Introduction to Time-Delay Systems. Emilia FRIDMAN

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

The simplest time-delay equation with constant delay h > 0 is ẋ(t) = x(t h). Time-Delay Systems (TDS) are also called systems with aftereffect or dead-time, hereditary systems, or differential-difference equations. They belong to the class of functional differential equations which are infinite-dimensional. Time-delays appear in many engineering systems - aircraft, chemical control systems, in lasers models, in internet, biology, medicine... There can be transport, communication or measurement delay.

Figure 1: Showering person.

A person wishes to achieve the desired water tempreture T d by rotating the mixer. Let T (t) denote the water temperature in the mixer output. Let h be the time needed by the water to go from the mixer output to the person s head. 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 proportional to T (t) T d. At time t the person feels the water temperature leaving the mixer at time t h, which results in the following equation with the constant delay h: T (t) = k[t (t h) T d ], k R.

Effects of delay on stability Delay may be destabilizing: ẋ(t) = x(t h) is stable for h < π/2 and unstable for h > π/2. Delay may be stabilizing: ẍ(t) = u(t), y(t) = x(t). The system is not stabilizable by the non-delayed u(t) = K 0 y(t)

The system is stabilizable by delayed [Niculescu, Michiels, Kharitonov 04] u(t) = K 0 y(t) + K 1 y(t r), r > 0. (1) since it is stabilizable by u(t) = K 0 y(t) + K 1 ẏ(t) and ẏ(t) y(t) y(t r), r > 0. r

The system is stabilizable by delayed [Niculescu, Michiels, Kharitonov 04] u(t) = K 0 y(t) + K 1 y(t r), r > 0. (1) since it is stabilizable by u(t) = K 0 y(t) + K 1 ẏ(t) and A time-delay approach to: ẏ(t) y(t) y(t r), r > 0. r DANCES [Fridman, Tutorial on TDS, Israel, 2011] Destabilizing delay in Venice Waltz Stabilizing delay in Tango

A time-delay approach to: Sampled-data control [Mikheev, Sobolev & Fridman, 1988] ẋ(t) = Ax(t) + BKx(t k ), t [t k, t k+1 ), lim t k = (2) k This system can be represented as a continuous system with time-varying delay τ(t) = t t k (note that τ = 1 for t = t k ) ẋ(t) = Ax(t) + BKx(t τ(t)), t [t k, t k+1 ). (3)

A time-delay approach became popular in Networked Control System is feedback control loop closed through communication network. ZOH Delay u(t) u(tk)=kx(tk ηk) Physical plant Network Medium Controller x Sampler Delay Figure 2: Networked state-feedback control system.

Packet dropouts result in variable sampling. The closed-loop ẋ(t) = Ax(t) + B 2 Kx(t k η k ), t [t k, t k+1 ). Input delay approach ẋ(t) = Ax(t) + B 2 Kx(t τ(t)), (4) where and. 0 τ(t) = t t k + η k t k+1 t k + η k τ M, t [t k, t k+1 ) τ(t) = 1, t = t k

Figure 3: Drilling pipe model

The drilling pipe can be modeled by the wave equation GJ 2 z L 2 σ 2 (σ, t) I 2 z z B (σ, t) β (σ, t) = 0, σ [0, 1], t2 t under the boundary conditions z(0, t) = 0, GJ L z σ (1, t) + I B 2 z t 2 (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, I B 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 z t (1, t).

Assuming β = I B = 0, the model reduces to 1-D wave equation where a = 2 z t 2 (σ, t) = a 2 z σ 2 (σ, t), σ (0, 1), t > 0, z z z(0, t) = 0, σ (1, t) = k t (1, t), (5) GJ I B L 2, k = GJ LT, r = GJ L 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 C 1 1 and s = a. This leads to time-delay system with c 0 = s k s+k and c 1 = ψ(t + s) = c 0 ψ(t s), t > 0, (7) r s+k.

18 cent.: 1-st eqs with delay by Bernoulli, Euler, Concordet 1940-... Systematical study by A. Myshkis, R. Bellman 1956-... Lyapunov method for stability by Krasovskii and by Razumikhin. Smith predictor. 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) 1995-... Robust control of systems with uncertain delay τ(t) 2000-... Delay boom

Solution concept. Initial value problem: ẋ(t) = x(t h), t 0, x(s) = φ(s), s [ h, 0]. (8) Step method for solution: h 0 h 2h t t [0, h], ẋ(t) = φ(t h), x(0) = φ(0), t [h, 2h], t [2h, 3h],...

1 0.5 x 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 t 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 simple infinite-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 + θ) = z(t, θ), θ [ h, 0] ẋ(t) = x(t h) t z(t, θ) = θ z(t, θ), θ [ h, 0), z(t, 0) = z(t, h). t TDS may be studied in the framework of abstract infinite-dimensional systems in the Hilbert/Banach spaces

The simplest equation with a single delay h has a form ẋ(t) = Ax(t) + A 1 x(t h) + f(t), (9) where h > 0, x(t) R n, A and A 1 are constant matrices, f : [0, ) R n is a given 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 A 1 = 0 is given by x(t) = e At φ(0) + t 0 e A(t s) f(s)ds. (11)

In order to extend (11) to A 1 = 0 we define the fundamental n n-matrix which satisfies the homogenous equation with the following initial conditions ẋ(t) = Ax(t) + A 1 x(t h) (12) 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 h X(t θ h)a 1φ(θ)dθ + t 0 X(t s)f(s)ds.

ẋ(t) = A(t)x(t) + A 1 (t)x(t h) + B(t)u(t). (i) Controllable on [t 0, t 1 ] if x t0 = x(t 0 + ) C[ h, 0], x t1 C[ h, 0] a piecewise-cont. u(t) (ii) Controllability to origin if x t1 0. Unlike the non-delay case, (ii) does not imply (i) t 1 t 0 > h Criteria: In terms of the Grammian [L. Weiss, 1967] For LTI an algebraic one [Kirillova & Churakova, 1967]

Input delay and predictor-based design. The LTI system with the delayed input: ẋ(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 system ẋ(t) = Ax(t) + Bv(t) [Manitius & Olbrot, 1979]. Then u(t) = Kx(t + h) and x(t + h) = e Ah x(t) + u(t) = K[e Ah x(t) + h Note that in (13) the proper state is (x(t), u t(θ)), θ [ h, 0). t+h e A(t+h s) Bu(s h)ds. (14) t 0 e Aξ Bu(t + ξ)dξ]. (15)

Drawbacks of predictor-based design: difficulties in the case of uncertain systems and delays= For uncertain systems, the LMI approach leads to efficient design algorithms. Systems with state delay usually lead to infinite-dimensional conditions. Thus, LQR ẋ(t) = Ax(t h) + Bu(t), J = 0 [xt (t)qx(t) + u T (t)ru(t)]dt, Q 0, R > 0 (16) leads to PDEs of Riccati type [N. Krasovskii, 1962].

Retarded systems Retarded system with N discrete delays and with a distributed delay has a form: N 0 ẋ(t) = A k x(t h k ) + A(θ)x(t + θ)dθ, k=0 h d (RS) where 0 = h 0 < h 1... < h K, x(t) R n, A k are constant matrices and A(θ) is a continuous matrix function. The characteristic equation of this system is given by det[λi N k=0 0 A k e λh k A(s)e λs ds] = 0. h d (17)

Location of the characteristic roots has a nice property: there is a finite number of roots to the right of any vertical line. Ims Res Retarded system is as. stable iff all the roots in the LHP. Solutions are given by x(t) = j p j(t)e λ j(t), where λ j are the characteristic roots and p j (t) are polynomials.

Consider a scalar TDS, which is stable for h = 0 for small h > 0 ẋ(t) = ax(t) a 1 x(t h), a + a 1 > 0. (18) For a a 1 (18) is as. stable for all delay, i.e. delay-independently stable. If a 1 > a, then the system is as. stable for h < h and becomes unstable for h > h, where h = arccos( a a 1 ). a1 2 a2 If a = 0, h = π 2a 1. For time-varying h(t), h = 3 2a 1.

Neutral systems NEUTRAL systems in Jack Hale s form d dt [x(t) F x(t h)] = Ax(t) + A 1x(t h), x 0 = φ C[ h, 0]. (NS) Another form ẋ(t) F ẋ(t h) = Ax(t) + A 1 x(t h), x 0 = φ, φ L 2 [ h, 0]. (NS1) Zero location: Ims 0 Res

There exist systems with all zeros in the LHP with unbounded solutions Small delays in neutral systems may destabilize the system: a > 1. Characteristic quasipolynomial In PID controller D may stand for disaster. ẋ(t) + x(t) = a[ẋ(t h) + x(t h)], (λ + 1)(1 ae hλ ), z k = 1/h(ln a + 2kπi) RHP, k = 0, ±1,... Assume A1: D(x t) = x(t) F x(t g) = 0 is as. stable g. A1 σ(f ) < 1. Under A1 stability of neutral systems is similar to stab. of retarded

Robustness of stability with respect to small delays: If (RS) is as. stable for h = 0, then it is as. stable for all small enough h. El sgol ts & Norkin, 1973, Hale & Lunel, 1993). 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). A finite dimensional system that may be destabilized by small delay is a singularly perturbed system [E. Fridman, Aut02]: εẋ(t) = x(t h), ε > 0 As. stable for h = 0, but for small h = εg with g > π/2 unstable. This led to a Descriptor approach to TDS [ Fridman, SCL01], [ Fridman & Shaked, TAC02]

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] Krasovskii method is a natural generalization of the direct Lyapunov method: a proper state for TDS is a function. Lyapunov-Razumikhin functions are simpler to use.

Delay-independent conditions Consider TDS with time-varying bounded delay τ(t) [0, h]: ẋ(t) = Ax(t) + A 1 x(t τ(t)), x(t) R n (19) (24) is as. stable if V (x(t)) > 0 such that along (24) d dt V α x(t) 2, α > 0. (20) Differentiate Lyapunov function V (x(t)) = x T (t)p x(t) along (24) d dt V (x(t)) = 2xT (t)p [ ẋ(t) = 2x T (t)p [Ax(t) + A 1 x(t τ(t))] = [x T (t) x T A (t τ)] T ] [ ] P + P A P A 1 x(t) A T 1 P 0 x(t τ) In order to guarantee (20) we need to compensate x(t τ(t))

Krasovskii method V (t, x t ) = x T (t)p x(t) + t t τ(t) xt (s)qx(s)ds, P > 0, Q > 0. V : R C[ h, 0] R + is a functional, x t = x(t + θ), θ [ h, 0]. Delay τ: differentiable, τ d < 1 ( slowly-varying delay). d dt t t τ(t) xt (s)qx(s)ds x T (t)qx(t) (1 d)x T (t τ)qx(t τ) d dt V < α x(t) 2, α > 0 if the delay-independent (h-independent) LMI holds: [ A W = T ] P + P A + Q P A 1 A T 1 P < 0. (21) (1 d)q

Razumikhin method d dt V (x(t)) < α x(t) 2 along (24) if Razumikhin s condition holds: V (x(t + θ)) < pv (x(t)) for some p > 1 The idea: if a solution begins inside the ellipsoid x T (t 0 )P x(t 0 ) δ, and is to leave this ellipsoid at some time t, then x T (t + θ)p x(t + θ) x T (t)p x(t), θ [ h, 0]. x( t) x( t0)

Razumikhin method For any q > 0, and p > 1 if d dt V (x(t)) = 2xT (t)p ẋ(t) 2x T (t)p [Ax(t) + A 1 x(t τ(t))] +q [px T (t)p x(t) x(t τ(t)) T P x(t τ(t))] }{{} 0 by Razumikhin [ condition ] x(t) = [x T (t) x T (t τ(t))]w R < α x(t) x(t τ(t)) 2, α > 0 [ A W R = T ] P + P A + qpp P A 1 A T 1 P qp < 0. (22) Sufficient: W R p=1 < 0 ( W R p=1+ε < 0 for small ε > 0)

Implications of the Delay-Independent MIs [ A T ] [ P + P A + qp P A 1 A A T 1 P qp < 0 T P + P A + qp P A 1 A T 1 P qp Hence, delay-ind MIs guarantee the stability of ] < 0 ẋ(t) = Ax(t) ± A 1 x(t τ(t)), τ(t) (23) A is Hurwitz (since P A + A T P < 0); A ± A 1 are Hurwitz (corresponds to τ 0 in (23)); σ( A 1 1 d A 1 ) < 1(Krasovskii) and σ(a 1 A 1 ) < 1 (Razumikhin)

Delay-dependent stability of linear systems with tvr delay Consider TDS with a time-varying bounded delay τ(t) [0, h]: The stability conditions may be ẋ(t) = Ax(t) + A 1 x(t τ(t)), x(t) R n. (24) delay-independent (h-independent) A is Hurwitz not applicable for stabilization of unstable systems by the delayed feedback; delay-dependent.

Descriptor method Delay-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 ẋ(t) = [A + A 1 ]x(t) A 1 t t τ(t) Conservatism: (25) (with double delay) is NOT to (24). [Kharitonov & Melchor-Aguilar, 2000], [Gu & Niculescu, 2001] A descriptor model transformation [Fridman SCL 2001]: [Ax(s) +A 1 x(s τ(s))]ds. (25) ẋ(t) = y(t), y(t) = Ax(t) + A 1 x(t τ(t)). The equivalent descriptor form (in the sense of stability) ẋ(t) = y(t), 0 = y(t) + (A+ A 1 )x(t) A 1 t t τ(t) y(s)ds.

Descriptor method V = x T P x, P > 0 Novelty in V : ẋ is not substituted by the RHS of eq.: V = 2x T (t)p ẋ(t) +2[x T (t)p2 T + ẋt (t)p3 T ][ ẋ(t) + (A+ A t 1)x(t) A 1 t τ(t) ẋ(s)ds] V < α( x(t) 2 + ẋ(t) 2 ), α > 0. Leads to slack variables P 2 and P 3. Advantages of the descriptor method: less conservative conditions for uncertain systems, efficient design (with P 3 = εp 2, ε R is tuning parameter), unifying LMIs for the discrete-time & the cont. systems; simple conditions for neutral systems, simple delay-dep. LMIs for diffusion PDEs.

m m ẋ = Ax, A = f i A i, f i = 1 i=1 i=1 V = x T (t)p x(t), P > 0. Adding to dt d V = 2xT (t)p ẋ(t) 0 = 2[x T (t)p2 T + ẋ T (t)p3 T ][ ẋ(t) + Ax(t)], we arrive at d dt V = [xt ẋ T ]Ψ[x T ẋ T ] T < 0 if [ Ψ = P 2 T A + AT P 2 P P 2 T + AT P 3 P 3 P 3 T ] < 0. LMIs in the vertices with different P (i) [ P T 2 A i + A T i P 2 P (i) P T 2 + AT i P 3 P 3 P T 3 ] < 0.

Example: robust stability of uncertain system Consider the uncertain system from Example 1: [ ] 0 1 ẋ(t) = x(t), g ḡ. 1 + g 1 g From the char. eq.: as. stable for all g < 1. The quadratic stability conditions P A i + A T i P < 0, i = 1, 2 at the two vertices that correspond to ±ḡ: g 0.6812. 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) V n (k) = x T (k)p x(k), P > 0 V n (k + 1) V n (k) = 2x T (k)p y(k) + y T (k)p y(k) +2[x T (k)p2 T + yt (k)p3 T ][ y(k) + (A I)x(k)] we arrive at LMIs which are similar to the continuous ones [ P T 2 (A I) + (A I) T P 2 P P T 2 + (A I)T P 3 P P 3 P T 3 ] < 0.

Delay-dependent via Lyapunov-Krasovskii We differentiate x T (t)p x(t) and apply the descriptor method with some n n-matrices P 2, P 3. d dt [xt (t)p x(t)] = 2x T (t)p ẋ(t) +2[x T (t)p2 T + ẋt (t)p3 T ][(A+A t 1)x(t) A 1 ẋ(s)ds ẋ(t)] t τ t To compensate ẋ(s)ds consider [Fridman & Shaked, IJC 03]: t τ(t) 0 t V R = h t+θ ẋt (s)rẋ(s)dsdθ t = (h + s t h t)ẋt (s)rẋ(s)ds, R > 0, d dt V R = hẋ T t (t)rẋ(t) t h ẋt (s)rẋ(s)ds = hẋ T (t)rẋ(t) t t τ(t) ẋt (s)rẋ(s)ds t τ(t) ẋ T (s)rẋ(s)ds t h }{{} is ignored

Delay-dependent via descriptor method We apply further Jensen s inequality (for τ > 0) [Gu, 03] t t τ(t) ẋt (s)rẋ(s)ds 1 h 1 Then, for Lyapunov functional and d dt V ηt (t)ψη(t) < 0 if Ψ = τ(t) t t τ(t) ẋt (s)dsr t t τ(t) ẋ(s)ds. V = x T (t)p x(t) + V R η(t) = col{x(t), ẋ(t), 1 h t t τ(t) ẋt (s)dsr t t τ(t) ẋ(s)ds t t τ ẋ(s)ds} [ P T 2 (A + A1) + (A + A1)T P 2 P P T 2 + (A + A1)T P 3 hp T 2 A1 P 3 P T 3 + hr hp T 3 A1 hr ] < 0.

Delay-dependent methods Delay-dependent without the descriptor: ẋ(t) is replaced by RHS of eq. +Schur complement to ẋ T Rẋ Important improvements: [Y. He et al. Aut 07]: The relation between x(t τ(t)) and x(t h) is taken into account: V (x t,ẋ t ) = x T (t)p x(t) + t t h xt (s)sx(s)ds + V R [P.G. Park et al. Aut11]: convex analysis. Treating stabilizing delay (no stability without delay) via LMIs: [K. Gu, IJC97]: discretized Lyapunov functional method; [Seuret & Gouaisbaut Aut13]: extended integral inequalities.

Illustrative example ẋ(t) = x(t τ(t)), τ d < 1. LMIs with d = 0 guarantee stability for constant τ [0, 1.41] [Fr& Sh 02] (compared with the analytical result τ < 1.57) For fast-varying delays the analytical result τ < 1.5 LMIs of [Fr& Sh 02] S = 0 τ(t) [0, 0.99] LMIs of [He et al 07] S = 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 of the non-delayed system ẋ(t) = (A + A 1 )x(t). [ ẋ(t) = 0 1 2 0.1 ] [ 0 0 x(t) + 1 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 sufficient conditions for stability [Y. M. Repin, 1965] V (x t ) = x(t) T P x(t) + 2x T (t) 0 h Q(ξ)x(t + ξ)dξ + 0 0 h h xt (t + s)r(s, ξ)x(t + ξ)dsdξ (26)

Let ẋ(t) = Ax(t) + A 1 x(t h) be as. stable. Find V : d dt V (x t) = x T (t)w x(t), W > 0, x(t) = X(t)φ(0) + 0 h X(t θ h)a 1φ(θ)ds V (φ) = 0 x T (s)w x(s)ds = φ T (0)U(0)φ(0) +2φ T (0) 0 h U( h θ)a 1φ(θ)dθ + 0 h φ(θ 2)A T 0 1 h U(θ 2 θ 1 )A 1 φ(θ 1 )dθ 1 dθ 2, U(θ) = 0 X T (s)w X(s + θ)ds < V (φ) > β φ(0) 3, β > 0 [Huang 89] d dt V = xt (t)w x(t) x T (t h)w 1 x(t h), W, W 1 > 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 design problems have been solved by this method due to some terms in V, which arise after substitution of ẋ(t). Descriptor discretized method [Fridman 06] avoids the substitution: V (x t ) = 2x T (t) 0 h Q(ξ)ẋ(t + ξ)dξ +2ẋ T (t)[p x(t) + 0 h Q(ξ)x(t + ξ)dξ] +2 0 0 h h ẋt (t + s)r(s, ξ)dsx(t + ξ)dξ +2 0 h ẋt (t + ξ)s(ξ)x(t + ξ)dξ +2[x T (t)p T 2 + ẋt (t)p T 3 ][Ax(t)+A 1x(t h)] Integrating by parts + discretization of Gu leads to LMIs for design.

Treating stabilizing delay (no stability without delay) via LMIs: [K. Gu, IJC97]: discretized Lyapunov functional method; [Seuret & Gouaisbaut Aut13]: extended integral inequalities. Some open problems: Suf. stability conds taking into account particular τ(t); analytical stability bounds and nec. Lyapunov-based stability conds for some classes of tvr delays.

For detailed introduction to time-delay systems with applications to sampled-data and network-based control see Thank you!