Ind. Eng. Chem. Res. 2004, 43, 8251-8256
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Sliding Mode Conditioning for Constrained Processes Ricardo J. Mantz,*† Herna´ n De Battista,‡ and Fernando D. Bianchi§ Laboratorio de Electro´ nica Industrial Control e Instrumentacio´ n, Universidad Nacional de La Plata, C.C.91 (1900) Argentina
A reference conditioning algorithm based on variable structure systems theory is developed to cope with actuator saturation. This sliding mode conditioning presents some distinctive features. In fact, the proposed compensation shapes a realizable reference, thus completely avoiding saturation. Then, the closed loop never leaves its prescribed linear behavior. Furthermore, as a consequence of this property as well as of the robustness features of sliding mode control, the conditioning circuit dynamics can be designed independently of the main loop dynamics. Although the paper mainly focuses on antiwindup compensation, the proposed conditioning algorithm can be exploited to address other problems of constrained processes control. 1. Introduction The control input to all industrial processes is subjected to physical constraints. Similarly, the internal variables of the processes have bounded ranges of operation. All these limitations may seriously degrade the desired closed-loop performance. Interest in developing control strategies to reduce the undesirable effects of restrictions has led to an increasing number of publications addressing the problem from different viewpoints, for instance, the use of concepts of LQ (linear quadratic) theory,1 LPV (linear parameter varying),2 LMIs (linear matrix inequalities),3 SM (sliding mode),4 robust control,5 etc. All control solutions fit within one of two possible design procedures: the socalled “one-step” and “two-step” methodologies.6 In the former, the controller design is carried out taking the restrictions into consideration from the beginning. In the latter, the controller is first designed neglecting the restrictions (first step), and a correction loop is incorporated afterward to deal with them (second step). The aim of this auxiliary loop is to guarantee graceful degradation from the performance of the system without restrictions. Particularly, this paper is concerned with the design of this correction loop. Generally, limitations occur during transient responses to external excitations, such as set-point changes and disturbances. Since the magnitude of set-point changes is larger than disturbance ones, their adverse effects are more notorious. Then, many proposals consist of shaping the reference signal to avoid problems caused by actuator saturation. For instance, one of the most cited contributions is the conditioning technique developed by Hanus and co-workers for PID controllers.7 Although the conditioning is applied to the integral state of the controller, it is deduced from concepts of “realizable reference”. Some years later, Walgama and coauthors8 made use of this concept to generalize the Hanus technique. More recently, other antiwindup (AW) algorithms based on reference shaping have been published (see ref 9 and references therein). * To whom correspondence should be addressed. Tel./Fax: +54 221 425 9306. E-mail:
[email protected]. † Prof. Mantz is member of CICpBA. ‡ Prof. De Battista is member of CONICET. § Prof. Bianchi is member of CONICET.
In this paper, a novel conditioning technique is developed within the framework of variable structure systems undergoing sliding regimes. This conditioning algorithm is based on realizable reference concepts. Certainly, the reference is continuously shaped via SM techniques to completely avoid actuator restrictions. The proposed conditioning provides a solution with very attractive features. In fact, the closed-loop operation of the main control loop is guaranteed, thus preserving the prescribed linear dynamics. Furthermore, the dynamics of the SM loop is completely robust to the process response. In addition, the implementation and tuning of the SM algorithm is straightforward, even in the case of a preexisting linear controller. This paper is organized as follows: The next section briefly outlines the main features of SM control. In the section after the next, the proposed SM conditioning technique is described, then its performance is evaluated through a pair of examples. Finally, conclusions are summarized.
2. Theoretical Framework: Sliding Regimes A variable structure system consists of a set of continuous subsystems with an associated switching function that determines a manifold on the state space, the so-called sliding surface. According to the sign of the switching function, the control signal takes one of two possible values, leading to a discontinuous control law. The basic idea is to enforce the state to reach the prescribed sliding surface and, henceforth, to slide on it through an adequate switching action. Once this particular mode of operation is established, known as sliding mode or sliding regime, the prescribed manifold imposes the new system dynamics. Among other attractive features, sliding regimes are easy to implement, reduce the order of the system dynamics, and provide robustness to matched uncertainties and external disturbances.10-12 On the other hand, the most common shortcomings of variable structure control are the chattering and the reaching mode previous to the establishment of the sliding regime. Chattering might damage the actuator and process devices, whereas the reaching phase may degrade the global performance unless it is not carefully designed.
10.1021/ie049494p CCC: $27.50 © 2004 American Chemical Society Published on Web 12/15/2004
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Because of its interesting features, a large number of papers proposing practical applications of SM control have been reported in the last years. For instance, in refs 13-17,13-17 the application of SM to chemical process control is discussed. In contrast with conventional variable structure controllers, the aim of the control strategy developed here is not to evolve in SM toward the equilibrium point. On the contrary, the sliding regime is intended as a transitional mode of operation. It is just aimed at conditioning the rate of change of the reference in order to avoid actuator saturation. Hence, once the closedloop system is able to evolve toward the linear operating region without reference conditioning, the SM compensation becomes inactive. Because of the nature of this SM compensation, the aforementioned drawbacks of variable structure control (i.e. chattering and reaching modes) are negligible in the current application. In fact, the sliding regime is confined to the low-power side of the system, hence allowing the use of fast electronic devices. Additionally, the conditioning loop is inactive until the system state reaches by itself the sliding surface, so there is no reaching mode in the sense that no control effort is done to enforce the state trajectories toward the surface.
3. AW Reference Conditioning Algorithm via Sliding Mode Windup is a problem frequently found in process control. Although it is sometimes caused by limitations inherent to the process (“plant windup”),9,18 it is commonly associated to restrictions in the actuator (“controller windup”).19,20 In the latter case, windup can be viewed as a discrepancy between the state of the controller and the input to the plant caused by the actuator nonlinearity. As a result, the control system performs as an open loop. This may lead to an undesirable transient behavior with large overshoot and long settling time. Consistency can be restored by means of shaping the input to the control loop. The ideal modified input signal is called “realizable reference”. Actually, if this realizable reference had been applied to the controller from the beginning, the actuator would not have fallen in saturation, i.e., the controller output would have coincided all the time with the input to the plant. In this context, Hanus and co-workers modified the state of the controller to approximate the response to the realizable reference.7 Among other limitations, the Hanus technique works only for biproper controllers. A modified version of the conditioning technique, the socalled generalized conditioning technique, was proposed by Walgama and co-workers8 to overcome this and other drawbacks. In this approach, a reference filter is incorporated, and conditioning is carried out on the filter output instead of on the reference. More recent contributions in this field are found in ref 9. On the basis of these ideas, an SM conditioning loop is developed to avoid actuator saturation, thus completely eliminating windup. Figure 1 illustrates the control system with the proposed reference conditioning. Two loops can be distinguished: the main control loop and the SM reference conditioning loop.
Figure 1. Proposed reference conditioning via sliding mode.
Main Control Loop. P is the process to be controlled, which is assumed stable. A is the constrained actuator. C ) (C1, C2) is the controller designed to accomplish the control specifications during linear operation (first step of the two-step design procedure). The broad case of two degree-of-freedom controllers, which allow accomplishing tracking and regulation specifications simultaneously, is addressed. One-degree-of-freedom controllers usually found in process control are special cases of the previous ones in which the feed-forward compensator reduces to C1 ) 1. To apply the proposed conditioning, C2 has to be a minimum phase controller, and C1 needs to be stable and strictly proper; i.e., it needs to have relative degree g1 with respect to its input r. (The relative degree of a linear system can be defined as the difference between the number of poles and the number of zeros of the transfer function.) So, when C1 is biproper (that is, C1 is a constant gain or its transfer function has the same number of zeros and poles), it is expanded with a state variable. This expanded dynamics must be much faster than the rest of the controller in order to affect negligibly the desired performance of the control system. (This expanded dynamics has not of itself the aim of avoiding saturation after set-point changes. In fact, this obvious solution would lead to an extremely conservative design. Contrarily, the compensation lies in the conditioning of the output of C1 as function of the restrictions.) The dynamic behavior of process P, as well as of controller (C1, C2), is represented in state space as follows:
P:
C 2:
C1:
{
x3 p ) Apxp + bpup y ) cTp xp
{ {
x3 c2 ) Ac2xc2 + bc2e u ) ccT2xc2 + dc2e
x3 c1 ) Ac1xc1 + bc1r + bww rf ) ccT1xc1
(1)
(2)
(3)
where e ) rf - y is the error signal. Without loss of generality, it is assumed that bw ) cc1 ) col (1, 0, ..., 0), so that ccT1bw ) 1. In addition, it is assumed that eqs 2 and 3 are minimal realizations of C2 and C1 and that the states xc2 and xc1 are accessible. Besides, the actuator presents a lower or upper saturation limit. In the general case, its nonlinearity is characterized by
{
up ) u j p if u > u jp jp A: up ) u if up e u e u up ) up if u < up
(4)
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SM Conditioning Loop. From Figure 1 and eqs 2 and 3, the open-loop conditioning circuit dynamics is given by
[ ] [ x3 c2 e˘
)
][ ] [
Ac2 bc2 xc 2 0T 0
e
weq ) ke-1[-kcT2(Ac2xc2 + bc2e) ke(ccT1Ac1xc1 + ccT1bc1r - y˘ )] (10)
+
] []
0 0 ccT1Ac1xc1 + ccT1bc1r - y˘ + w
(5)
The input and output of the conditioning circuit are the signal w and the control u, respectively. The conditioning signal w is governed by
being
It readily follows that
{
w+ if s > 0 w ) 0 if s ) 0 w- if s < 0
(6)
s(xc2, e, rs) ) kcT2xc2 + kee - rs
(7)
As a result of this switching law, the conditioning signal w is discontinuous on the surface S defined by S ) {(xc2, e)|s(xc2, e, rs) ) 0}. The surface coordinate rs is a constant value proportional to the actuator bound that is about to be reached. Surface S divides the state space into two regions. In one of these regions (characterized, without loss of generality, by s < 0), there is no risk of saturation. Hence, the conditioning takes no action therein, i.e., w) 0. In contrast, in the other region (characterized by s > 0), the input to the plant is likely to fall into limitations. In this region, the conditioning signal takes the value w ) w+ with the aim of steering the state trajectory toward the safety region, s < 0. The conditioning loop works as follows. Let us assume that the system is initially in the linear region (w ) w- ) 0) but the state trajectory points toward surface S, that is, toward the limit of the safety region. When S is reached, the conditioning loop becomes active (w ) w+), enforcing the system state to reenter the safety region. Once on the safety region, the conditioning signal switches again to w ) 0. This sequence of discontinuities is repeated, establishing a sliding regime on S until the system state (with w ) 0) points no more from the safety region toward the dangerous one. That is, the sliding regime is a transitional mode of operation that is maintained until the system state evolves by itself toward the interior of the safety region. Note also that there is no reaching phase previous to the sliding regime in the sense that no control action is applied to drive the system state from its initial condition in the linear region toward S. Effectively, operation in the interior of the safety region is desired, and the SM controller is inactive until the sliding surface is reached. SM Conditioning Loop Dynamics. The dynamics of the SM conditioning loop can be obtained using the equivalent control method.21 The so-called equivalent control is a fictitious continuous signal that is an invariant control for the system on the sliding surface. Accordingly, the equivalent conditioning signal weq is derived from the invariance condition
s(xc2, e, rs) ) 0
(8)
s˘ (xc2, e, rs) ) 0
(9)
This equivalent control exists provided the transversality condition ke * 0 holds.10 Replacing w in the open loop dynamics (5) by weq, the SM dynamics yields
[][ x3 c2 e˘
)
Ac2 -ke-1kcT2Ac2
][ ]
bc2
x c2 -ke-1kcT2bc2 e
(11)
Actually, eq 11 describes the SM dynamics redundantly. Effectively, the state variables xc2 and e satisfy eq 8 on the sliding surface, i.e they are linearly dependent. Consequently, solving eq 8 for e and replacing in the right-hand side of eq 11, the following reduced-order description of the SM dynamics yields
x3 c2 ) [Ac2 - bc2ke-1kcT2]xc2 + bc2ke-1rs
(12)
Observations. 1. As is observed in eq 12, the SM conditioning dynamics is insensitive to the process output response (and its derivative y˘ ). This property is attained because the process output satisfies the matching condition with respect to w in eq 5.10 Moreover, the SM dynamics is completely governed by the sliding surface design. In fact, since (Ac2, bc2) is a controllable T pair, the eigenvalues of [Ac2 - bc2k-1 e kc2] can be freely assigned through the sliding gains ke and kc2. Appropriately selecting these gains, a stable and fast response of the conditioning loop is achieved. 2. The dynamics of the main loop, with input rf and output y, is also independent of the conditioning loop dynamics. Two facts make this possible: (a) The conditioning circuit acts on the filtered reference rf, taking no action within the main loop. (b) The conditioning circuit avoids saturation, thus guaranteing closed-loop linear operation of the main loop. Then, the dynamics of the main loop is always described by
[][
][ ] [ ]
Ap - bpdc2cTp bpccT2 xp bpdc x3 p + b 2 rf x3 c2 ) -b cT x A c2 c2 c2 c2 p
[ ]
xp y ) [cTp 0T ] x c2
(13)
with eigenvalues exclusively governed by the process and the controller C2. 3. Since the transfer function Q(s) between rf and u (Q(s) ) u(s)/rf(s) ) C2(s)/(1+C2(s)P(s))) is minimum phase, the shaped reference rf will be stable during stable SM conditioning regimes, so stability of the main loop and SM dynamics directly implies stability of the overall system. In fact, rf will be a bounded input to a stable loop, thus leading to a stable response. Note that as rf approaches the reference step value r, less control effort is necessary to cross the sliding surface and reenter the linear region. Then, the state trajectory will finally point toward the interior of the linear region from both sides of the surface, thus abandoning the sliding mode and recovering the unconstrained operation.
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Figure 2. Smith predictor PI control of example 4.1.
4. To shape the reference rf, the signals w+ and wmust be designed to satisfy the SM existence condition w+ e weq e w-.21 Actually, one of these values (say, w-) is 0 because the conditioning loop takes no action within the linear region (say, s < 0). In fact, there is no intention to modify the state trajectory in the linear region. The other signal (w+) must be designed to satisfy w+ e weq during the sliding regimes in order to avoid saturation. Bounds on the signals in eq 10 could be used to check this condition and to design w+. Anyway, SM correction is carried out on the low-power side of the system, so there are not practical limitations to use conservatively large values of w+. As a rule of thumb, values on the order of the expected reference step r will be sufficient unless the SM dynamics is designed excessively fast. 4. Examples 4.1. SM Reference Conditioning Algorithm with Antiwindup Features. Consider a high-vacuum distillation column, which is a typical long dead-time process, together with a Smith predictor PI controller (Figure 2). The parameters of the real process, the model, and the controller are taken from ref 22. (The design of the controller corresponds to the first step of the control system design, which falls beyond the scope of this work. In fact, the proposed AW compensation is independent of the design methodology used in the first step. Then, both the plant and the Smith predictor PI controller are given, and an AW loop is designed to approximate the linear response as much as possible.)
Gp: gp(s)e-Lps )
0.57 e-18.70s (8.60s + 1)2
Gm: gm(s)e-Lms )
(
PI: Kp 1 +
)
0.57 e-18.80s 2 (7.99s + 1)
(
)
1 1 ) 1.75 1 + sTi 7.99s
(14) (15) (16)
The performance of the closed loop in the absence of actuator constraints is observed in Figure 3 (dotted line). Additionally, the dashed lines depict the response of the closed loop with the actuator restriction up e UP ) 2. It is confirmed in Figure 3b that the controller output u exceeds the available control effort for a long time, leading to a long settling time in the controlled variable y (Figure 3a). To avoid windup, the proposed SM reference conditioning algorithm is used. Considering that C1 ) 1 in the original controller design, a first-order reference filter is included.
F:
{
x˘ f ) λf(-xf + r + w) rf ) xf
(17)
Its time constant λ-1 f ) 1 has been chosen much faster than the closed-loop dynamics so that it does not appreciably affect by itself the linear response. The
Figure 3. (a) Controlled variable y and (b) controller output u for the linear system (dotted) and for the constrained system without (dashed) and with (solid) SM conditioning. (c) State trajectories in the plane (u, xq).
auxiliary control signal w switches according to the switching law s ) u - UP. Since the controller has a proportional term, the error e is implicit in u. Then a sliding regime can be established on the surface S defined by s ) 0, completely avoiding actuator saturation. The response of the closed loop with the proposed AW conditioning is also shown in Figure 3 (solid line). The control signal confirms that the actuator never saturates, whereas the process output response shows the graceful degradation from the unconstrained system response. In Figure 3c, the control/state trajectories are illustrated in the plane (u, xq), being xq the integral state of the PI controller. In the absence of AW compensation, the control signal u enters the saturation region (at tA) after the set-point step. The system returns to linear operation at time tC, leading to a large overshoot in the integral state (i.e., windup) and, therefore, a long settling time in y. The state trajectory of the system with the proposed SM conditioning coincides with the previous one until tA. At that moment, the conditioning loop produces an abrupt change in the trajectory direction through the discontinuous control action w. Actually, this control action forces the state to return to the linear region where w ) 0. This situation is repeated at high frequency (ideally infinite), leading to a sliding regime on the surface S. At tB, the trajectory of the uncompensated system (i.e., with w ) 0) points no more toward the saturation region, but in the opposite direction. Then, the sliding regime is left, and the system evolves in the linear region without conditioning toward the equilibrium point. In Figure 4, the performance of the closed loop without and with the proposed AW reference conditioning are compared for different saturation levels (UP ) [2.5, 2.2, 2, 1.9]). It is seen that as the available control effort is reduced, the settling time of the uncompensated system increases notoriously. Conversely, the closedloop response with the proposed SM reference conditioning degrades gracefully from the linear response in all cases. Remarks. (a) Evolution within the linear region is the desired mode of operation, so there exists no reaching mode in the sense that no control effort is done to enforce the state to reach the sliding surface from any initial condition.
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Figure 4. Performance of the constrained system without (a, b) and with (c, d) SM conditioning for different saturation levels. (a, c) Controlled variable y, (b, d) controller output u (dashed) and process input up (solid).
(b) As a result of the SM antiwindup conditioning, the controller and the actuator outputs coincide at every time. Since no saturation occurs, the main control loop remains closed all the time. (c) Since the conditioning acts on the reference, the SM protection does not alter the prescribed linear behavior of the main control loop. In fact, the closedloop poles of the main loop are independent of the conditioning circuit. 4.2. SM Reference Conditioning Algorithm for Safety Operation. Consider now the following example. The equations
[ ] [ ]
0 .01 0 0 x˘ ) 0 -1 1 x ) 50 (r - y) 0 0 0 .001 y ) x1
(18) (19)
correspond to a closed-loop system. The first two equations of eq 18 describe the dynamics of the process, whereas the third one is the integral state of the PI controller. It is supposed here that, for safety operation of the process, the variable x2 must be upper-bounded to X2 ) 20. To fit within the general framework previously developed, the controller and process are redefined. Effectively, eq 18 is reinterpreted as a process P′ described by the first differential equation and a strictly proper controller C′2 described by the last two differential equations. To meet with this constraint on x2 (which is seen here as a control signal), an SM conditioning of the reference is proposed. Unfortunately, the obvious switching law s(x) ) x2 - X2 cannot be used because the transversality condition ke * 0 does not apply (see eq 10), so a sliding regime cannot be established on s(x) ) 0, and x2 might cross the surface and evolve toward dangerous operating regions. To overcome this obstacle, a new switching law defined by s(x) ) x2 + τx˘ 2 - X2 ) 0 is proposed. Note that the error is implicit in x˘ 2 ) - x2 + 50e + x3. To ensure safety operation, the SM reference conditioning starts even before x2 reaches its upper bound. During the sliding regime, x2 converges exponentially toward its limit value with time constant τ. Figure 5a
Figure 5. Closed-loop response with (solid) and without (dashed) SM conditioning, (a) variable x2, (b) variable x1, and (c) phase trajectories in (x2, x˘ 2).
shows the smooth convergence of x2 to its bound at X2 ) 20 with the fast sliding dynamics. Figure 5c displays the associated system trajectory in the phase plane (x2, x˘ 2). The SM protection acts along the line AB. Obviously, the slope of the sliding surface as well as the sliding dynamics is given by the sliding gain τ. Once the system trajectory points toward the safety region from both sides of the sliding surface, the system naturally evolves in this region without reference conditioning toward its equilibrium point. Finally, Figure 5b shows the graceful degradation of the controlled variable x1 caused by the SM conditioning. Remark: As this example shows, the proposed SM algorithm originally developed to cope with actuator nonlinearities can be extended to overcome problems associated to other kind of restrictions. 5. Conclusions A novel reference conditioning algorithm to cope with restrictions in process control is developed using tools of sliding mode control. The most distinctive feature of the proposed methodology is that the main loop dynamics and the SM conditioning dynamics are designed independently one of each other. In fact, due to the robustness properties of SM control, the conditioning loop dynamics is insensitive to the process output evolution. Additionally, since the conditioning loop acts on the reference and nonlinearities are completely avoided, the prescribed closed-loop linear behavior of the main loop is never abandoned. Moreover, stability of the overall system is accomplished because the shaped reference is bounded during the SM conditioning. Consequently, no further analysis is necessary to guarantee the stability of the system. Finally, the implementation and tuning of the SM conditioning algorithm is extremely simple. Acknowledgment This work was funded by CICpBA, ANPCyT, CONICET and UNLP. Literature Cited (1) Turner, M. C.; Walker, D. J. Linear quadratic bumpless transfer. Automatica 2000, 36, 1089-1101.
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Received for review June 10, 2004 Revised manuscript received October 12, 2004 Accepted October 13, 2004 IE049494P
