Ind. Eng. Chem. Res. 2008, 47, 2721-2727
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Collective Sliding-Mode Technique for Multivariable Bumpless Transfer Fabricio Garelli,*,† Ricardo J. Mantz,‡ and Herna´ n De Battista§ Laboratorio de Electro´ nica Industrial Control e Instrumentacio´ n, UniVersidad Nacional de La Plata, C.C.91 (1900) Argentina
This paper proposes a strategy for the reduction of the undesired effects caused by manual-automatic or controller switching in multivariable process control. The proposal takes advantage of dynamic sliding mode properties to avoid inconsistency between the off-line controller outputs and the plant inputs. As a consequence, jumps at the plant inputs are prevented (which is known as bumpless transfer) and undesired transients on controlled variables are significantly reduced. Some advantages of the proposed algorithm are that (1) its implementation is extremely simple, (2) it presents distinctive robustness properties, which are characteristic of sliding regimes, and (3) it does not need the model of the plant. 1. Introduction A common practice in automatic control, especially in the chemical industry, is to take the plant manually to the operating point and just then to connect the controller so that the system starts operating automatically. As is well-known, such a mode switch may cause jumps at the plant inputs and a deterioration of the system response if no action is taken to avoid it. The suppression of the jumps at the plant inputs and their associated transient effects is referred to as bumpless transfer. Because of the practical importance of this topic, there has been a lot of research in this area. Many contributions have dealt with bumpy transfers together with windup problem (caused by plant input constraints) because of their similarities. One of the earliest published methodologies was proposed by Hanus et al.,1 which is based on the concept of “realizable reference”, and it has been applied to many real-life projects. Among the large number of articles that have been subsequently reported in the literature, concepts of linear quadratic theory,2 linear matrix inequalities,3 L2 bounds on state mismatch,4 state/ output feedback,5,6 and H∞ optimization7 have also been exploited to find solutions to windup and bumpy transfers. Contributions on this field also allow achieving smooth commutations between multiple linear controllers, which is particularly significant when switched control of nonlinear systems is considered.8 This work introduces concepts of variable structure system theory and the associated sliding regimes to solve the problems that arise from switching between open-loop (OL) and closedloop (CL) operation or from commutations between controllers for different operating points in MIMO systems. One of the main advantages of the resulting proposal is that it is applicable to controllers for which conventional bumpless algorithms were not conceived, like multivariable controllers with general transfer matrix, and even then it requires minimal design and implementation effort. It also presents distinctive robustness properties, which are characteristic of sliding regimes. Furthermore, the chattering phenomenonswhich usually degrades the performance of variable structure controlsdoes not affect at all the present application, and the model of the plant is not necessary for the methodology to be applied. * Corresponding author. E-mail:
[email protected]. Tel./ Fax: +54 221 425 9306. † Prof. Garelli is a member of CONICET. ‡ Prof. Bianchi is member of CICpBA. § Prof. De Battista is a member of CONICET.
The paper is organized as follows. Section 2 reviews some basic concepts on sliding-mode control for multi-input/multioutput (MIMO) systems. In Section 3, the sliding-mode (SM) algorithm proposed in this article to achieve bumpless transfer is described. This section also gives sufficient conditions for assuring the reach of the corresponding surface and analyzes the hidden dynamics of the conditioning loop once SM is established. The approach properties are verified through simulations on a benchmark MIMO process in Section 4. Finally, some final comments and concluding remarks are given. 2. Basic Concepts on Sliding Mode A variable structure system comprises a set of continuous subsystems with a switching logic that is a function of the system state. A particular operation is achieved when switching occurs at a very high frequency constraining the system state to a surface, named a sliding surface. This kind of operation is called sliding mode (SM) and has many attractive properties. It is robust to parameter uncertainties and external disturbances, it reduces the order of the sliding dynamics that become dependent on the designer-chosen sliding surface, and it is easy to implement.9 Because of its interesting features, a large number of papers presenting practical applications of SM control have been reported. For instance, in refs 10-14, the application of SM to chemical process control is discussed. Consider the following dynamical system, m
x3 ) Ax +
biwi ) Ax + Bw ∑ i)1
(1)
where x ∈ Rn is the system state and w ∈ Rm is the control vector. Matrices A and B (and its column vectors bi) are of consistent dimensions. The variable structure control law is defined componentwise as
wi )
{
w+ i if si(x) > 0 , i ) 1 ‚‚‚ m wi if si(x) < 0
(2)
according to the sign of the scalar switching functions si(x) ) Ri - kTi x. The sliding surface S is defined as the intersection of the m so-called individual sliding surfaces Si defined by
Si ) {x ∈ Rn: si(x) ) 0}
10.1021/ie070870q CCC: $40.75 © 2008 American Chemical Society Published on Web 03/12/2008
(3)
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Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 Table 1. Quadruple Tank Parameters for Operating Point Pfixed parameters
value
A1, A3 A2, A4 a1, a3 a2, a4 kc g
28 cm2 32 cm2 0.071 cm2 0.057 cm2 0.5 V/cm 981 cm/s2
parameters Figure 1. Dynamic SM bumpless strategy for multivariable systems.
(h01; h02; (V01; V02) (k1; k2)
h03;
value at Ph04)
(12.4; 12.7; 1.8; 1.4) cm (3.00; 3.00) V (3.33; 3.35) cm3/(V s) (0.7; 0.6)
(γ1; γ2)
controllers commutation, for which most of the following analysis is also valid. Although multivariable square processes are considered here, the method can be obviously applied to single-input/single-output (SISO) systems as a particular case. P(s) ∈ Rm×m represents the plant to be controlled, while C(s) ∈ Rm×m is a multivariable controller (centralized or decentralized) designed for stable closed-loop operation near the operating point P0. It is assumed that this controller is biproper; thus, it has a minimal realization like
C(s):
Figure 2. Schematic diagram of the quadruple tank.
Thus, n T S) m ∩ Si ) {x ∈ R : s(x) ) R - K x ) 0} i)1
(4)
with s(x) being the vector of switching functions, i.e., s(x) ) [s1(x) s2(x) ‚‚‚ sm(x)]T. Besides, R ) [R1 R2 ‚‚‚ Rm]T is the SM reference vector. Finally, the columns of the matrix K are the feedback gain vectors ki, i.e., K ) [k1 k2 ‚‚‚ km]. A sliding motion locally exists on a particular individual sliding surface Sj if, as a result of the switching logic (eq 2), the following reaching condition is satisfied
{
s˘ j < 0 if sj > 0 s˘ j > 0 if sj < 0
A necessary condition for eq 5 to be satisfied is that the switching function sj has a relative degree of one with respect to the discontinuous signal wj.9 Once the individual surface Sj is reached, the control action wj switches at a high frequency constraining the state trajectory to Sj. If the individual sliding motion converges toward the intersection surface S, where all controllers induce sliding motions on their individual surfaces, then the combination of individual sliding motions results in a collective sliding regime. The intersection surface S can also be reached without arriving first at any individual surface Sj, as we will see in the following section. 3. Development of the Bumpless Algorithm Figure 1 presents a schematic diagram of a control system with the proposed manual-automatic bumpless strategy. A slight changed scheme will be suggested in Section 4 to address
x3 c ) Acxc + Bcec u ) Ccxc + Dcec
(6)
with Dc being a nonsingular matrix. Observe that this assumptionswhich significantly simplifies the method explanations does not impose a severe restriction, since in multivariable design biproper controllers are aimed to avoid unnecessary delays in closed-loop transfer matrices. In any way, the extension of the method to controllers with singular Dc can be made including additional controller states in the sliding functions that are proposed in eq 9, as was done in ref 15. The loop around C(s), inside the dotted box of Figure 1, is the proposed correction via dynamic sliding mode.16 Therein, the filter F(s) is aimed to smooth the signal ws added at the controller input and to guarantee the necessary condition for the establishment of sliding regimes, i.e., that the switching function s(x) has relative degree of one with respect to the discontinuous action w. This filter can be represented in statespace as
F(s): (5)
{
{
x3 f ) Afxf + Bfw ws ) Cfxf
(7)
with Af ) -Cf ) λfIm and Bf ) Im. The discontinuous signal w ∈ Rm is given by
w(x) ) M(x)sign(s(x))
(8)
where x ) [xc xf]T, M(x) is a diagonal gain matrix, and
s(x) ) [s1(x)s2(x) . . . sm(x)]T ) uˆ - u(xc, ec)
(9)
is the switching function, with u ∈ Rm and uˆ ∈ Rm being vectors that contain the controller outputs and plant inputs, respectively. The objective of the variable structure loop added to the original controller C(s) is the establishment of a dynamic sliding regime over the surface S ) {x: s(x) ) 0}, in such a way that the controller outputs u are forced to coincide with the plant inputs uˆ , thus avoiding jumps at the plant input when the main control loop is closed. Naturally, in order to guarantee the stability of the closed-loop system, commutation must be
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2723
Figure 3. Jumps at the plant inputs and transients in the controlled variables when commutation is performed without compensation.
Figure 4. Multivariable bumpless transfer by means of the proposed SM method.
performed within the domain of attraction17 of the operating point P0, which may be characterized as
D0 ) {xcl0 : lim φ(t, t0, xcl) ) P0} tf∞
(10)
where xcl is the closed-loop state vector and φ(t, t0, xcl) denotes the state trajectory corresponding to the initial condition xcl(t0) ) xcl0 evaluated at time t. 3.1. Reaching Condition for Collective SM. For the existence of collective sliding regimes in the intersection of the surfaces Sj (sj(x) ) 0, j ) 1, ..., m), it is not necessary that condition 5 holds for each individual surface.18 In this case, the existence conditions of MIMO sliding modes can
be formulated in terms of the stability theory given by Lyapunov functions. Particularly, it has been demonstrated that, if the derivative of the switching function s(x) is expressed as
s3 ) d(x) - D(x)sign(s)
(11)
then a sufficient condition for the existence of SM (with convergence to the surface in finite time) is that for x ∈ Sd ⊂ S, with Sd being the domain of interest in S ) {x: s(x) ) 0},
(c1) D(x) is positive definite, D(x) + DT(x) > 0 λ0 , with λmin (x) > λ0 > 0 (c2) ||d(x)|| < d0 < xm
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Figure 5. Auxiliary signals ws added by the dynamic SM at controller input, and switching function s(t) ) uˆ - u.
Figure 6. Verification of the sufficient condition ||d(x)||2 < d0 for SM existence.
where λmin(x) is the minimum eigenvalue of 1/2(D(x) + DT(x)).18 In the proposed bumpless scheme, replacing the expressions of eqs 6 and 7 in the derivative of the switching function s ) uˆ - u yields
d(x) ) uˆ - CcAcxc - CcBcec - Dce3 - DcCfAfxf ) uˆ - CcAcxc - CcBcec - Dce3 - λfDcws
(12)
D(x) ) DcCfBfM ) -λfDcM
(13)
Therefore, SM establishment will be assured provided d(x) and D(x) satisfy conditions c1 and c2. Clearly, the matrix D(x) depends on the control law (eq 8), and consequently condition c1 can be satisfied by choosing the signs of the diagonal entries in M so that every leading principal minor of M is positive.19 Moreover, the value λmin(x) can be incremented by enlarging the gains in M. This allows, once verified the condition c1, adjusting these diagonal entries in order to satisfy the reaching condition c2. In this manner, the SM
establishment will be assured; consequently, the error between the controller output vector u and the plant input uˆ will reduce to zero, thus eliminating potential jumps in the signal at the plant input when commutation is performed. It is important to remind that, from a theoretical point of view, for bounded initial conditions, the time in which the system reaches the surface S (reaching time) can be made arbitrarily short.18 Comment: As mentioned, a positiVe definite D(x) can be obtained by a proper choice of the elements signs in M. HoweVer, for high-dimensional multiVariable systems, selecting by inspection each entry sign of the matrix M may be quite tedious. An alternatiVe procedure for this kind of systems is to take
M ) ηDc-1, η ∈ R+
(14)
D(x) ) -λfηIm
(15)
so that
which is always positiVe definite.
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2725
3.2. Hidden SM Dynamics. An open-loop representation of the SM compensation can be derived from the state-space representations of the controller C(s) and the filter F(s), eqs 6 and 7, respectively, resulting in
[] [
x3 c Ac Bc e3 c ) 0 λfIm
][ ] [
] [
]
xc 0 0 ec + r3 - y3 - λf(r - y) + - λfw (16) (17) u ) Ccxc + Dcec
Since C(s) is biproper (Dc nonsingular), during SM -1
ec ) Dc (uˆ - Ccxc)
Figure 7. Bumpless transfer topology for controller switching.
(18)
which results from eq 17 after making eq 9 equal to zero. Thus, the last row of eq 16 becomes redundant. Replacing eq 18 in the first rows of eq 16 results in the following reduced dynamics for the SM
x3 c ) Qcxc + BcDc-1uˆ Qc ) Ac - BcDc-1Cc
(19)
where the eigenvalues of Qc are the zeros of the controller C(s). Therefore, the SM dynamics given by eq 17 will be stable as long as the controller is minimum phase. It is important to remark that this SM dynamics does only depend on the controller parameters, and it is not seen from the controller output because u ) uˆ during the sliding regime. Thus, all the dynamics associated to SM are hidden dynamics. Another interesting feature of the proposed compensation is that the second and third terms of eq 16 satisfy the matching condition,20 i.e., they are collinear vectors. Consequently, because of SM robustness properties, the variable structure loop will present strong invariance to the references r and output disturbances that might appear in y. This distinctive property of the present algorithm is verified by eq 17 obtained for SM dynamics.
c11(s) )
The laboratory process known as quadruple tank, originally proposed in ref 21, is considered as an example for the application of the method to nonlinear MIMO processes. A schematic diagram of this plant, which has been used for the evaluation of many multivariable strategies in the last years, is presented in Figure 2. The nonlinear model of this plant, derived from physical data, mass balances, and Bernoulli’s law, is given by
a1 a3 γ1k1 dh1 ) - x2gh1 + x2gh3 + V dt A1 A1 A1 1 (20)
(1 - γ2)k2 a3 dh3 V2 ) - x2gh3 + dt A3 A3 (1 - γ1)k1 dh4 a4 V1 ) - x2gh4 + dt A4 A4 where hi represents the water level in each tank. Ai and ai are the cross sections of the tanks and the outlet holes, respec-
2.385(s + 0.043)(s + 0.033)(s + 0.016) (21) s(s + 0.059)(s + 0.017)
c12(s) )
-0.08(s + 0.033)(s + 0.011) s(s + 0.059)(s + 0.017)
(22)
c21(s) )
-0.039(s + 0.043)(s + 0.016) s(s + 0.059)(s + 0.017)
(23)
c22(s) )
4. Simulation Results
dh2 a2 a4 γ2k2 ) - x2gh2 + x2gh4 + V dt A2 A2 A2 2
tively. The constants γ1, γ2 ∈ (0, 1) are determined from how two flow-divider valves are set. The process outputs are the signals in volts generated by the sensors in the lower tanks (y1 ) kch1 and y2 ) kch2), while the inputs to the system are the voltage Vi applied to the two pumps (the corresponding flow is kiVi), which must be between 0 and 12 V to avoid damaging the pumps. The values of the system parameters for the operating point P-, at which the system shows minimum phase (MP) characteristics, are given in Table 1.21 For this operating point, a controller that achieves closed-loop dynamic decoupling (diagonal complementary sensitivity T(s) around P-) was designed following the procedures described in ref 22. The resulting centralized controller consists of the individual transfer functions
3.214(s + 0.043)(s + 0.033)(s + 0.011) (24) s(s + 0.059)(s + 0.017)
and it has all its multivariable zeros in the left half-plane. 4.1. Manual-Automatic Switching. Simulations were run on the nonlinear model of the plant with the controller given by eqs 21-24. The system was taken manually close to P-, and the time for commutation to automatic mode was set as that one in which both water levels surpass the 90% of the value corresponding to P-. Hence, ts ) 255 s turned out to be the instant for switching from open- to closed-loop operation. Figure 3 plots the results obtained by simulations carried out without the proposed methodology. The jumps at the plant inputs caused by the inconsistency between u and uˆ were limited to 12 V with an amplitude limiter, but even then they produced unacceptable transients in the water levels whose control is aimed. In effect, if this commutation had been performed on the real system, it would have overflowed the lower tanks. In order to improve the system response, the proposed algorithm was added. Taking λf ) -0.05 in F(s) and M ) 20 I2, from eqs 21-24 we have
D(x) ) Dc )
[
2.385 0 0 3.214
]
(25)
which is clearly positive definite. The responses obtained with
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Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008
Figure 8. Bumpy (a) and bumpless (b) transfer between decoupling controllers for the quadruple tank at different operating points.
the dynamic SM correction of the MIMO controller are shown in Figures 4-6. Figure 4 reveals how the controlled variables reach the desired operating point without overshooting it after the mode switching. It also verifies that the controller output u is always close to um during manual operation (until ts). It is important to remark that, although the chattering effect in u previous to commutation can always be reducedsas was done in simulationssfor being the SM confined to the low-power side of the system, in the present application this phenomenon would not affect at all any component or variable of the system. Indeed, the chattering in u may only occur when this signal is not connected to the plant. Finally, Figure 5 illustrates the temporal evolution of the auxiliary signal ws and the switching function s ) uˆ - u corresponding to Figure 4, while Figure 6 verifies that the SM loop design satisfies the reaching condition ||d(x)|| < d0 < λ0/ xm ) 1.686, where this latter bound was obtained from eq 25. 4.2. Automatic-Automatic Commutation. Now, the proposed methodology is evaluated for being applied to avoid bumpy transfers when switching between different controllers is performed. To this end, the scheme of Figure 1 is slightly modified, as shown in Figure 7 for the case of the quadruple tank. Observe that the analysis of Section 3 is also valid for this configuration, except for eq 15. Since the off-line controller (C2(s)) is now disconnected from the main control loop before the commutation, neither the reference r nor the output y will affect the open-loop dynamics of the SM compensation. In addition, it is worth mentioning that, like occurs in general with bumpless algorithms when nonarbitrary switching is considered (see refs 2 and 4 and references therein), the present proposal does not guarantee the stability of the switched closed-loop system. This depends on the conditions under which controllers are switched, and it might be considered, for example, in a supervisory level. However, the proposed method reduces the risk of instability by making the off-line controller states consistent with the actual plant inputs. Controller C1(s) in Figure 7 corresponds to the one described by eqs 21-24 that was designed to decouple the system around P-, while controller C2(s) is a decoupler controller obtained
from the model linearization at another operating point P/-, with (V01; V02) ) (3.9; 3.9) V and (h01; h02; h03; h04) ) (20.7; 21.6; 2.8; 2.4) cm. In order to preserve a good degree of decoupling of the nonlinear system, a commutation to C2(s) was scheduled for the case in which both controlled water levels h1 and h2 are greater than 17 cm. C1(s) and C2(s) have the same relative degree structure and direct feed-forward matrix Dc, but different poles and zeros locations. Therefore, the SM compensation can be implemented with the same filter F and matrix M that were used for the manual-automatic switching. Once at P- (it can be viewed as being at t ) 1000 s in the top plot of Figure 4), two positive step references were applied just to take the system to P/-, and as a consequence, the condition for controller switching was reached at t ) 1306.2 s. The responses to the step references and the controller commutation at t ) 1306.2 s is depicted in Figure 8a for the system without the proposed methodology and in Figure 8b for the case in which the scheme of Figure 7 was implemented. The SM bumpless compensation considerably reduces the transient caused by the controller switching, and it helps to preserve the decoupling of the system. The response of the closed loop to the last two negative step references verifies that C2(s) achieves the dynamic decoupling of the quadruple tank at P/-. 5. Conclusions In this paper, a variable structure loop was proposed to be added to the original controller in order to reduce the effects caused by manual-automatic or controller switching in MIMO process control. The proposal takes advantage of dynamic sliding regimes to avoid discrepancies between the off-line controller outputs and the plant inputs prior to the mode switching. In this manner, jumps at the plant inputs are eliminated and undesired transients on controlled variables are significantly reduced. The chattering phenomenon, a common drawback of SM, does not affect the present application, while the design effort in order to implement the algorithm is minimum. In the development of the proposal, biproper and minimum phase controllers were considered. The first assumption was
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2727
made for the sake of simplicity, and it can be relaxed at the cost of a greater design complexity. In order to apply the algorithm to nonminimum phase controllers, future extensions of the methodology could take into account the conditioning of controller states rather than controller inputs. Acknowledgment This work was supported by ANPCyT (PICT 11-14111), CONICET (PIP 5532/05), CICpBA, and UNLP. Literature Cited (1) Hanus, R.; Kinnaert, M.; Henrotte, J. Conditioning technique, a general anti-windup and bumpless transfer method. Automatica 1987, 23, 729-739. (2) Turner, M.; Walker, D. Linear quadratic bumpless transfer. Automatica 2000, 36, 1089-1101. (3) Mulder, E.; Kothare, M.; Morari, M. Multivariable anti-windup controller synthesis using LMI. Automatica 2001, 37, 1407-1416. (4) Zaccarian, L.; Teel, A. The L2 (l2) bumpless transfer problem for linear plants: Its definition and solution. Automatica 2005, 41, 1273-1280. (5) Wu, W. Anti-windup schemes for a constrained continuous stirred tank reactor process. Ind. Eng. Chem. Res. 2002, 41, 1796-1804. (6) Zheng, K.; Lee, A.; Bentsman, J.; Taft, C. Steady-state bumpless transfer under controller uncertainty using the state/output feedback topology. IEEE Trans. Control Syst. Technol. 2006, 14, 3-17. (7) Edwards, C.; Postlethwaite, I. Anti-windup and bumpless transfer schemes. Automatica 1998, 34, 199-210. (8) Liberzon, D. Switching in Systems and Control; Systems & Control: Foundations and Applications series; Birkha¨user: Boston, 2003. (9) Edwards, C.; Spurgeon, S. Sliding Mode Control: Theory and Applications, 1st ed.; Taylor & Francis: London, 1998. (10) Camacho, O.; Smith, C.; Moreno, W. Development of an internal model sliding mode controller. Ind. Eng. Chem. Res. 2003, 42, 568-573.
(11) Herrmann, G.; Spurgeon, S.; Edwards, C. A model-based sliding mode control methodology applied to the HDA-plant. J. Process Control 2003, 13, 129-138. (12) Mantz, R.; De Battista, H.; Bianchi, F. Sliding mode conditioning for constrained processes. Ind. Eng. Chem. Res. 2004, 43, 8251-8256. (13) Chen, C.; Peng, S. Design of a sliding mode control system for chemical processes. J. Process Control 2005, 15, 515-530. (14) Garelli, F.; Mantz, R.; De Battista, H. Limiting interactions in decentralized control of MIMO systems. J. Process Control 2006, 16, 473483. (15) Garelli, F.; Mantz, R.; De Battista, H. Sliding mode reference conditioning to preserve decoupling of stable systems. Chem. Eng. Sci. 2007, 62, 4705-4716. (16) Sira-Ramı´rez, H. On the dynamical sliding mode control of nonlinear systems. Int. J. Control 1993, 57, 1039-1061. (17) Sastry, S. Nonlinear Systems: Analysis, stability and control; Springer-Verlag: New York, 1999. (18) Utkin, V.; Guldner, J.; Shi, J. Sliding Mode Control in Electromechanical Systems, 1st ed.; Taylor & Francis: London, 1999. (19) Chen, C. Linear system theory and design, 3rd ed.; Oxford University Press: New York, 1999. (20) Sira-Ramı´rez, H. Differential geometric methods in variable structure systems. Int. J. Control 1988, 48, 1359-1390. (21) Johansson, K. The quadruple-tank process: A multivariable laboratory process with an adjustable zero. IEEE Trans. Control Syst. Technol. 2000, 8, 456-465. (22) Goodwin, G.; Graebe, S.; Salgado, M. Control System Design, 1st ed.; Prentice Hall: Upper Saddle River, NJ, 2001.
ReceiVed for reView June 25, 2007 ReVised manuscript receiVed January 2, 2008 Accepted January 23, 2008 IE070870Q
