [h, 2h], t ∈ [2h, 3h], ...

−1 0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

t

x

Figure 4: Solutions with h = 1,ϕ ≡ 1 (plain) or ϕ ≡ 0.5t (dotted).

In spite of their complexity, time-delay systems (TDS) often appear as simpleinfinite-dimensional models of more complicated PDEs.E.g. D’Alambert transformation for the wave eq. leads to a TDS.Conversely, TDS can be represented by a classical transport PDE:

x(t) = −x(t− h)

x(t+ θ) = z(t, θ), θ ∈ [−h, 0] ⇒

∂∂tz(t, θ) =

∂∂θ z(t, θ), θ ∈ [−h, 0),

∂∂tz(t, 0) = −z(t,−h).

TDS may be studied in the framework of abstract infinite-dimensional systems in theHilbert/Banach spaces

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

The simplest equation with a single delay h has a form

x(t) = Ax(t) +A1x(t− h) + f(t), (9)

where h > 0, x(t) ∈ Rn, A and A1 are constant matrices, f : [0,∞)→ Rn is agiven piecewise-continuous function. The initial condition is given by

x(θ) = φ(θ), θ ∈ [−h, 0], (10)

where φ is supposed to be a piecewise-continuous initial function.The solution to the finite-dimensional (9) with A1 = 0 is given by

x(t) = eAtφ(0) +∫ t

0eA(t−s)f(s)ds. (11)

In order to extend (11) to A1 6= 0 we define the fundamental n× n-matrix whichsatisfies the homogenous equation

x(t) = Ax(t) +A1x(t− h) (12)

with the following initial conditions

X(t) =

{0, t < 0,I, t = 0.

Then the solution to (9), (10) is given by [Bellman & Cooke, 1963]

x(t) = X(t)φ(0) +∫ 0−hX(t− θ− h)A1φ(θ)dθ+

∫ t0 X(t− s)f(s)ds.

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

x(t) = A(t)x(t) +A1(t)x(t− h) +B(t)u(t).

(i) Controllable on [t0, t1] if ∀xt0 = x(t0 + ·) ∈ C[−h, 0], ∀xt1 ∈ C[−h, 0] ∃ apiecewise-cont. u(t)(ii) Controllability to origin if xt1 ≡ 0.Unlike the non-delay case,

I (ii) does not imply (i)I t1 − t0 > h

Criteria:I In terms of the Grammian [L. Weiss, 1967]I For LTI an algebraic one [Kirillova & Churakova, 1967]

Input delay and predictor-based design. The LTI system with the delayed input:

x(t) = Ax(t) +Bu(t− h). (13)

Objective: seek a state-feedback controller (for stabilization, LQR, etc). Denote v(t) = u(t− h)and find v(t) = Kx(t) for non-delay systemx(t) = Ax(t) + Bv(t) [Manitius & Olbrot, 1979].Then u(t) = Kx(t+ h) and

x(t+ h) = eAhx(t) +

∫ t+h

t

eA(t+h−s)Bu(s− h)ds.⇒ (14)

u(t) = K[eAhx(t) +

∫ 0

−he−AξBu(t+ ξ)dξ]. (15)

Note that in (13) the proper state is (x(t),ut(θ)), θ ∈ [−h, 0).

I Drawbacks of predictor-based design: difficulties in the case of uncertainsystems and delays=⇒ For uncertain systems, the LMI approach leads toefficient design algorithms.

I Systems with state delay usually lead to infinite-dimensional conditions.Thus, LQR

x(t) = Ax(t− h) +Bu(t),J =

∫∞0 [xT (t)Qx(t) + uT (t)Ru(t)]dt, Q ≥ 0, R > 0 (16)

leads to PDEs of Riccati type [N. Krasovskii, 1962].

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

Retarded systems

Retarded system with N discrete delays and with a distributed delay has a form:

x(t) =

N∑k=0

Akx(t− hk) +∫ 0

−hd

A(θ)x(t+ θ)dθ, (RS)

where 0 = h0 < h1... < hK , x(t) ∈ Rn, Ak are constant matrices and A(θ) is a continuous matrixfunction.The characteristic equation of this system is given by

det[λI −N∑k=0

Ake−λhk −

∫ 0

−hd

A(s)eλsds] = 0. (17)

I Location of the characteristic roots has a nice property:there is a finite number of roots to the right of any vertical line.

Im s

Re s

I Retarded system is as. stable iff all the roots in the LHP.I Solutions are given by x(t) =

∑j pj(t)e

λj (t),where λj are the characteristic roots and pj(t) are polynomials.

Consider a scalar TDS, which is stable for h = 0⇒ for small h > 0

x(t) = −ax(t)− a1x(t− h), a+ a1 > 0. (18)I For a ≥ |a1| (18) is as. stable for all delay, i.e. delay-independently stable.I If a1 > |a|, then the system is as. stable for h < h∗ and becomes unstable forh > h∗, where

h∗ =arccos(− a

a1)√

a21 − a2

.

I If a = 0, h∗ = π2a1

. For time-varying h(t), h∗ = 32a1

.

Neutral systems

NEUTRAL systems in Jack Hale’s formd

dt[x(t)− Fx(t− h)] = Ax(t) +A1x(t− h), x0 = φ ∈ C[−h, 0]. (NS)

Another formx(t)− F x(t− h) = Ax(t) +A1x(t− h), x0 = φ, φ ∈ L2[−h, 0]. (NS1)

Zero location:

Im s

Re s0

I There exist systems with all zeros in the LHP with unbounded solutionsI Small delays in neutral systems may destabilize the system:

x(t) + x(t) = a[x(t− h) + x(t− h)],

a > 1. Characteristic quasipolynomial

(λ+ 1)(1− ae−hλ),

zk = 1/h(ln a+ 2kπi) ∈ RHP , k = 0,±1, ...

In PID controller ”D” may stand for disaster.I Assume

A1: D(xt)=x(t)−Fx(t−g)=0 is as. stable ∀g.A1 ⇐⇒ σ(F ) < 1.Under A1 stability of neutral systems is similar to stab. of retarded

I Robustness of stability with respect to small delays: If (RS) is as. stable forh = 0, then it is as. stable for all small enough h. El’sgol’ts & Norkin, 1973,Hale & Lunel, 1993).

I In the case of infinite-dimensional systems (e.g. wave eq. or neutral system),even arbitrarily small delays in the feedback may destabilize the system (Datko 88; Logemann, Rebarber & G. Weiss 96; Hale & Lunel 99).

I A finite dimensional system that may be destabilized by small delay is asingularly perturbed system [E. Fridman, Aut02]:

εx(t) = −x(t− h), ε > 0

As. stable for h = 0, but for small h = εg with g > π/2 unstable. This led to aDescriptor approach to TDS[ Fridman, SCL01], [ Fridman & Shaked, TAC02]

Plan

1 Models with time-delay and effects of delay on stability

2 A brief history of Time-Delay System (TDS)

3 Solution concept, the step method and the state of TDS

4 Solution to linear TDS and fundamental matrix

5 On controllability, observability, LQR and Kalman filter

6 Linear Time-Invariant (LTI) systems and characteristic equation

7 Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

Lyapunov-Krasovskii and Lyapunov-Razumikhin methods

For TDS there are two main direct Lyapunov methods:1. Krasovskii method of Lyapunov functionals [Krasovskii, 1956]2. Razumikhin method of Lyapunov functions [Razumikhin,1956]

I Krasovskii method is a natural generalization of the direct Lyapunov method: aproper state for TDS is a function.

I Lyapunov-Razumikhin functions are simpler to use.

Delay-independent conditions

Consider TDS with time-varying bounded delay τ (t) ∈ [0,h]:

x(t) = Ax(t) +A1x(t− τ (t)), x(t) ∈ Rn (19)

(24) is as. stable if ∃V (x(t)) > 0 such that along (24)

d

dtV ≤ −α|x(t)|2, α > 0. (20)

Differentiate Lyapunov function V (x(t)) = xT (t)Px(t) along (24)

ddtV (x(t)) = 2xT (t)Px(t) = 2xT (t)P [Ax(t) +A1x(t− τ (t))]

= [xT (t) xT (t− τ )][ATP + PA PA1

AT1 P 0

] [x(t)x(t− τ )

]In order to guarantee (20) we need to ”compensate” x(t− τ (t))

Krasovskii method

V (t,xt) = xT (t)Px(t) +∫ tt−τ (t) x

T (s)Qx(s)ds, P > 0, Q > 0.

V : R×C[−h, 0]→ R+ is a functional, xt∆= x(t+ θ), θ ∈ [−h, 0].

Delay τ : differentiable, τ ≤ d < 1 ( slowly-varying delay).

ddt

∫ tt−τ (t) x

T (s)Qx(s)ds ≤ xT (t)Qx(t)− (1− d)xT (t− τ )Qx(t− τ )ddtV ≤ < −α|x(t)|

2, α > 0

if the delay-independent (h-independent) LMI holds:

W =

[ATP + PA+Q PA1

AT1 P −(1− d)Q

]< 0. (21)

Razumikhin methodddtV (x(t)) < −α|x(t)|2 along (24) if Razumikhin’s condition holds:

V (x(t+ θ)) < pV (x(t)) for some p > 1The idea: if a solution begins inside the ellipsoid xT (t0)Px(t0) ≤ δ, and is to leavethis ellipsoid at some time t, then

xT (t+ θ)Px(t+ θ) ≤ xT (t)Px(t), ∀θ ∈ [−h, 0].

0( )x t

( )x t

Razumikhin method

For any q > 0, and p > 1

ddtV (x(t)) = 2xT (t)Px(t) ≤ 2xT (t)P [Ax(t) +A1x(t− τ (t))]+q [pxT (t)Px(t)− x(t− τ (t))TPx(t− τ (t))]︸ ︷︷ ︸

≥0 by Razumikhin condition

= [xT (t) xT (t− τ (t))]WR

[x(t)x(t− τ (t))

]< −α|x(t)|2, α > 0

ifWR =

[ATP + PA+ qpP PA1

AT1 P −qP

]< 0. (22)

Sufficient: WR|p=1 < 0 (⇒WR|p=1+ε < 0 for small ε > 0)

Implications of the Delay-Independent MIs

[ATP + PA+ qP PA1

AT1 P −qP

]< 0⇔

[ATP + PA+ qP −PA1

−AT1 P −qP

]< 0

Hence, delay-ind MIs guarantee the stability of

x(t) = Ax(t)±A1x(t− τ (t)), ∀τ (t)⇒ (23)

I A is Hurwitz (since PA+ATP < 0);I A±A1 are Hurwitz (corresponds to τ ≡ 0 in (23));I σ(A

−1A1√1−d ) < 1(Krasovskii) and σ(A−1A1) < 1 (Razumikhin)

Delay-dependent stability of linear systems with tvr delay

Consider TDS with a time-varying bounded delay τ (t) ∈ [0,h]:

x(t) = Ax(t) +A1x(t− τ (t)), x(t) ∈ Rn. (24)

The stability conditions may beI delay-independent (h-independent) ⇒ A is Hurwitz ⇒ not applicable for

stabilization of unstable systems by the delayed feedback;I delay-dependent.

Delay-dependent conditions use the relation

x(t− τ (t)) = x(t)−∫ t

t−τ (t)x(s)ds,

which leads tox(t) = [A+A1]x(t)−A1

∫ t

t−τ (t)x(s)ds.

Delay-dependent stability of linear systems with tvr delay

Consider TDS with a time-varying bounded delay τ (t) ∈ [0,h]:

x(t) = Ax(t) +A1x(t− τ (t)), x(t) ∈ Rn. (24)

The stability conditions may beI delay-independent (h-independent) ⇒ A is Hurwitz ⇒ not applicable for

stabilization of unstable systems by the delayed feedback;I delay-dependent.

Delay-dependent conditions use the relation

x(t− τ (t)) = x(t)−∫ t

t−τ (t)x(s)ds,

which leads tox(t) = [A+A1]x(t)−A1

∫ t

t−τ (t)x(s)ds.

Descriptor methodDelay-dependent cond-s were based on model transformations and on bounding[Li & De Souza et al., 1995], [Goubet, Dambrine & Richard, 1995],[Kolmanovskii & Richard, 1999], [Niculescu, 2001].

1-st Model Transformtaion

x(t) = [A+A1]x(t)−A1

∫ t

t−τ (t)[Ax(s) +A1x(s−τ (s))]ds. (25)

Conservatism: (25) (with double delay) is NOT ⇔ to (24).[Kharitonov & Melchor-Aguilar, 2000], [Gu & Niculescu, 2001]

A descriptor model transformation [Fridman SCL 2001]:

x(t) = y(t), y(t) = Ax(t) +A1x(t− τ (t)).

The equivalent descriptor form (in the sense of stability)

x(t) = y(t),0 = −y(t) + (A+A1)x(t)−A1

∫ tt−τ (t) y(s)ds.

Descriptor methodDelay-dependent cond-s were based on model transformations and on bounding[Li & De Souza et al., 1995], [Goubet, Dambrine & Richard, 1995],[Kolmanovskii & Richard, 1999], [Niculescu, 2001].

1-st Model Transformtaion

x(t) = [A+A1]x(t)−A1

∫ t

t−τ (t)[Ax(s) +A1x(s−τ (s))]ds. (25)

Conservatism: (25) (with double delay) is NOT ⇔ to (24).[Kharitonov & Melchor-Aguilar, 2000], [Gu & Niculescu, 2001]

A descriptor model transformation [Fridman SCL 2001]:

x(t) = y(t), y(t) = Ax(t) +A1x(t− τ (t)).

The equivalent descriptor form (in the sense of stability)

x(t) = y(t),0 = −y(t) + (A+A1)x(t)−A1

∫ tt−τ (t) y(s)ds.

Descriptor method

V = xTPx, P > 0⇒ Novelty in V : x is not substituted by the RHS of eq.:

V = 2xT (t)Px(t)+2[xT (t)PT2 + xT (t)PT3 ][−x(t) + (A+A1)x(t)−A1

∫ tt−τ (t) x(s)ds]⇒

V < −α(|x(t)|2 + |x(t)|2), α > 0.

Leads to ”slack variables” P2 and P3.

Advantages of the descriptor method:I less conservative conditions for uncertain systems,I efficient design (with P3 = εP2 , ε ∈ R is tuning parameter),I ”unifying” LMIs for the discrete-time & the cont. systems;I simple conditions for neutral systems,I simple delay-dep. LMIs for diffusion PDEs.

Descriptor method

V = xTPx, P > 0⇒ Novelty in V : x is not substituted by the RHS of eq.:

V = 2xT (t)Px(t)+2[xT (t)PT2 + xT (t)PT3 ][−x(t) + (A+A1)x(t)−A1

∫ tt−τ (t) x(s)ds]⇒

V < −α(|x(t)|2 + |x(t)|2), α > 0.

Leads to ”slack variables” P2 and P3.

Advantages of the descriptor method:I less conservative conditions for uncertain systems,I efficient design (with P3 = εP2 , ε ∈ R is tuning parameter),I ”unifying” LMIs for the discrete-time & the cont. systems;I simple conditions for neutral systems,I simple delay-dep. LMIs for diffusion PDEs.

x = Ax, A =m∑i=1

fiAi,m∑i=1

fi = 1

V = xT (t)Px(t), P > 0. Adding to ddtV = 2xT (t)Px(t)

0 = 2[xT (t)PT2 + xT (t)PT3 ][−x(t) +Ax(t)],

we arrive at ddtV = [xT xT ]Ψ[xT xT ]T < 0 if

Ψ =[

P T2 A + AT P2 P − P T

2 + AT P3∗ −P3 − P T

3

]< 0.

LMIs in the vertices with different P (i)[PT2 Ai +ATi P2 P (i) − PT2 +ATi P3

∗ −P3 − PT3

]< 0.

Example: robust stability of uncertain system

Consider the uncertain system from Example 1:

x(t) =

[0 1

−1 + g −1− g

]x(t), |g| ≤ g.

I From the char. eq.: as. stable for all |g| < 1.I The quadratic stability conditions PAi +ATi P < 0, i = 1, 2 at the two vertices

that correspond to ±g: |g| ≤ 0.6812.I The descriptor LMI with the vertex-dependent P (1) and P (2): analytical bound|g| ≤ 0.9999.

Descriptor method: discrete-time case

Extension to the discrete-time: LMIs are almost like the continuous

x(k+ 1) = Ax(k)⇒Descriptor form: x(k + 1) = y(k) + x(k), y(k) = (A− I)x(k)

Vn(k) = xT (k)Px(k), P > 0⇒Vn(k+ 1)− Vn(k) = 2xT (k)Py(k) + yT (k)Py(k)+2[xT (k)PT2 + yT (k)PT3 ][−y(k) + (A− I)x(k)]

we arrive at LMIs which are similar to the continuous ones[PT2 (A− I) + (A− I)TP2 P − PT2 + (A−I)TP3

∗ P − P3 − PT3

]< 0.

Delay-dependent via Lyapunov-Krasovskii

We differentiate xT (t)Px(t) and apply the descriptor methodddt[xT (t)Px(t)] = 2xT (t)P x(t)

+2[xT (t)PT2 + xT (t)PT3 ][(A+A1)x(t)−A1∫ tt−τ x(s)ds− x(t)]

with some n× n-matrices P2,P3.

To ”compensate”∫ tt−τ (t) x(s)ds consider [Fridman & Shaked, IJC 03]:

VR =∫ 0

−h

∫ tt+θ

xT (s)Rx(s)dsdθ

=∫ tt−h(h+ s− t)xT (s)Rx(s)ds, R > 0,

ddtVR = hxT (t)Rx(t)−

∫ tt−h x

T (s)Rx(s)ds

= hxT (t)Rx(t)−∫ tt−τ (t) x

T (s)Rx(s)ds−∫ t−τ (t)

t−hxT (s)Rx(s)ds︸ ︷︷ ︸

is ignored

Delay-dependent via Lyapunov-Krasovskii

We differentiate xT (t)Px(t) and apply the descriptor methodddt[xT (t)Px(t)] = 2xT (t)P x(t)

+2[xT (t)PT2 + xT (t)PT3 ][(A+A1)x(t)−A1∫ tt−τ x(s)ds− x(t)]

with some n× n-matrices P2,P3.

To ”compensate”∫ tt−τ (t) x(s)ds consider [Fridman & Shaked, IJC 03]:

VR =∫ 0

−h

∫ tt+θ

xT (s)Rx(s)dsdθ

=∫ tt−h(h+ s− t)xT (s)Rx(s)ds, R > 0,

ddtVR = hxT (t)Rx(t)−

∫ tt−h x

T (s)Rx(s)ds

= hxT (t)Rx(t)−∫ tt−τ (t) x

T (s)Rx(s)ds−∫ t−τ (t)

t−hxT (s)Rx(s)ds︸ ︷︷ ︸

is ignored

Delay-dependent via descriptor method

We apply further Jensen’s inequality (for τ > 0) [Gu, 03]

−∫ tt−τ (t) x

T (s)Rx(s)ds ≤ − 1τ (t)

∫ tt−τ (t) x

T (s)dsR∫ tt−τ (t) x(s)ds

≤ − 1h

∫ tt−τ (t) x

T (s)dsR∫ tt−τ (t) x(s)ds.

Then, for Lyapunov functional

V = xT (t)Px(t) + VR

andη(t) = col{x(t), x(t), 1

h

∫ t

t−τx(s)ds}

⇒ ddtV ≤ η

T (t)Ψη(t) < 0 if

Ψ =

[P T

2 (A + A1) + (A + A1)T P2 P − P T

2 + (A + A1)T P3 −hP T

2 A1∗ −P3 − P T

3 + hR −hP T3 A1

∗ ∗ −hR

]< 0.

Delay-dependent methods

I Delay-dependent without the descriptor:x(t) is replaced by RHS of eq. +Schur complement to xTRx

I Important improvements:I [Y. He et al. Aut 07]:

The relation between x(t− τ (t)) and x(t− h) is taken into account:

V (xt,xt) = xT (t)Px(t) +∫ t

t−hxT (s)Sx(s)ds+ VR

I [P.G. Park et al. Aut11]: convex analysis.I Treating stabilizing delay (no stability without delay) via LMIs:

I [K. Gu, IJC97]: discretized Lyapunov functional method;I [Seuret & Gouaisbaut Aut13]: extended integral inequalities.

Illustrative example

x(t) = −x(t− τ (t)), τ ≤ d < 1.

I LMIs with d = 0 guarantee stability for constant τ ∈ [0, 1.41] [Fr& Sh 02](compared with the analytical result τ < 1.57)

I For fast-varying delays the analytical result τ < 1.5LMIs of [Fr& Sh 02] S = 0⇒ τ (t) ∈ [0, 0.99]LMIs of [He et al 07] S 6= 0⇒ τ (t) ∈ [0, 1.22]Convex analysis [Park et al. 11] τ (t) ∈ [0, 1.33]

Conditions in terms of LMIs are sufficient only ⇒ they can be improved.

General Lyapunov functional for constant delays

Nec. condition for the application of simple Lyapunov functionals is the stability ofthe non-delayed system x(t) = (A+A1)x(t).

x(t) =

[0 1−2 0.1

]x(t) +

[0 01 0

]x(t− h), h constant

This system is unstable for h = 0 and is as. stable for h ∈ (0.1002, 1.7178) [Gu03].Here stabilization by using delay.⇒ A general Lyapunov functional that corresponds to necessary and sufficientconditions for stability [Y. M. Repin, 1965]

V (xt) = x(t)TPx(t) + 2xT (t)∫ 0−hQ(ξ)x(t+ ξ)dξ

+∫ 0−h∫ 0−h x

T (t+ s)R(s, ξ)x(t+ ξ)dsdξ(26)

Let x(t) = Ax(t) +A1x(t− h) be as. stable. Find V :

ddtV (xt) = −xT (t)Wx(t), W > 0,x(t) = X(t)φ(0) +

∫ 0−hX(t− θ− h)A1φ(θ)ds⇒

V (φ) =∫∞

0 xT (s)Wx(s)ds = φT (0)U(0)φ(0)+2φT (0)

∫ 0−h U(−h− θ)A1φ(θ)dθ

+∫ 0−h φ(θ2)AT1

∫ 0−h U(θ2 − θ1)A1φ(θ1)dθ1dθ2,

U(θ) =

∫ ∞0

XT (s)WX(s+ θ)ds <∞⇒

V (φ) > β|φ(0)|3, β > 0 [Huang 89]ddtV = −xT (t)Wx(t)−xT(t− h)W1x(t− h), W ,W1 > 0⇒ V (φ) > β|φ(0)|2 ⇒complete LKF [Kharitonov & Zhabko, 03].

Let x(t) = Ax(t) +A1x(t− h) be as. stable. Find V :

ddtV (xt) = −xT (t)Wx(t), W > 0,x(t) = X(t)φ(0) +

∫ 0−hX(t− θ− h)A1φ(θ)ds⇒

V (φ) =∫∞

0 xT (s)Wx(s)ds = φT (0)U(0)φ(0)+2φT (0)

∫ 0−h U(−h− θ)A1φ(θ)dθ

+∫ 0−h φ(θ2)AT1

∫ 0−h U(θ2 − θ1)A1φ(θ1)dθ1dθ2,

U(θ) =

∫ ∞0

XT (s)WX(s+ θ)ds <∞⇒

V (φ) > β|φ(0)|3, β > 0 [Huang 89]ddtV = −xT (t)Wx(t)−xT(t− h)W1x(t− h), W ,W1 > 0⇒ V (φ) > β|φ(0)|2 ⇒complete LKF [Kharitonov & Zhabko, 03].

LMI conditions via general V and discretization were found in Gu 97]. No designproblems have been solved by this method due to some terms in V , which arise aftersubstitution of x(t).Descriptor discretized method [Fridman 06] avoids the substitution:

V (xt) = 2xT (t)∫ 0−hQ(ξ)x(t+ ξ)dξ

+2xT (t)[Px(t) +∫ 0−hQ(ξ)x(t+ ξ)dξ]

+2∫ 0−h∫ 0−h x

T (t+ s)R(s, ξ)dsx(t+ ξ)dξ

+2∫ 0−h x

T (t+ ξ)S(ξ)x(t+ ξ)dξ

+2[xT(t)PT2 + xT(t)PT

3 ][Ax(t)+A1x(t− h)]

Integrating by parts + discretization of Gu leads to LMIs for design.

I Treating stabilizing delay (no stability without delay) via LMIs:I [K. Gu, IJC97]: discretized Lyapunov functional method;I [Seuret & Gouaisbaut Aut13]: extended integral inequalities.

Some open problems:I Suf. stability conds taking into account particular τ (t);I analytical stability bounds and nec. Lyapunov-based stability conds for

some classes of tvr delays.

For detailed introduction to time-delay systems with applications tosampled-data and network-based control see

Thank you!