Design of controllers for time delay systems: Integrating and unstable 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. www.intechopen.com Time-Delay Systems 114 2. Approximate transfer functions The transfer functions in the sequence ITDS, UFOTDS, USOTDS, SFOPITDS and UFOPITDS have these forms: 1() ds K Gs e s τ − = (1) 2() 1ds K Gs e s τ τ − =− (2) 312 () (1)(1) ds K Gs e ss τ ττ − =−+ (3) 4,5() (1) ds 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 dsd d s es τ τ τ −− ≈+. (5) Then, the approximate transfer functions take forms 01 121 (2 ) () (2 ) d A d Ksbbs Gsss sas τ τ −− == ++ (6) where 02 d K b τ = , 1 bK = and 12 d a τ = for the ITDS, 01 2210 (2 ) () (1)(2 ) d A d Ks bbs Gs ss sasa τ ττ −− == −+ + + (7) with 02 d K b τ τ =,1K b τ = , 02 d a τ τ =− , 12d d a τ τ ττ − = and τd ≠ 2τ for the UFOTDS, 312 (2 ) () (1)(1)(2) d A d Ks Gs ss s τ ττ τ − =−++ 01 32 210 bbs sasasa − = + +− (8) where 012 2 d K b τ ττ =, 112 K b τ τ =, 012 2 d a τ ττ =, 12 112 2( ) d d a τ ττ τττ −− =, 12 1 2 212 2dd d a τ τττττ τττ +− = and τd ≠ 2τ1 for the USOTDS, and, 01 4,5 32 21 (2 ) () (1)(2 ) d A d Ks bbs Gs ss s sasas τ ττ −− == ±+ ++ (9) www.intechopen.com Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 115 where 02 d K b τ τ =, 1K b τ = , 12 d a τ τ =± , 22d d a τ τ ττ ± = and τd ≠ 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, u0 is the controller output, y is the controlled output, u is the control input and GA represents one of the approximate transfer functions (6) – (9) in the general form (10). Remark: Here, the approximate transfer function GA is used only for a controller derivation. For control simulations, the models G1 – G5 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. www.intechopen.com 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 www.intechopen.com 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 () ni i i ts ts = = ∑ , 0 () ni i i rs rs = = ∑ , 1 1 () ni 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 ti can be calculated, unknown coefficients ri and qi 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 γi 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 www.intechopen.com 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 21 211 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 gs g = ++++ (38) www.intechopen.com Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 119 for USOTDS, SFOPITDS and UFOPITDS, where 2 012 22 222 12 12 12 12 22 2 12 213 3 2 222 2 2 2 12 12 1 2 21 4 1 4 , 4( ) 241 2411 , dd d d ddd KKK ggg K ggg g g τττ τττ ϕ ϕτττττ ττ τ ϕτττϕ τ ττ ϕ τ τ τ ⎛⎞ ==++ ⎜⎟ ⎜⎟ ⎝⎠ ++ =− + =+++ (39) for the USOTDS, and, 2 012 2 213 3 2 22 24 11 ,, 41 1 2 41 2, dd dd d KK K ggg ggg K g g ττ ϕ ττ ϕτ ττ ττ ϕ ϕττ ⎛⎞ ==+ ⎜⎟ ⎜⎟ ⎝⎠ ⎛⎞ =+ − =++ ⎜⎟ ⎜⎟ ⎝⎠ (40) for both SFOPITDS ans UFOPITDS. The transfer functions of controllers are 21 0 () qs q Qs sp + =+, 2 210 0 () () rs rs r Rs ss p + + =+ (41) for both ITDS and UFOTDS, and, 232 321 3210 22 10 10 () , () () qs qs q rs rs rs r Qs Rs spsp sspsp + ++++ == ++ ++ (42) for the USOTDS, SFOPITDS and UFOPITDS. 6. Controller parameters For the sake of limited space, formulas derived from (20) for all considered systems together with conditions of the controllers’ stability are introduced in the form of tables. Parameters ri and qi in (41) and (42) can then be calculated from ti according to (28). 02 1 0 (2 ) 4 dd p ggg ττ =+ + , 00 1 tg K = 110 1() d tgg K τ =+ , 210 (2 ) 4 dd tgg K ττ =+ p0 > 0 for all τd Table 1. Controller parameters for the ITDS www.intechopen.com Time-Delay Systems 120 210 0 2( )2 2 2 d d d ggg p τ ττ ττ ⎡⎤ + ++ ⎢⎥ ⎣⎦ =− 00 tg K τ =, 1 010 1() d tpgg K ττ =++ ⎡ ⎤ ⎣ ⎦, 202 1()1tpg K τ = −− ⎡ ⎤ ⎣ ⎦ p0 > 0 for τd < 2τ Table 2. Controller parameters for the UFOTDS 31 2 1 0 1 01 2 22( ) 2 2 d d d gggg p τ ττ τ ττ ⎡⎤ ++++ ⎢⎥ ⎣⎦ =−, 131 1 pg τ =+ 1 00 tg K τ =, () 101120 1() d tpg g K τττ ⎡ ⎤ =+++ ⎣ ⎦ 12 2 2120312121 1 14 4 1 1 dd tpggg K ττ τ ττ τ ττ τττ ⎡ ⎤ ⎡ ⎤ ⎛⎞⎛⎞ ⎛⎞ ⎢ ⎥ =+−−+++− ⎢ ⎥ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎢ ⎥ ⎢ ⎥ ⎝⎠ ⎝⎠⎝⎠ ⎣ ⎦ ⎣ ⎦ 2 31023 1 1 ()tpgg K ττ τ ⎡ ⎤ =−−− ⎢ ⎥ ⎣ ⎦ p1 > 0 for all τd , p0 > 0 for τd < 2τ1 Table 3. Controller parameters for the USOTDS 02 1 0 (2 ) 4 dd pg g g ττ =+ + , 13 1 pg τ = + 00 1 tg K =, 11 0 1() d tg g K ττ =++ ⎡ ⎤ ⎣ ⎦, 2100 1(2 )(2 ) 2 4ddd t gg g K ττ τ ττ =+++ ⎡ ⎤ ⎣ ⎦ 310 2 4 dd t gg K ττ τ =+ ⎡ ⎤ ⎣ ⎦ p1, p0> 0 for all τd Table 4. Controller parameters for the SFOPITDS 2 3210 0 4 4(2 ) 24 2 dd d d gggg p ττ ττ τ ττ ⎛⎞ + ++++ ⎜⎟ ⎜⎟ ⎝⎠ =−, 13 2 pg τ = + 00 1 tg K =, 11 0 1() d tg g K ττ =++ ⎡ ⎤ ⎣ ⎦, 20321 44 18 8 11 ddd d tpggg K ττ τ τ ττ ττ ⎡⎤ ⎛⎞ ⎛⎞ =−−−+−− ⎢⎜⎟ ⎜⎟ ⎥ ⎜⎟ ⎜⎟ ⎢⎝⎠ ⎝⎠ ⎦ ⎣, 3023 12 ()2tpgg K τ τ ⎡ ⎤ =−−− ⎢ ⎥ ⎣ ⎦ p1 > 0 for all τd , p0 > 0 for τd < 2τ Table 5. Controller parameters for the UFOPITDS www.intechopen.com Design of Controllers for Time Delay Systems: Integrating and Unstable Systems 121 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. 7.1 ITDS In the transfer function (1), let K = 1. The responses in Fig. 2 for τd = 5 show the effect of ϕ upon the control quality. An increasing value ϕ improves control stability, and, by choosing its value higher, aperiodic responses can be obtained. Simulation results shown in Fig. 3 demonstrate the influence of parameters γ on the control responses. Their smaller values accelerate step reference responses but they do not affect load disturbance responses. Higher values of γ can lead to overshoots and oscillations. The effect of parameters γ on the control 0 20 40 60 80 100 120 140 160 180 0.0 0.2 0.4 0.6 0.8 1.0 1.2 y(t) Time ϕ = 100 ϕ = 400 ϕ = 900 w Fig. 2. ITDS: controlled output responses (τd = 5, v = - 0.1, γ1 = γ2 = 0) 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 y(t) Time γ1 = γ2 = 0 γ1 = γ2 = 0.25 γ1 = γ2 = 0.4 w Fig. 3. ITDS: controlled output response (τd = 5, v = - 0.1, ϕ = 900). 0 50 100 150 200 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 y(t), u(t) Time γ1 = γ2 = 0 γ1 = γ2 = 0.4 y(t) 5xu(t)w Fig. 4. ITDS: Control input and controlled output responses (τd = 5, ϕ = 900) www.intechopen.com