Reset-and-Hold Control of Systems With Time Delay

Reset control is a kind of hybrid control which is capable of overcoming fundamental limitations on the performance intrinsic to linear time invariant (LTI) systems. In this work, we develop a novel control strategy for first order LTI plants with time delay, based on the proportional-integral plus Clegg integrator (PI+CI) controller, a hybrid extension of the proportional-integral (PI) controller, augmented with a new mechanism for keeping its state constant during a given interval of time (reset-and-hold strategy). This strategy is capable of producing an approximation of a flat response which greatly improves the performance in comparison with non-hybrid linear strategies, extending previous results developed for first order systems without delay. Well-posedness and closed-loop stability of the resulting control system are analyzed under the Hybrid Inclusions (HI) framework, and a set of sufficient stability conditions are provided. Furthermore, a case study is developed that showcases the possibilities of this new approach.

In its original meaning, reset control involves linear time 23 invariant controllers endowed with a means to reset their 24 states to zero. A fundamental example of reset controller, now 25 called the Clegg integrator (CI), was introduced by Clegg in 26 his seminal work [2]. In the works of Horowitz and others [3], 27 [4] another basic example of reset controller was introduced, 28 the first order reset element (FORE), and design rules for 29 The associate editor coordinating the review of this manuscript and approving it for publication was Chao-Yang Chen . the CI and FORE were developed. Since then, the meaning 30 of reset system in practice has been expanded to encompass 31 different triggering conditions such as error bands [5], [6] 32 or periodic reset instants [7], as well as systems that are 33 nonlinear or in which the resetted states do not necessarily 34 go to zero. 35 The field of reset control has proven to be very fruitful, 36 with a multitude of reset strategies having been successfully 37 devised and applied in practice. More concretely, reset control 38 strategies specific to systems with time delay have been 39 developed and studied e.g. in [8], [9], [10], and [11]. 40 In this work, we consider the problem of reset control for 41 LTI first order plants with time delay (also called FOPDT 42 systems), extending previous results pertaining to first order 43 plants without delay [12]. With the introduction of delay, first 44 order systems are sufficiently generic to capture the essential 45 dynamics of many practical processes in industry [1]. Our 46 approach is based on the proportional-integral plus Clegg 47 integrator (PI+CI) controller, together with a new reset mech-48 anism (reset-and-hold) in which the controller is augmented 49 where the delay is characterized by a single parameter h, 99 such that the first order dynamics of the output is dependent with three parameters a, b, h. Despite its simplicity, it is 105 known that many practical processes in industry can be well 106 approximated by such a dynamics [1]. 107 B. THE HI FRAMEWORK 108 The formalism of Hybrid Dynamical Systems, also called 109 the Hybrid Inclusions (HI) framework, which is developed 110 in [13], is the mathematical basis upon which the devel-111 opments of this work rest. This subsection includes a brief 112 description of the HI framework for systems with inputs and 113 for systems with memory. 114

115
This section briefly describes the formalism of Hybrid 116 Dynamical Systems adapted to systems with inputs [14] as 117 will be used in this work. A hybrid system with inputs for 118 a state x ∈ R n x and input w ∈ W for some set W ⊆ R n w is 119 given by 120 : (1) 121 where f : R n x ⇒ R n x is the flow map, g : R n x ⇒ R n x is the 122 jump map (f and g are both set-valued maps in general), and 123 C, D ⊆ R n x × R n w are the flow and jump sets respectively. 124 Note that as in [14], we do not regard w as a hybrid signal 125 (as otherwise its domain must be known in advance, which 126 is unrealistic in practice); instead, the space of admissible 127 inputs w(t) will be taken as the set of piecewise continuous 128 144 A hybrid system = (C, F, D, G), with memory of size 145 , is given by and output u ∈ R is given by where A r , B r , C r , D r are constant matrices of the appropriate 172 dimensions, and A ρ is a diagonal matrix that sets to zero the 173 last n ρ elements of x r . The flow and jump sets C and D are 174 given by (3) and (4), respectively, where

197
A new hybrid controller, referred to as reset-and-hold, 198 inspired by the distributed state resetting approach of [10], 199 will be shown to be specially useful for systems with time 200 delays. The main motivation has been to overcome the per-201 formance of reset controllers for systems with time delays. 202 This is based on the fact that the performance improvement 203 due to resetting crucially depends on a balance between the 204 after-reset states of both plant and controller. However, the 205 presence of delay in the feedback path destroys that balance, 206 typically producing an undesired undershooting, which limits 207 the potential performance improvement. To avoid this prob-208 lem, besides resetting the basic idea is to hold the control 209 signal after a jump, for some time interval.

210
The reset-and-hold controller R H , with state (x r , q, m, τ ) ∈ 211 O H := O n r × {0, 1} × R ≥0 , and input e ∈ R, is defined as a 212 hybrid system with inputs, given by: where the output is u = C r x r + mD r e, the flow set is C H = 216 Note that, in comparison with the reset controller (3), R H 224 includes a extra discrete state m ∈ {0, 1}, that will be used 225 to switch between two operating modes, and a timer τ that 226 will be in charge of regulating the time interval in which 227 the controller output is held constant after every jump due 228 to reset. The two operating modes are:

229
• m = 1 (resetting mode). The controller output 230 corresponds to that of the base linear controller 231 (A r , B r , C r , D r ). Jumps are enabled only when the reset-232 ting law q S(e) ≤ 0 is triggered: a crossing is detected 233 and thus the sign of q is changed, x r is reset, the timer is 234 reset to zero, and m switches to 0 (holding mode). defined by the delay-differential equation with state x ∈ R n p , input v ∈ R and output v ∈ R, 259 and the feedback connection between P and a reset-and-hold  7) and (8), the 265 closed loop system (without exogenous inputs) is given by where the flow and jump sets are given by respectively, and

283
Now, let z = (x, q, m, τ ) be a hybrid arc and z [t,j] ∈ M a 284 hybrid memory arc. One way of interpreting the closed-loop 285 hybrid system (9)-(13) as a hybrid dynamical system with 286 memory = (C, F, D, G) is to use the following data 287 (some similar cases are described in [15] and [16]):

328
From their definition in (14) and (10)   In this section, stability of the closed loop hybrid system , 348 given by (14), in investigated. More specifically, a closed 349 set W is asymptotically stable for the hybrid system 350 if there exists a candidate Lyapunov-Krasovskii functional 351 V : M → R ≥0 , α 1 , α 2 ∈ K ∞ and a continuous positive 352 definite function ρ such that the three following conditions 353 are satisfied: whereV denotes the upper right hand derivative of the func-360 tional V , and φ + γ is the new hybrid arc obtained after a single 361 jump to the value γ ; the set of all possible φ + γ is denoted 362 G + (D) (the reader is referred to [15] for precise definitions 363 and technical details).

372
(Delay-dependent stability conditions) Consider the reset-373 and-hold control system given by (14). The set W is 374 asymptotically stable for if there exist matrices P > 0, 375 Q > 0, X = X , Y and Z > 0 such that the following 376 conditions are satisfied: 2) where u t is the value maximizing the norm |x φ (t, j)| among 391 the j such that (t, j) ∈ dom φ (this choice is related to the defi-392 nition of upper right hand derivative), and for any t ∈ (−h, 0)

396
Note that (15a) easily follows from the fact that V , as given (becomes identically zero) after the first reset instant (Fig. 2).

423
It is known that a PI+CI controller is able to produce a flat 424 response for first order systems without delay [17]; this result 425 was recently extended to MISO plants in [22]. It will be 426 shown that a flat response can also be attained for FOPDT 427 systems (to a very good approximation) using a well-tuned 428 reset-and-hold controller.

429
The reset-and-hold PI+CI controller with state (x I , x CI , q, 430 m, τ ) is given by (5), with the controller parameters given in 431 section II.B.2. Note that the design parameters are k P and T I 432 (corresponding to the base PI controller), and p r , τ H , and θ 433 corresponding to the reset-and-hold strategy. It is assumed 434 that the base PI controller, that is, the parameters k P and 435 T I , are designed to produce a fast oscillatory response (note 436 that an oscillatory response occurs whenever the base linear 437 control system has a pair of complex poles in the frequency 438 domain; in terms of the parameters of the plant and controller, 439 this happens whenever the inequality T I < 4bk P /(bk P +a) 2 is 440 satisfied. The base controller can be designed in the usual 441 way using any common tuning method, taking into account 442 this constraint). The role of the reset-and-hold strategy will be 443 to reduce the overshoot as much as possible to obtain a flat 444 response without decreasing the initial speed of the response. 445 In the following, the tuning strategy for the parameters p r , τ H , 446 and θ is detailed. The first design choice is to use a reset band equal to the 459 delay, that is to make θ = h. In this way, the reset action 460 will occur at t 1 ≈ t c − h, where a first order approximation 461 e(t +δ, j) ≈ e(t, j)+δė(t, j), for t ∈ [t c −h, t c ] has been used. 462 As a result, it is obtained that x I (t 1 , 1) = x I (t c −h, 1) = x I (t 1 ) 463 and x CI (t 1 , 1) = 0, and directly from (23)-(24) that Here, note that the proportional part is zeroed out after a 466 jump under the reset-and-hold strategy. Note that although in (26) and (30) the amplitudes w 10 and 508 w 20 appear explicitly, these values are cancelled since they 509 are also a factor of x I (t 1 ) due to the linearity of the base 510 system. As a result, the proposed values for p r are constant 511 and intrinsic to the hybrid control system, that is, they depend 512 only on the plant and the base PI controller. However, since 513 an explicit computation is x I (t 1 ) is hard to obtain, in practice 514 the value of p r may be simply computed by using the value 515 of the integrator state at the first reset instant.

516
The reset-and-hold strategy is also applicable in cases 517 where the delay h is not known precisely, but has some 518 associated degree of uncertainty. If it is known that h ∈ 519 [h min , h max ], then a simple conservative approach is to take 520 τ H = θ = h min (and the same reset ratios than in (26) 521 or (30)); this maximizes the time the controller spends in 522 flowing mode, producing a flat response only in the best case, 523 but improving the performance in all cases with respect to 524 more standard reset approaches.

525
On the other hand, the proposed reset-and-hold strategy 526 may not be suitable in those cases in which the appearance of 527 several disturbances or reference changes in a short time span, 528 lower than the delay h, is expected. This is because the con-529 troller outputs are held constant during a fixed time interval, 530 so they will not be able to reject any incoming disturbances 531 until after this interval has passed. In such cases, an additional 532 supervisory mechanism is needed so that holding can be 533 disabled whenever a new disturbance or reference change is 534 detected.

536
To demonstrate the capabilities of the proposed tuning rule, 537 a simulated example, consisting of a delayed first-order pro-538 cess model of a heat exchanger in an experimental food 539 processing pilot plant, is considered (see Section 6.1 of [17] 540 for a detailed description). For a given operation point, the 541 plant is given by the transfer function 542 P(s) = 0.49 e −139s 1 + 106s , (32) 543 In the following, a PI+CI reset-and-hold controller (as 544 given by (5) with the data of Section II.B.2) will be designed. 545 The base PI controller parameters have been tuned to produce 546 an oscillatory base closed-loop response. The chosen param-547 eters are 548 k P = 1.3, T I = 118.
(33) 549 Well-posedness of the closed-loop hybrid control system 550 directly follows (see Section IV.A). Two design cases are 551 considered: tracking step references, and rejecting step distur-552 bances. Both sections V.A and V.B will be closely followed. 553

554
A step reference change of amplitude w 10 = 2 starting 555 at time t = 30 is considered. The tuning rule (26) has 556 then been used to determine the reset ratio. Both the instant 557 been found for the matrices (P, Q, X , Y , Z ). As a result, the 567 set W is asymptotically stable for the hybrid control system. 568 Fig. 3a shows the step response of the closed-loop hybrid 569 system, together with the response of the base linear system.

570
The controller output of the controller is shown in Fig. 3b.

571
For the purposes of comparison, a PI+CI reset controller is 572 also considered; two different cases will be compared: (i) a 573 case with the same parameters p r and θ of the reset-and-hold 574 controller, as given by (27), and (ii) a case with value p r = 575 0.350 (and θ = 139).

576
As can be observed, in the first PI+CI reset controller case, 577 the response is actually worse than the base linear control Note that the designed controller parameters are only 601 slightly different to the reference tracking parameters given 602 by (34). Closed-loop stability has been again checked by 603 solving (17)-(20) for some matrices (P, Q, X , Y , Z ) and the 604 parameters (31). Fig. 4a shows the closed-loop response with 605 the designed PI+CI reset-and-hold controller, and the with 606 the base linear controller. The corresponding controller out-607 puts are shown in Fig. 4b. As expected, the reset-and-hold 608 controller produces an (almost) flat response after rejecting 609 the step disturbance (note that the step disturbance has an 610 amplitude w 20 = 3).    Optimization (AMIGO) [24].

641
The combination of a unit step reference change at t = 642 0 and a negative unit step disturbance at t = 1100 is consid-643 ered. Again the amplitudes and controller parameters for the 644 PI+CI are the same as in the previous cases, with p r being 645 changed from (34) to  The results are shown in Figure 7 and Table 1. As expected, 651 the PI+CI controller achieves better performance indices 652 than its linear counterparts.   The closed loop response to a unit step reference change 679 at t = 0 plus a negative unit step disturbance at t = 10 has 680 been simulated. The results of the comparison are shown in 681 Figure 8 and Table 2. As can be seen, the performance in 682 all metrics is greatly improved in reference tracking without 683 causing any significant overshoot. In contrast, the overshoot 684 in disturbance rejection is degraded, but the settling time is 685 improved, in such a way that most other metrics remain of 686 similar magnitude. Note that the strategy in [25] deals with 687 reference tracking and disturbance rejection using a combi-688 nation of controllers, where as our proposed setup utilizes 689 only one controller. It is possible that a similar setup using 690 combined PI+CI controllers with the reset-and-hold strategy 691 would achieve a better handling of disturbances; however, this 692 is out of the scope of the current work.

694
A new hybrid controller for systems with time delays is 695 proposed, consisting of a combination of reset and hold 696 strategies. This reset-and-hold controller has been analyzed 697 in detail in the framework of Hybrid Inclusions, equipping 698 the resulting closed-loop hybrid system with good structural 699 properties. Besides well-posedness, which has been shown to 700 be guaranteed for any LTI plant with time delays, stability 701 conditions have been developed. Moreover, for the specific 702 case of a PI+CI reset-and-hold controller and a FOPDT 703 system, a set of design rules have been proposed. An (almost) 704 closed-loop flat response is obtained both in step tracking 705 and in rejecting step disturbances, notably improving the 706 performance of PI+CI reset controllers.

707
The developments in this work apply only to FOPDT 708 systems. A possible idea for extending the current strategy to 709 deal with more general time-delayed processes, such as sec-710 ond order plus dead time (SOPDT) or integrating first order 711 plus dead time (IFOPDT), is to combine the reset-and-hold 712 strategy with the on-line PI+CI tuning method for second 713 order plants from [19]. In this way, the parameters p r (t k ), 714 and possibly θ(t k ), τ H (t k ), would become functions of the 715 kth reset instant, computed on-line using a simple quadratic 716 optimization algorithm. This possibility will be explored in 717 future work.