One can easy verify that k 1 = Γ  2 v 1 = 0, k 2 = Γ  1 v 2 = 0, k 3 = Γ  2 v 3 = 0, k 4 = Γ  1 v 4 = 0. Denote M 1 =  I − k −1 1 v 1 Γ  2  e A 1 T 1 , M 2 =  I − k −1 2 v 2 Γ  1  e A 2 T 2 , M 3 =  I − k −1 3 v 3 Γ  2  e A 1 T 3 , M 4 =  I − k −1 4 v 4 Γ  1  e A 2 T 4 . and M =      4 ∏ i=1 M i      ≈ 0.3033 < 1. So, as ds k+1 1 = Mds k 1 , the periodic solution under consideration is orbitally asymptotically stable. Similar results can be obtained in case of nonlinearity (3). 6. Perturbed system Consider a system: ˙ x = Ax + c  ϕ(t)+u(t − τ)  , (10) where ϕ (t) is scalar T ϕ -periodic continuous function of time. Let f is given by (3). Consider a special case of the previous system (see Nelepin (2002), Kamachkin & Shamberov (1995)). Let n = 2, ¨ y + g 1 ˙ y + g 2 y = u(t − τ)+ϕ(t), (11) here y (t) ∈ R is sought-for time variable, g 1, 2 are real constants, σ = α 1 y + α 2 ˙ y, α 1, 2 are real constants. Let us rewrite system (11) in vector form. Denote z  =  y ˙ y  ,inthatcase ˙ z = Pz + q ( ϕ(t)+u( t − τ) ) , (12) u (t − τ)= f ( σ(t − τ) ) , σ = α  z, where P =  01 −g 2 −g 1  , q =  0 1  , α =  α 1 α 2  . Suppose that characteristic determinant D (s)=det ( P − sI ) has real simple roots λ 1, 2 ,and vectors q, Pq are linearly independent. In that case system (12) may be reduced to the form (10), where A =  λ 1 0 0 λ 2  , c =  1 1  , by means of nonsingular linear transformation z = Tx, T = ⎛ ⎝ N 1 ( λ 1 ) D  ( λ 1 ) N 1 ( λ 2 ) D  ( λ 2 ) N 2 ( λ 1 ) D  ( λ 1 ) N 2 ( λ 2 ) D  ( λ 2 ) ⎞ ⎠ , D  (λ j )= d ds D (s)     s=λ j , N j (s)= 2 ∑ i=1 q i D ij (s), (13) D ij (s) is algebraic supplement for element lying in the intersection of i-th row and j -th column of determinant D (s). 109 On Stable Periodic Solutions of One Time Delay System Containing Some Nonideal Relay Nonlinearities Note that σ = γ  x, γ = T  α. Furthermore, since γ i = −  D  ( λ i )  −1 2 ∑ j=1 α j N j ( λ i ) , i = 1, 2. then γ 1 = ( λ 1 − λ 2 ) −1 ( α 1 + α 2 λ 1 ) , γ 2 = ( λ 2 − λ 1 ) −1 ( α 1 + α 2 λ 2 ) . Transformation (13) leads to the following system:  ˙ x 1 = λ 1 x 1 + f ( σ(t − τ) ) + ϕ(t), ˙ x 2 = λ 2 x 2 + f ( σ(t − τ) ) + ϕ(t). (14) If, for example, α 1 = −λ 1 α 2 , then γ 1 = 0, γ 2 = α 2 , σ = γ 2 x 2 . Function f in that case is independent of variable x 1 ,and ˙ σ = λ 2 σ + γ 2 ( f ( γ 2 x 2 (t − τ)) + ϕ(t) ) . Solution of the latest equation when f = u (where u = m 1 , m 2 or 0) has the following form: σ ( t, t 0 , σ 0 , u ) = e λ 2 (t−t 0 ) σ 0 + γ 2 e λ 2 t  t t 0 e −λ 2 s  u + ϕ(s)  ds. Let us trace out necessary conditions for existing of periodic solution of the system (10), (3) having four switching points ˆ s i : σ 2 = σ ( t 1 , t 0 + τ, ˆ σ 1 ,0 ) , ˆ σ 2 = σ ( t 1 + τ, t 1 , σ 2 ,0 ) , σ 3 = σ ( t 2 , t 1 + τ, ˆ σ 2 , m 1 ) , ˆ σ 3 = σ ( t 2 + τ, t 2 , σ 3 , m 1 ) , σ 4 = σ ( t 3 , t 2 + τ, ˆ σ 3 ,0 ) , ˆ σ 4 = σ ( t 3 + τ, t 3 , σ 4 ,0 ) , σ 1 = σ ( t 4 , t 3 + τ, ˆ σ 4 , m 2 ) , ˆ σ 1 = σ ( t 4 + τ, t 4 , σ 1 , m 2 ) , for some positive T i , i = 1, 4, and t i = t i−1 + T i .Denoteu 1 = 0, u 2 = m 1 , u 3 = 0, u 4 = m 2 ,then σ i+1 = σ ( t i , t i−1 + τ, σ ( t i−1 + τ, t i−1 , σ i , u i−1 ) , u i ) = = e λ 2 (T i −τ)  e λ 2 τ σ i + γ 2 e λ 2 (t i−1 +τ)  t i−1 +τ t i−1 e −λ 2 t ( u i−1 + ϕ(t) ) dt  + + γ 2 e λ 2 t i  t i t i−1 +τ e −λ 2 t ( u i + ϕ(t) ) dt = e λ 2 T i σ i + K i , where K i = γ 2 e λ 2 t i   t i t i−1 e −λ 2 t ϕ(t) dt +  t i−1 +τ t i−1 e −λ 2 t u i−1 dt +  t i t i−1 +τ e −λ 2 t u i dt  . 110 Time-Delay Systems So, ⎛ ⎜ ⎜ ⎝ σ 1 σ 2 σ 3 σ 4 ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ 000e λ 2 T 4 e λ 2 T 1 000 0 e λ 2 T 2 00 00e λ 2 T 3 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ σ 1 σ 2 σ 3 σ 4 ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ K 1 K 2 K 3 K 4 ⎞ ⎟ ⎟ ⎠ and σ 1 =  1 − e λ 2 T  K 2 e λ 2 (T 2 +T 3 +T 4 ) + K 3 e λ 2 (T 3 +T 4 ) + K 4 e λ 2 T 4 + K 1  = l 0 , σ 2 =  1 − e λ 2 T  K 3 e λ 2 (T 1 +T 3 +T 4 ) + K 4 e λ 2 (T 1 +T 4 ) + K 1 e λ 2 T 1 + K 2  = −l, σ 3 =  1 − e λ 2 T  K 4 e λ 2 (T 1 +T 2 +T 4 ) + K 1 e λ 2 (T 1 +T 2 ) + K 2 e λ 2 T 2 + K 3  = −l 0 , σ 4 =  1 − e λ 2 T  K 1 e λ 2 (T 1 +T 2 +T 3 ) + K 2 e λ 2 (T 2 +T 3 ) + K 3 e λ 3 T 3 + K 4  = l, here T = T 1 + T 2 + T 3 + T 4 is a period of the solution (let it is multiple of T ϕ ). Consider the latest system as a system of linear equations with respect to γ 2 , m (for example), i.e. σ 1 = Ψ 1 (m, γ 2 )=l 0 , σ 2 = Ψ 2 (m, γ 2 )=− l, σ 3 = Ψ 3 (m, γ 2 )=− l 0 , σ 4 = Ψ 4 (m, γ 2 )=l. Suppose Ψ i ≡−Ψ i+2 (it can be if the solution is origin-symmetric). Denote ˆ ψ i (t)=σ ( t i + t, t i , σ i , u i−1 ) , t ∈ [ 0, τ ) , ψ i (t)=σ ( t i + τ + t, t i + τ, ˆ σ 1 , u i ) , t ∈ [ 0, T i − τ ) Following result may be formulated. Theorem 6. Let the system  Ψ 1 (m, γ 2 )=l 0 , Ψ 2 (m, γ 2 )=−l. has a solution such as f or given γ =  0, γ 2   and m conditions ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆ ψ 1 (t) > −l, t ∈ [0, τ), ψ 1 (t) > −l, t ∈ [0, T 1 − τ), ˆ ψ 2 (t) > −l 0 , t ∈ [0, τ), ψ 2 (t) > −l 0 , t ∈ [0, T 2 − τ), ˆ ψ 3 (t) < l, t ∈ [0, τ), ψ 3 (t) < l, t ∈ [0, T 3 − τ), ˆ ψ 4 (t) > l 0 , t ∈ [0, τ), ψ 4 (t) > l 0 , t ∈ [0, T 4 − τ) (15) are satisfied. In that case system (14) has a stable T-periodic solution with switching points ˆ s i ,if λ 1 < 0 and TT −1 ϕ ∈ N. Proof In order to prove the theorem it is enough to note that under above-listed conditions system (14) settles self-mapping of switching lines σ = l i .Moreover,foranyx (i) lying on switching line, x (i+1) 1 = e λ 1 T x (i) 1 + Θ, Θ ∈ R, 111 On Stable Periodic Solutions of One Time Delay System Containing Some Nonideal Relay Nonlinearities and in general case (Θ = 0) the latter difference equation has stable solution only if λ 1 < 0.  Inordertopassontovariablesz i it is enough to effect linear transform (13). Note that conditions (15) may be readily verified using mathematical symbolic packages. Of course the statement Theorem 6 is just an outline. Further investigation of the system (11) requires specification of ϕ function, detailed computations are quite laborious. On the analogy with the previous section a case of multiple delays can be observed. 7. Conclusion The above results suppose further development. Investigation of stable modes of the forced system (10) is an individual complex task (systems with several delays may also be considered). Results similar to obtained in the last part can be outlined for periodic solutions of the system (10) having a quite complicated configuration (large amount of control switching point etc.). Stabilization problem (i.e. how to choose setup variables of a system in order to put its steady state solution in a prescribed neighbourhood of the origin) was not discussed. This problem was elucidated in Zubov (1999), Zubov & Zubov (1996) for a bit different systems. 8. References Zubov, V.I. (1999). Theory of oscillations, ISBN: 978-981-02-0978-0, Singapore etc., World Scientific. Zubov, S.V. & Zubov, N.V. (1996). Mathematical methods for stabiliozation of dynamical systems, ISBN: 5-288-01255-5, St Petersburg univ. press, ISBN, St Petersburg. In Russian. Petrov, V.V. & Gordeev, A.A. (1979) Nonlinear servomechanisms, Moscow, Mashinostroenie Publishers. In Russian. Kamachkin, A.M. & Shamberov, V.N. (1995) Automatic systems with essentially n onlinear characteristics, St Petersburg, St Petersburg state marine technical univ. press. In Russian. Nelepin, R.A. (2002). Methods of Nonlinear Vibrations Theory and their Application for Control Systems Investigation, ISBN: 5-288-02971-7, St. Petersburg, St. Petersburg Univ. Press. In Russian. Varigonda, S. & Georgiou, T.T. (2001) Dynamics of relay relaxation oscillators, IEEE Trans. on Automatic Control, 46(1): pp. 65-77, January 2001. ISSN: 0018-9286. Kamachkin, A.M. & Stepanov, A.V. (2009) Stable Periodic Solutions of Time Delay Systems Containing Hysteresis Nonlinearities, Topics in Time Delay Systems Analysis, Algorithms and Control, Vol.388, pp. 121-132, ISBN: 978-3-642-02896-0, Springer-Verlag Berlin Heidelberg. 112 Time-Delay Systems 6 Design of Controllers for Time Delay Systems: Integrating and Unstable Systems Petr Dostál, František Gazdoš, and Vladimír Bobál Faculty of Applied Informatics, Tomas Bata University in Zlín Nad Stráněmi 4511, 760 05 Zlín 5, Czech Republic 1. Introduction The presence of a time delay is a common property of many technological processes. In addition, a part of time delay systems can be unstable or have integrating properties. Typical examples of such processes are e.g. pumps, liquid storing tanks, distillation columns or some types of chemical reactors. Plants with a time delay often cannot be controlled by usual controllers designed without consideration of the dead-time. There are various ways to control such systems. A number of methods utilise PI or PID controllers in the classical feedback closed-loop structure, e.g. (Park et al., 1998; Zhang and Xu, 1999; Wang and Cluett, 1997; Silva et al., 2005). Other methods employ ideas of the IMC (Tan et al., 2003) or robust control (Prokop and Corriou, 1997). Control results of a good quality can be achieved by modified Smith predictor methods, e.g. (Åström et al., 1994; De Paor, 1985; Liu et al., 2005; Majhi and Atherton, 1999; and Matausek and Micic, 1996). Principles of the methods used in this work and design procedures in the 1DOF and 2DOF control system structures can be found in papers of authors of this article (Dostál et al., 2001; Dostál et al., 2002). The control system structure with two feedback controllers is considered (Dostál et al., 2007; Dostál et al., 2008). The procedure of obtaining controllers is based on the time delay first order Padé approximation and on the polynomial approach (Kučera, 1993). For tuning of the controller parameters, the pole assignment method exploiting the LQ control technique is used (Hunt et al., 1993). The resulting proper and stable controllers obtained via polynomial Diophantine equations and spectral factorization techniques ensure asymptotic tracking of step references as well as step disturbances attenuation. Structures of developed controllers together with analytically derived formulas for computation of their parameters are presented for five typical plant types of integrating and unstable time delay systems: an integrating time delay system (ITDS), an unstable first order time delay system (UFOTDS), an unstable second order time delay system (USOTDS), a stable first order plus integrating time delay system (SFOPITDS) and an unstable plus integrating time delay system (UFOPITDS). Presented simulation results document usefulness of the proposed method providing stable control responses of a good quality also for a higher ratio between the time delay and unstable time constants of the controlled system. Time-Delay Systems 114 2. Approximate transfer functions The transfer functions in the sequence ITDS, UFOTDS, USOTDS, SFOPITDS and UFOPITDS have these forms: 1 () d s K Gs e s τ − = (1) 2 () 1 d s K Gs e s τ τ − = − (2) 3 12 () (1)(1) d s K Gs e ss τ ττ − = −+ (3) 4,5 () (1) d s K Gs e ss τ τ − = ± . (4) Using the first order Padé approximation, the time delay term in (1) – (4) is approximated by 2 2 d s d d s e s τ τ τ − − ≈ + . (5) Then, the approximate transfer functions take forms 01 1 2 1 (2 ) () (2 ) d A d Ksbbs Gs ss sas τ τ −− == + + (6) where 0 2 d K b τ = , 1 bK = and 1 2 d a τ = for the ITDS, 01 2 2 10 (2 ) () (1)(2 ) d A d Ks bbs Gs ss sasa τ ττ −− == −+ + + (7) with 0 2 d K b τ τ = , 1 K b τ = , 0 2 d a τ τ =− , 1 2 d d a τ τ ττ − = and τ d ≠ 2τ for the UFOTDS, 3 12 (2 ) () (1)(1)(2) d A d Ks Gs ss s τ ττ τ − = −++ 01 32 210 bbs sasasa − = + +− (8) where 0 12 2 d K b τ ττ = , 1 12 K b τ τ = , 0 12 2 d a τ ττ = , 12 1 12 2( ) d d a τ ττ τττ −− = , 12 1 2 2 12 2 dd d a τ τττττ τττ +− = and τ d ≠ 2τ 1 for the USOTDS, and, 01 4,5 32 21 (2 ) () (1)(2 ) d A d Ks bbs Gs ss s sasas τ ττ −− == ±+ ++ (9) Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 115 where 0 2 d K b τ τ = , 1 K b τ = , 1 2 d a τ τ =± , 2 2 d d a τ τ ττ ± = and τ d ≠ 2τ for the SFOPITDS and UFOPTDS, respectively. All approximate transfer functions (6) – (9) are strictly proper transfer functions () () () A bs Gs as = (10) where b and a are coprime polynomials in s that fulfill the inequality de g de g ba< . The polynomial a(s) in their denominators can be expressed as a product of the stable and unstable part () () ()as asas +− = (11) so that for ITDS, UFOTDS, USOTDS and SFOPITDS the equality de g de g 1aa + = − (12) is fulfilled. 3. Control system description The control system with two feedback controllers is depicted in Fig. 1. In the scheme, w is the reference, v is the load disturbance, e is the tracking error, u 0 is the controller output, y is the controlled output, u is the control input and G A represents one of the approximate transfer functions (6) – (9) in the general form (10). Remark: Here, the approximate transfer function G A is used only for a controller derivation. For control simulations, the models G 1 – G 5 are utilized. Both w and v are considered to be step functions with Laplace transforms 0 () w Ws s = , 0 () v Vs s = . (13) The transfer functions of controllers are assumed as () () () q s Qs p s =   , () () () rs Rs p s =  (14) where ,andqr p  are polynomials in s. v - - y u u 0 e w R Q G A Fig. 1. The control system. Time-Delay Systems 116 4. Application of the polynomial method The controller design described in this section follows the polynomial approach. General requirements on the control system are formulated as its internal properness and strong stability (in addition to the control system stability, also the controller stability is required), asymptotic tracking of the reference and load disturbance attenuation. The procedure to derive admissible controllers can be performed as follows: Transforms of basic signals in the closed-loop system from Fig.1 take following forms (for simplification, the argument s is in some equations omitted) () () () b Ys rWs pVs d =+ ⎡ ⎤ ⎣ ⎦  (15) 1 () ( ) () ()Es ap bqWs bpVs d =+ − ⎡ ⎤ ⎣ ⎦   (16) () () () a Us rWs pVs d =+ ⎡ ⎤ ⎣ ⎦  (17) where () ()() () () ()ds asps bs rs qs=+ + ⎡ ⎤ ⎣ ⎦  (18) is the characteristic polynomial with roots as poles of the closed-loop. Establishing the polynomial t as () () ()ts rs qs = +  (19) and substituting (19) into (18), the condition of the control system stability is ensured when polynomials p  and t are given by a solution of the polynomial Diophantine equation ()() ()() ()asps bsts ds + =  (20) with a stable polynomial d on the right side. With regard to transforms (13), the asymptotic tracking and load disturbance attenuation are provided by divisibility of both terms ap bq +  and p  in (16) by s. This condition is fulfilled for polynomials p  and q  having forms () () p ssps =  , () ()qs sqs =  . (21) Subsequently, the transfer functions (14) take forms () () () q s Qs p s = , () () () rs Rs s p s = (22) and, a stable polynomial p(s) in their denominators ensures the stability of controllers (the strong stability of the control system). The control system satisfies the condition of internal properness when the transfer functions of all its components are proper. Consequently, the degrees of polynomials q and r must fulfil these inequalities Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 117 de g de g qp ≤ , de g de g 1rp ≤ + . (23) Now, the polynomial t can be rewritten to the form () () ()ts rs sqs=+ . (24) Taking into account solvability of (20) and conditions (23), the degrees of polynomials in (19) and (20) can be easily derived as de g de g de g tra==, de g de g 1qa = − , de g de g 1pa≥−, de g 2de g da≥ . (25) Denoting deg a = n, polynomials t, r and q have forms 0 () n i i i ts ts = = ∑ , 0 () n i i i rs rs = = ∑ , 1 1 () n i i i qs qs − = = ∑ (26) and, relations among their coefficients are 00 rt = , iii rq t + = for 1, ,in = (27) Since by a solution of the polynomial equation (20) only coefficients t i can be calculated, unknown coefficients r i and q i can be obtained by a choice of selectable coefficients 0,1 i γ ∈ such that iii rt γ = , (1 ) iii qt γ = − for 1, ,in = . (28) The coefficients γ i divide a weight between numerators of transfer functions Q and R. Remark: If 1 i γ = for all i, the control system in Fig. 1 reduces to the 1DOF control configuration (Q = 0). If 0 i γ = for all i, and, both reference and load disturbance are step functions, the control system corresponds to the 2DOF control configuration. The controller parameters then result from solutions of the polynomial equation (20) and depend upon coefficients of the polynomial d. The next problem here is to find a stable polynomial d that enables to obtain acceptable stabilizing and stable controllers. 5. Pole assignment The polynomial d is considered as a product of two stable polynomials g and m in the form () () ()ds gsms= (29) where the polynomial g is a monic form of the polynomial g ′ obtained by the spectral factorization () () ()() () ()sa s sa s b s b s g s g s ϕ ∗ ∗∗ ′ ′ += ⎡⎤⎡⎤ ⎣⎦⎣⎦ (30) where ϕ > 0 is the weighting coefficient. Remark: In the LQ control theory, the polynomial g ′ results from minimization of the quadratic cost function Time-Delay Systems 118 {} 22 0 () ()Jetutdt ϕ ∞ =+ ∫  (31) where ()et is the tracking error and ()ut  is the control input derivative. The second polynomial m ensuring properness of controllers is given as 2 () () d ms a s s τ + ==+ (32) for both ITDS and UFOTDS, 2 21 () () d ms a s s s τ τ + ⎛⎞ ⎛⎞ ==+ + ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (33) for the USOTDS, and, 21 () d ms s s τ τ ⎛⎞ ⎛⎞ =+ + ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ . (34) for both UFOPITDS and SFOPITDS. The coefficients of the polynomial d include only a single selectable parameter ϕ and all other coefficients are given by parameters of polynomials b and a. Consequently, the closed loop poles location can be affected by a single selectable parameter. As known, the closed loop poles location determines both step reference and step load disturbance responses. However, with respect to the transform (13), it may be expected that weighting coefficients γ influence only step reference responses. Then, the monic polynomial g and derived formulas for their parameters have forms 32 210 () g ss g s g s g = +++ (35) for both ITDS and UFOTDS, where 2 01221 2 21 14 2 4 ,, dd d KK gggKgg τϕ ϕτ ϕ τ ⎛⎞ ==+=+ ⎜⎟ ⎜⎟ ⎝⎠ (36) for the ITDS, and, 22 01 2 22 2 2 21 2 11 1 1 ,4 1, 11 24 dd dd dd d K ggKgK gg ττ τ τ τττϕ ϕ ϕ ττ τ τ ττ ϕ ⎛⎞ == ++ ⎜⎟ ⎝⎠ =++ (37) for the UFOTDS, and, 432 3210 ()gs s gs g s g s g = ++++ (38) [...]... of nonlinear time- varying delay systems, where the nonlinear time- delay functions are bounded by known functions In Shyu et al (2005), a decentralized state-feedback variable structure controller was proposed for large-scale systems with time delay and dead-zone nonlinearity However, in Shyu et al (2005), the time delay is constant and the parameters of the dead-zone are 128 Time- Delay Systems known... stochastic systems with time delay was proposed in Wu et al (2006) In Hua et al (20 07) a result of backstepping adaptive tracking in the presence of time delay was established In Zhou (2008), we develop a totally decentralized controller for large scale time- delays systems with dead-zone input In Zhou et al (2009), adaptive backstepping control is developed for uncertain systems with unknown input time- delay. .. unstable time delay systems using LQ control theory Proceedings of European Control Conference ECC’01, pp 3026-3031, Porto, Portugal Dostál, P., Bobál, V., and Sysel, M (2002) Design of controllers for integrating and unstable time delay systems using polynomial method Proceedings of 2002 American Control Conference, pp 277 3- 277 8, Anchorage, Alaska, USA Dostál, P., Gazdoš, F., and Bobál, V (20 07) Design... processes with time delay J Process Control, 13, pp 203-213 Wang, L., and Cluett, W.R (19 97) Tuning PID controllers for integrating processes IEE Proc Control Theory Appl., 144, pp 385-392 Zhang, W.D., and Xu, X.M (1999) Quantitative performance design for integrating processes with time delay Automatica, 35, pp 71 9 -72 3 7 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time- Delay. .. 0.0 0 20 40 Time 60 80 Fig 6 UFOTDS: controlled output responses (τd = 2, v = - 0.1, γ1 = γ2 = 0) γ1 = γ2 = 0 (2DOF) 2.0 y(t) 2.5 γ1 = γ2 = 1 (1DOF) 1.5 w 1.0 0.5 0.0 0 10 20 30 40 Time 50 60 70 Fig 7 UFOTDS: controlled output responses (τd = 2, v = - 0.05, ϕ = 400) 80 123 Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 1.0 w y(t) 0.8 0.6 0.4 0.2 0.0 0 50 100 Time 150 200... nonlinear system with unknown but constant time delays Jiao & Shen (2005) and Wu (2002) considered the control problem of the class of time- invariant large-scale interconnected systems subject to constant delays In Chou & Cheng (2003), a decentralized model reference adaptive variable structure controller was proposed for a large-scale time- delay system, where the time- delay function is known and linear In... Dostál, P., Gazdoš, F., and Bobál, V (20 07) Design of controllers for processes with time delay by polynomial method Proceedings of European Control Conference ECC' 07, pp 4540-4545, Kos, Greece Dostál, P., Gazdoš, F., and Bobál, V (2008) Design of controllers for time delay systems Part II: Integrating and unstable systems Journal of Electrical Engineering, 59, No 1, pp 38 Hunt, K.J., Šebek, M., and... time- delay In fact, the existence of time- delay phenomenon usually deteriorates the system performance The stabilization and control problem for time- delay systems is a topic of great importance and has received increasing attention Due to the difficulties on considering the effects of interconnections and time delays, extension of single-loop results to multi-loop interconnected systems is still a challenging... parameters for the UFOPITDS 121 Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 7 Simulation results The simulations were performed by MATLAB-Simulink tools For all simulations, the unit step reference w was introduced at the time t = 0 and the step load disturbance v after settling of the step reference responses y(t) 7. 1 ITDS In the transfer function (1), let K = 1 The responses... control problem for time- delay systems have received much attention, see for examples, Jankovic (2001); Luo et al (19 97) ; Wu (1999), etc The Lyapunov-Krasovskii method and Lyapunov-Razumikhin method are always employed The results are often obtained via linear matrix inequalities Some fruitful results have been achieved in the past when dealing with stabilizing problem for time- delay systems using backstepping . integrating and unstable time delay systems: an integrating time delay system (ITDS), an unstable first order time delay system (UFOTDS), an unstable second order time delay system (USOTDS),. pp. 121-132, ISBN: 978 -3-642-02896-0, Springer-Verlag Berlin Heidelberg. 112 Time- Delay Systems 6 Design of Controllers for Time Delay Systems: Integrating and Unstable Systems Petr Dostál,. 65 -77 , January 2001. ISSN: 0018-9286. Kamachkin, A.M. & Stepanov, A.V. (2009) Stable Periodic Solutions of Time Delay Systems Containing Hysteresis Nonlinearities, Topics in Time Delay Systems