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A swarm intelligence-based tuning method for the sliding mode generalized predictive control J.B. Oliveira n,1, J. Boaventura-Cunha, P.B. Moura Oliveira, H. Freire INESC TEC - INESC Technology and Science (formerly INESC Porto, UTAD pole) Department of Engineering, School of Sciences and Technology 5001-811 Vila Real, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2013 Received in revised form 9 June 2014 Accepted 11 June 2014 This paper was recommended for publication by Dr. R. Dubay
This work presents an automatic tuning method for the discontinuous component of the Sliding Mode Generalized Predictive Controller (SMGPC) subject to constraints. The strategy employs Particle Swarm Optimization (PSO) to minimize a second aggregated cost function. The continuous component is obtained by the standard procedure, by Quadratic Programming (QP), thus yielding an online dual optimization scheme. Simulations and performance indexes for common process models in industry, such as nonminimum phase and time delayed systems, result in a better performance, improving robustness and tracking accuracy. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model Predictive Control Sliding Mode Particle Swarm Optimization Soft computing Robustness
1. Introduction Controllers play a central role in industrial plants, since they are designed to regulate process variables in accordance with some performance criteria. A low energy consumption is also of practical interest, thus arising the well known trade-off: tracking or regulation accuracy versus energy consumption. To cope with these issues, controller parameters must be appropriately adjusted and research in this subject keeps its relevance since the work by [1], regarding Proportional Integral Derivative (PID) controller. This tuning method allows getting an easier initial set for PID parameters. When some intelligence is incorporated into the closed loop, one has an online or automatic tuning method [2]. Metaheuristic algorithms, such as Particle Swarm Optimization (PSO) are quite feasible in such cases. PSO is a natural inspired computation technique introduced by [3] and it is characterized for its simplicity and high efficiency in searching global optimal solutions in problem spaces. This feature attracted the attention of control
n
Corresponding author: Tel.: þ 55 84 33424836. E-mail addresses: [email protected] (J.B. Oliveira), [email protected] (J. Boaventura-Cunha), [email protected] (P.B. Moura Oliveira), [email protected] (H. Freire). 1 Permanent address: Agricultural School of Jundiai - Federal University of Rio Grande do Norte 59280-000 Macaiba, Brazil.
engineers in the sense of a simple way of searching optimal or semi-optimal tuning of controller's parameters [4]. Besides PID, known for its simplicity due to just three parameters to tune, Model Predictive Controllers (MPC) are interesting for linear, nonlinear, time-delayed and nonminimum phase systems [5]. It offers a straightforward design method to anticipate future control actions within some time horizon (control horizon), in order to track a future behavior (in some prediction horizon), predicted by an explicit model. The most common model forms in the various MPC products rely on convolution (Finite Step Response FSR and Finite Impulse Response FIR) models, such as Dynamic Matrix Control (DMC), but recent controllers suggest a trend toward state space formulations which provides flexibility in representing stable, unstable, integrating and unmeasured disturbances, just as the Controller Auto-Regressive Integrated MovingAverage (CARIMA) model in the Generalized Predictive Control (GPC) [6,7]. According to [8,9], the objective function of GPC is very similar to that of DMC, with the fundamental difference of using a Diophantine equation and CARIMA model to formulate the dynamic matrix. Abu-Ayyad and Dubay [7] showed that GPC and Extended Predictive Control (EPC) can handle the system matrix ill-conditionality better than other MPC methods and, therefore, it still motivates the development and applications of GPC, as in [6]. The direct treatment of practical constraints such as actuator and output limits is carried out by the minimization or maximization of some objectives, expressed in its simpler form as an aggregated
http://dx.doi.org/10.1016/j.isatra.2014.06.007 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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quadratic cost function. For this optimization scenario, Quadratic Programming (QP) based on active-set algorithm is usual and PSO can be also applied to tune its parameters, as proposed by [10–13]. Considering its wide application in environments subject to disturbances, robustness is a necessary feature and must be taken into account. An attempt to aggregate robustness into GPC by combining GPC with Sliding Mode Control (SMC) was firstly reported in [14]. SMC is a nonlinear control scheme known to be robust to model uncertainties, disturbances and unmodeled dynamics, being quite suitable for industrial environments. Since the considerations by [15], the research on SMC theory and its applications have been of increasing interest, providing an engineering look at SMC. Key aspects were clarified, such as the chattering phenomena, both in continuous and discrete time. The key idea consists in choosing a state variables function (sliding surface) in which all trajectories must reach in finite time (reaching phase) and, once reached, cannot escape, sliding to the desired final value (sliding phase). A control law is then designed to force the trajectories towards this surface (corrective action) and, moreover, to keep them thereafter (equivalent control [16]). This control law must be discontinuous or, at least, it must contain a discontinuous component. In classical SMC, a possibility is for instance the tuning of sliding surface parameters, as obtained by [17,18]. A GPC based on PSO was compared with the traditional QP-based GPC in a greenhouse experiment, giving better results without great increase of computational burden [19]. The well succeeded melting of Sliding Mode Predictive Controllers (SMPC) motivated other works and applications [20–30]. Most of the work referenced is concerned with common process control problems, such as delayed and nonminimum phase systems, often represented by a First Order Plus Dead Time (FOPDT) transfer function. For this type of continuous-time models, Camacho and Smith [20] proposed a set of tuning equations for the initial values of the discontinuous component of the control law, as a function of the characteristic parameters of the FOPDT model. When other model structures are considered, including discrete and higher order systems, these equations are no longer valid and computational intelligent approaches are interesting in order to help online tuning of the controller. In [6] it is stated that, for the MPC controllers used in process industries today, the tuning emphasis is on disturbance rejection and suggest as trends and research directions the development of improved disturbance estimators and robust controllers, by means of randomized algorithms which would rely on extensive offline simulation. Tuning is therefore commonly based on offline simulation and the actual performance of the online controller. It is typically carried out using the nominal model and via trial and error try do determine steady state behavior, providing initial tuning values for the parameters. Such simulations need to consider expected model errors and incorporate the characteristics of unmeasured (stochastic) disturbances obtained, for instance, from the actual controller. In this sense, a tuning strategy incorporated into the control loop can provide adaptability and robustness, without significant increase of the computational load, considering the hardware technology currently available. In [2], it also corroborates that while tuning guidelines for initial tuning values may be found in the literature, these rules are not general and do not learn from the controller operation and system response. Following some design steps of the SMPC presented in [27], here named Sliding Mode Generalized Predictive Controller (SMGPC), this paper keeps QP active-set for the optimization of the continuous component of the control law responsible for the sliding phase, but proposes PSO as an optimization tool for selecting optimal parameters for the discontinuous component of the control law, thus yielding a dual optimization scheme (henceforth named Dual-SMGPC), applicable to a wider class of systems. Moreover, now
its parameters are adaptive, providing robustness during reaching phase. In order to test the way PSO can find the optimum solutions, besides the common approach of getting the results from the initial population, two other variations are compared: restarting the population after some iterations while keeping a member in the next population, and a totally random new population after restarting, to avoid possible local minimum. In traditional SMGPC, these parameters are kept constant and calculated offline, normally through simulations. Simulations on common process models will be presented and the results compared with SMGPC without PSO, with a fixed pair ðK d ; δÞ. The remaining of this paper is organized as follows: Section 2 states the GPC and both optimization problems (QP and PSO), discussing adjustments criteria; Section 3 presents and comment simulations results; Section 4 provides some conclusions and encourage future works.
2. Controller design The SMGPC presented here is based on a Controller AutoRegressive Integrated Moving-Average model (CARIMA), considered linear around each operating point and described as
ΔAðq 1 ÞyðkÞ ¼ Bðq 1 ÞΔuðk d 1Þ þ ξðkÞ;
ð1Þ
where d is the delay from input to output (here considered as a multiple of the sampling time), u is the input signal, q 1 is the backward-shift operator, Δ : 1 q 1 and ξ is the zero mean white noise. A and B are polynomials in q 1 defined as Aðq 1 Þ ¼ 1 þ a1 q 1 þa2 q 2 þ … þ ana q na ;
ð2Þ
Bðq 1 Þ ¼ b0 þ b1 q 1 þ b2 q 2 þ …þ anb q nb :
ð3Þ
According to the SMC theory [16], the first step to design the controller is to define a sliding surface, S(t), along which the process can slide to find its desired final value. Very often, S(t) is chosen in such a way that represents a desired system dynamics and/or control objective. For instance, S(t) could be the tracking error eo ¼ y w, with w being some reference signal. The problem of tracking a reference value can be reduced to keeping S(t) at zero. From [25,27], the j-step ahead prediction of S(k) with information until the actual instant t¼ k is given by ^ þjÞ ¼ P s ðq 1 Þðyðk ^ þjÞ wðk þ jÞÞ þ Q s ðq 1 ÞΔuðk þ j 1 dÞ: Sðk
ð4Þ
Polynomials P s ðq 1 Þ, Q s ðq 1 Þ have degree np and nq respectively, and allow to design the desired dynamics in the sliding condition. A common adjustment is choosing P s ðq 1 Þ and Q s ðq 1 Þ as a first order system: Q s ðq 1 Þ ð1 αÞq 1 ¼ ; 1 αq 1 P s ðq 1 Þ
ð5Þ
with 0 o α r 1, since all roots of P s ðq 1 Þ must be inside the unit circle [27]. As α-0 the dynamic is faster. The cost function aggregates two simultaneous objectives: Ny
Nu
j ¼ N1
j¼1
^ þ jÞ2 þ ∑ λ½Δuðk þj 1Þ2 J C ¼ ∑ ½Sðk
ð6Þ
where λ is set constant and N 1 N y is the period of time in which one desires the output tracks the reference signal and Nu is the control horizon. For these parameters, [27] suggested some intuitive relations, which can be used as initial values. Other online tuning strategies for these specific parameters are available in the literature [2].
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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Likewise GPC, to minimize (6) the output prediction within the interval j ¼ ½N 1 ; N y is formed by two parts: ^ þ jÞ ¼ yf þ yl ; yðk
ð7Þ
where yf is the forced response (considering the initial conditions null but subject to future control actions) and yl is the free response (natural system response from the initial conditions with no future control actions).
3
qs0 ¼ 0; qs1 ¼ 1 α; np ¼ 1; nq ¼ 1, 2 3 2 ps0 qs0 0 … 0 6p 7 6q p … 0 6 s1 7 6 s0 7; Q s ¼ 6 s1 Ps ¼ 6 6 ⋮ 6 ⋮ ⋮ ⋱ ⋮ 7 4 4 5 0 psnp … ps0 0
0 qs0
… …
⋮
⋱
qsnq
…
3 0 0 7 7 7; ⋮ 7 5 qs0
with order ðN y N 1 þ 1Þ ðN y N1 þ 1Þ and ðNy N 1 þ 1Þ N u , respectively. Substituting (17) into (6), the cost function becomes J C ¼ 12 ΔuTc GΔuc þ ΔuTc b þf 0 ;
2.1. Forced response, yf
ð19Þ
ð8Þ
with G ¼ 2ðλ1 Z Z þ λ2 IÞ, Z ¼ P s H þ Q s , f 0 ¼ λ1 ðP s ðyl wÞÞ ðP s ðyl wÞÞ and b ¼ 2λ1 Z T P s ðyl wÞ. A necessary and sufficient condition for discrete time reaching ^ þ 1Þj r SðkÞ ^ motion is jSðk [27] and a control law satisfying this condition does guarantee that all trajectories will enter and remain within at least a non-increasing domain. In order to guarantee the reaching condition of (17), a constraint is added into the optimization problem of (19), namely:
ð9Þ
SðkÞ1 þ P s ðyl wÞ r ðP s H þ Q s ÞΔuc r SðkÞ1 P s ðyl wÞ;
T
The forced response may be calculated from the step response of the parametric model (1): yf ¼ H Δu; where, 2
H ðNy N1 þ 1ÞNu
h0 6 h 6 1 ¼6 6 ⋮ 4 hNy 1
0
…
0
h0 ⋮
... ⋱
0 ⋮
hNy 2
…
hNy Nu
3 7 7 7; 7 5
ΔuðNu 1Þ ¼ ½ΔuðkÞ Δuðk þ 1Þ … Δuðk þ N u 1ÞT : j1
i¼1
i¼0
hj ¼ ∑ aj hj i þ ∑ bi ;
hj ¼ 0; 8 ðj iÞ o 0:
ð10Þ
ð11Þ
Although yf and yl might be calculated by solving a Diophantine equation [27], as usual, here the available parametric model is used, thus simplifying the design. The concept of free response states that the future control signal increments after t ¼ k 1 are zero, then Δuðk þ jÞ ¼ 0; j Z 0. Another assumption is that ^ þ jÞ; j r0, since yl takes into account the initial yl ðk þjÞ ¼ yðk conditions. Considering these conditions and after some manipulation in (1), one has for the general case: T
ð12Þ
A~ l ¼ ½ a~ 1 a~ 2 … a~ na þ 1 ; T ;
ð13Þ
Y l ¼ ½yl ðk þ jÞ yl ðk þj 1Þ … yl ðk þ j naÞT ;
ð14Þ
Bl ¼ ½b0 b1 … bnb T ;
ð15Þ
Δul ¼ ½Δuðk d þ jÞ Δuðk d þ j 1Þ … Δuðk d þ j nbÞT ;
ð16Þ
with
and a~ 1 ¼ a1 1; a~ 2 ¼ a2 a1 ; …; a~ na ¼ ana ana 1 ; a~ na þ 1 ¼ ana . 2.3. Prediction of the sliding surface SMGPC control law, ΔuSMGPC ðkÞ, is the combination of two additive parts: a continuous part Δuc ðkÞ ¼ ΔuðkÞ (10) developed like a GPC by the minimization of (6) using QP, which is responsible for keeping the process variable on the reference value, and a discontinuous part Δud ðkÞ to be detailed further, responsible for guiding the system to the sliding surface. To calculate Δuc ðkÞ, (7) is substituted into (4) and, after putting it into matrix form, one has ^ SðkÞ ¼ ðP s H þQ s ÞΔuc ðkÞ þ P s ðyl ðkÞ wðkÞÞ;
ð20Þ
2.4. Discontinuous control signal component The discontinuous part Δud is given by
2.2. Free response, yl
yl ðk þj þ 1Þ ¼ A~ l Y l þ BTl Δul ;
T
where 1Nu 1 is a vector whose entries are ones.
The parameters that compound H may be obtained from: j
ð18Þ
ð17Þ
with the matrices Ps and Qs composed from (5), ps0 ¼ 1; ps1 ¼ α;
^ þ jÞ Sðk
Δud ðk þ jÞ ¼ K d ^ ; jSðk þ jÞj þ δ
δ Z 0;
ð21Þ
where Kd is a gain responsible for the velocity of the reaching mode but also increases chattering, reduced by an appropriate δ. The pair ðK d ; δÞ directly affects system performance and then are commonly selected through simulations. For continuous FOPDT models, [20] proposed some initial tuning equations. However, this work proposes a new methodology based on PSO for a wider class of systems which is model independent, providing an adaptive online scheme for tuning these parameters. PSO is a popular population based metaheuristic search algorithm which is inspired by the mechanism of biological swarm social behavior, such as bird flocking and fish schooling, which can be used to search for optima [3]. In this case, besides (19), a second quadratic aggregated bi-objective cost function must be minimized by PSO, as follows: J D ¼ λ3 ½ðyl wÞT ðyl wÞ þ λ4 ½ΔuTd Δud ;
ð22Þ
where
λ3 ðkÞ ¼ signð sin ð2π k=FÞÞ;
λ4 ðkÞ ¼ 1 λ3 ðkÞ;
ð23Þ
with F is a user-defined adaptation frequency. This strategy named Bang-Bang Weighted Aggregation (BWA) [31,32] is based on sign function and force the algorithm to promote movements towards the optimal solutions. The weights change periodically depending on F. For example, if F¼ 200, the weights behave as shown in Fig. 1. One entity of the Np individuals is named particle and its position in the swarm represents a possible solution for the problem. Therefore, a pair ðK d ; δÞ represents a particle in a 2-dimensional search space and modifies its movement according to its own experience and its neighboring particle experience. In classical PSO, only two equations are used and updated in each iteration: position update (X) and velocity update (V). For the ith particle of the swarm (of size Np), one has for this case the vectors X i ¼ ðxi1 ; xi2 Þ and V i ¼ ðvi1 ; vi2 Þ. The previously best visited position of the ith particle is denoted by P i ¼ ðpi1 ; pi2 Þ, and g is the index of the best particle of the swarm (global best). The movement equations are vid ðtÞ ¼ ωðtÞvid ðt 1Þ þ c1 r 1 ðpid ðtÞ xid ðtÞÞ þ c2 r 2 ðpgd ðtÞ xid ðtÞÞ;
ð24Þ
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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4
λ3 BWA 1.5 1 0.5 0 −0.5 −1 −1.5
0
50
100
150
100
150
time (s) λ4 BWA
2.5 2 1.5 1 0.5 0 −0.5
0
50
time (s) Fig. 1. Bang-Bang weighted aggregation for λ3 and λ4.
Table 3 Performance Indexes G2 ðzÞ – average Values for 21 runs – nondisturbed case. Performance Index
QP-SMGPC
Dual1
Dual2
Dual3
Eu σu IAE ISE ITAE
141.3 0.2083 13.2 10.6 105.6 0.4000
142.9 0.1989 11.8 9.8 82.7 0.5236
144.0 0.1956 11.7 9.5 87.0 0.5249
143.6 0.1954 11.4 9.5 77.7 0.4462
– 3.0000 –
0.0218 3.2345 0.1590
0.2740 3.2036 1.5937
0 2.2965 0
Kd σK d δ σδ Fig. 2. Block diagram – dual SMGPC.
Table 1 Performance Indexes G1 ðzÞ – average values for 21 runs – nondisturbed case. Performance Index
QP-SMGPC ð103 Þ
Dual1 ð103 Þ
Dual2 ð103 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
1.24 0.0002 0.0292 0.0197 0.5918 0.0001
1.26 0.0002 0.0153 0.0116 0.1507 0.0005
1.27 0.0001 0.0126 0.0094 0.1159 0.0005
1.26 0.0002 0.0144 0.0110 0.1342 0.0004
– 0.0010 –
0 0.0032 0.0001
0.0003 0.0033 0.0016
0 0.0023 0
Kd σKd δ σδ
Table 2 Performance Indexes G1 ðzÞ – average values for 21 runs – disturbed case.
Table 4 Performance Indexes G2 ðzÞ – average values for 21 runs – disturbed case. Performance Index
QP-SMGPC ð103 Þ
Dual1 ð103 Þ
Dual2 ð103 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
0.32 0.0013 0.1231 0.3436 9.6530 0.0004
0.28 0.0011 0.0977 0.2387 7.3 0.0005
0.27 0.0010 0.0910 0.1966 6.7 0.0005
0.28 0.0011 0.0952 0.2271 7.1 0.0004
– 0.0030 –
0 0.0032 0.0012
0.0003 0.0032 0.0016
0 0.0023 0
Kd σK d δ σδ
Table 5 Performance indexes G3 ðzÞ – average values for 21 runs – nondisturbed case.
Performance Index
QP-SMGPC ð104 Þ
Dual1 ð104 Þ
Dual2 ð104 Þ
Dual3 ð104 Þ
Performance Index
QP-SMGPC
Dual1
Dual2
Dual3
Eu σu IAE ISE ITAE
0.26 0.0002 0.0389 0.0802 8.0653 0.00001
0.26 0.0002 0.0141 0.0187 3.0694 0.0001
0.27 0.0003 0.0122 0.0154 2.3814 0.0001
0.26 0.0003 0.0131 0.0173 2.7276 0.00004
Eu σu IAE ISE ITAE
172.6 0.3462 24.37 20.07 342.6 0.4000
176.0 0.3385 23.13 19.03 319.6 0.2719
177.6 0.3422 19.02 19.0 368.0 0.2771
180.6 0.3315 23.39 17.87 388.4 0.3113
– 0.0001 –
0 0.0003 0
0 0.0003 0.0002
0 0.0002 0
0.0197 2.2440 0.1639
0.1296 2.2328 0.9863
0 1.6433 0
Kd σKd δ σδ
Kd σK d δ σδ
– 4.500 –
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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xid ðtÞ ¼ xid ðt 1Þ þ vid ðtÞ;
Table 6 Performance Indexes G3 ðzÞ – average values for 21 runs – disturbed case. Performance Index
QP-SMGPC ð104 Þ
Dual1 ð104 Þ
Dual2 ð104 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
0.0217 0.0002 0.0176 0.0419 1.55 0.00004
0.0229 0.0002 0.0167 0.0388 1.45 0
0.0225 0.0002 0.0172 0.0409 1.51 0
0.2688 0.0018 0.1069 0.2209 7.8 0.0003
– 0.00045 –
0 0.0002 0.0001
0 0.0002 0
0 0.0011 0.0005
Kd σKd δ σδ
ð25Þ
where d¼1,2, i ¼ 1; …; N p . c1 ; c2 are constants, called respectively cognitive and social parameters, r 1 ; r 2 are random numbers obtained from a uniform distribution in the set ½0; 1, and ω is the inertia weight, balancing the effect of initial velocity in the exploration and exploitation process. After studying 15 different strategies for selecting ω, Bansal et al. [33] indicated that a fixed ω or a linear decaying ω would give good results. In this paper, both approaches are tested. The limits for the search space (therefore, for the controller parameters) are expressed by Xmax and the maximum velocity Vmax, normally associated with Xmax, controls the global exploration of the particles in the swarm.
QP−SMGPC x Dual−SMGPC
1.5
Process Variable
5
1 QP SetPoint Dual1 Dual2 Dual3
0.5 0 −0.5
0
50
100
150
time (s) Control Signal 2
u
1.5 1
QP Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Fig. 3. G1 ðzÞ: nondisturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
Kd Dual1
0.65
Kd
Kd
1.5
0.4 0
50
100
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 4. G1 ðzÞ: nondisturbed, Kd parameter.
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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6
13. Solve quadratic optimization problem for the continuous component subject to constraints (19,20) 14. // Discontinuous control component evaluation, Δu d 15. // PSO Loop, to obtain Kd and δ 16. Define nIter Reinitialize 17. t¼ 0 18. Initialize swarmX(t) 19. While(!(Iter)) 20. if(Rem(t; nIterReinitialize) ¼0) Reinitialize Swarm// Dual 1 and Dual 2 21. evaluate the discontinuous weighting cost
Therefore, the complete control signal increment for the SMGPC at t¼ k is
ΔuSMGPC ðkÞ ¼ Δuc ðkÞ þ Δud ðkÞ:
ð26Þ
It is noteworthy that at each instant k, although Nu components are calculated for Δuc and Δud , only the first component of each (uc(k), ud(k)) is considered, neglecting the others components, obeying the receding horizon principle. Then, the actual control signal sent to the process is given by uSMGPC ðkÞ ¼ uSMGPC ðk 1Þ þ uc ðkÞ þ ud ðkÞ:
ð27Þ
A generalized block diagram for the proposed Dual SMGPC may be seen in Fig. 2.
22. 23.
2.5. Control algorithm - summary
24. 25. 26. 27. 28.
In order to facilitate the simulation and implementation of the proposed technique, the aforementioned steps may be summarized by the following algorithm with the respective equations numbers: 1. 2. 3. 4.
Initialize SMGPC parameters: α, N1, Ny, Nu, F Define system polynomials and delay: A, B, d (2,3) Define polynomials Qs and Ps with a initialized in step 1 (5) Define control signal saturation and slew rate limits
Define PSO search intervals for Kd and δ 6. Define Swarm size, Np, number of iterations, Iter 7. Control Loop begins, k ¼1 8. //Continuous control component evaluation, Δuc 9. Evaluate the forced response component, yf (8)–(11) 10. Evaluate the free response component, yl (12)–(16) 11. Evaluate the output prediction (7) 12. Evaluate the prediction of the sliding surface (17) 5.
29. 30.
3. Simulations, results and discussion All simulations were carried out on an Intel Core i7 2.3 GHz, 4 GB DDR3 RAM, MacOS X 10.7.3, Matlab R-2011 64-bit and represent a time of 150 s. Nondisturbed and disturbed scenarios are tested. The disturbances are a load step change of 0.3 acting between 60 s and 90 s and a parameter variation of 20% equally applied to all discrete model parameters after t¼ 100 s. The initial conditions are set to zero. The following general parameters are set for PSO: c1 ¼ c2 ¼ 1:4962, Np ¼100, Iter ¼80. For comparison
Delta QP
2
3.5
Delta
Delta
Delta Dual1
4
1.5 1 0.5 0
functions: λ3, λ4 (23) evaluate X(t) (22) update individual and global best particles, pi, pg evaluate velocity, v(t) and position x(t) (24,25) update particles velocity and position t ¼ t þ1 End while Evaluate the discontinuous control component, Δud (21) Evaluate the complete control signal uSMGPC (27) End control loop
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
time (s)
50
time (s) Fig. 5. G1 ðzÞ: nondisturbed, δ parameter.
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purposes, besides traditional SMGPC with QP active-set, three PSO variations are considered, changing the way the population is reinitialized. Therefore, four cases are simulated as follows:
7
Dual2: Every 20 epochs, the population is restarted, totally random.
Dual3: Without population restart.
QP active-set: Traditional SMGPC with fixed ðK d ; δÞ defined offline and tuned to avoid overshoot.
Dual1: Every 20 epochs (NIterRenitialize¼20, found appropriate both for Dual1 and Dual 2, by prior experimentation), the population is restarted keeping the average of the present population position and velocity vectors as a new individual in the next randomly generated swarm population.
For the initial population, in order to guarantee reproducibility, all cases used the same seed for the random number generator (Mersenne Twister, seed 0). For the restarts, to provide some variability, different seeds based on the computer clock are used to force new members. Besides, the control signal amplitude is restricted to 7 10 and its slew rate to 7 1.
QP−SMGPC x Dual−SMGPC Process Variable
6 QP SetPoint Dual1 Dual2 Dual3
4 2 0 −2
0
50
100
150
time (s) Control Signal 10 8
u
6 QP Dual1 Dual2 Dual3
4 2 0
0
50
100
150
time (s) Fig. 6. G1 ðzÞ: disturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
Kd Dual1
0.65
Kd
Kd
1.5
100
0.4
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 7. G1 ðzÞ: disturbed, Kd parameter.
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8
Integral of Square Error (ISE):
3.1. System description and design parameters
k
Three common plant models were chosen for simulation in each scenario (disturbed, nondisturbed) and controlled by the four algorithms listed above. For each system and scenario four plots are presented: set point tracking, control signal magnitude, Kd evolution (for each algorithm) and δ evolution (for each algorithm). Considering the stochastic nature of PSO, each algorithm was run 21 times and the following average performance indexes are used:
ISE ¼ ∑ ðwðjÞ yðjÞÞ2
Integral of Time Multiplied Absolute Error (ITAE): k
ITAE ¼ ∑ jjwðjÞ yðjÞj
k
k
IAE ¼ ∑ jwðjÞ yðjÞj
Eu ¼ ∑ juðjÞj
ð28Þ
j¼0
Delta QP
Delta Dual1
4 3.5
Delta
Delta
ð31Þ
j¼0
1.5 1 0.5 0
ð30Þ
j¼0
Energy Consumption (Eu):
Integral of Absolute Error (IAE):
2
ð29Þ
j¼0
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2.5 2
2
1.5 0
0
50
100
1
150
0
50
time (s)
time (s) Fig. 8. G1 ðzÞ: disturbed, δ parameter.
QP−SMGPC x Dual−SMGPC
Process Variable
1.5 1
QP SetPoint Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Control Signal 1.5
u
1 QP Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Fig. 9. G2 ðzÞ: nondisturbed.
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Standard Deviation – Control Signal: σu ¼
9
Kd Mean:
k 1 ∑ ðuðjÞ uÞ2 ; k 1 j ¼ 1
ð32Þ
Kd ¼
1 k ∑ K ðjÞ kj¼1 d
ð34Þ
Standard Deviation – Kd: u¼
k
1 ∑ uðjÞ kj¼1
Kd QP
1.5 1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
k 1 ∑ ðK d ðjÞ K d Þ2 k1 j ¼ 1
100
0.4
150
0
50
time (s)
0.6
0.5
Kd
Kd
1
0.4
0
0.2
−0.5 0
50
150
100
150
Kd Dual3
1.5
0.8
0
100
time (s)
Kd Dual2
1
ð35Þ
Kd Dual1
0.65
Kd
Kd
σKd ¼
ð33Þ
100
−1
150
0
50
time (s)
time (s) Fig. 10. G2 ðzÞ: nondisturbed, Kd parameter.
Delta QP
4
3.5
Delta
Delta
3.5 3 2.5 2
Delta Dual1
4
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
time (s)
50
time (s) Fig. 11. G2 ðzÞ: nondisturbed, δ parameter.
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10
δ Mean: δ¼
with a sampling rate of T ¼ 0:2 s, as simulated by [25], arising a nonminimum phase system described as
k
1 ∑ δðjÞ kj¼1
ð36Þ G1 ðzÞ ¼
Standard Deviation – δ: k 1 ∑ ðδðjÞ δ Þ2 σδ ¼ k1 j ¼ 1
ð37Þ
The first model represents the isothermal Van de Vussen reaction system, physically described in [34]. Here, it is used in its discretized form (Zero Order Hold) around an operating point
ð38Þ
General design parameters: N 1 ¼ 1; N y ¼ 30; N u ¼ 12; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:1; δ ¼ 1. For PSO: 0:05 rK d r 1, 0:5 r δ r 6, ω ¼ 0:4 fixed. The second model represents a FOPDT discrete transfer function, commonly associated with processes in the chemical industry, with a delay of 2 sampling times and time constant 1: G2 ðzÞ ¼
0:6321z 1 2 z 1 0:3679z 1
ð39Þ
QP−SMGPC x Dual−SMGPC
6
Process Variable
0:0939 þ 0:1745z 1 : 1 1:2573z 1 þ 0:3951z 2
QP SetPoint Dual1 Dual2 Dual3
4 2 0
0
50
100
150
time (s) Control Signal 6 QP Dual1 Dual2 Dual3
u
4 2 0
0
50
100
150
time (s) Fig. 12. G2 ðzÞ: disturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
Kd Dual1
0.65
Kd
Kd
1.5
100
0.4
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 13. G2 ðzÞ: disturbed, Kd parameter.
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General design parameters: N 1 ¼ 3; N y ¼ 30; N u ¼ 12; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:4; δ ¼ 3. For PSO: 0:05 r K d r 1, 0:5 r δ r 6, ω ¼ 0:4 fixed. The third model represents also a FOPDT discrete transfer function, but with a higher time delay [22,26] and a controllability ratio bigger than one, known for producing control difficulties [35]: G3 ðzÞ ¼
0:3069z 1 5 z 1 0:6065z 1
ð40Þ
General design parameters: N 1 ¼ 6; N y ¼ 30; N u ¼ 30; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:4; δ ¼ 4:5. For PSO: 0:05 rK d r 0:5, 0:5 r δ r 4, 0:4 r ω r5 linear decay.
3.2. Results and discussion When compared to conventional GPC, it is well known that the better robustness properties obtained with SMGPC with fixed, manually tuned Kd and δ, as may be seen in [14,24–26]. In this work, an even better robustness is obtained, since PSO is used to automatic tuning of Kd and δ, in order to minimize tracking error and control effort. For all simulated systems, in both scenarios, the error indexes for PSO approaches are lower, without significant increase in control effort and control slew rate. This fact is shown in Tables 1–6, observing Eu, σu, IAE, ISE and ITAE. The evolution of Kd and δ in Dual1 and Dual2 is similar (4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20) with a lower oscillation amplitude in Dual1 due to the use of some elitism. This term is inherited from evolutionary
Delta QP
4
Delta Dual1
4 3.5
Delta
3.5
Delta
11
3 2.5
3 2.5
2 0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
50
time (s)
time (s) Fig. 14. G2 ðzÞ: disturbed, δ parameter.
QP−SMGPC x Dual−SMGPC
Process Variable
1.5 1
QP SetPoint Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Control Signal
0
QP Dual1 Dual2 Dual3
u
−0.5 −1 −1.5 −2
0
50
100
150
time (s) Fig. 15. G3 ðzÞ: nondisturbed.
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12
algorithms and it is related with keeping one or more members of the present population (and thus its characteristics) in the next population. When no population restart is employed, Dual3 with fixed inertia weight for G1 ðzÞ and G2 ðzÞ gets a pair ðK d ; δÞ which satisfies de minimization problem and keeps that value. For the higher time delay in G3 ðzÞ, the varying inertia weight was shown to be better, providing a better disturbance rejection in Figs. 18–20. The choice of which strategy is more suitable depends on the engineering requirements. If some small overshoot is allowed, Dual2 presented the best disturbance rejection for G1 ðzÞ
(Figs. 3 and 6). The same occurs for Dual3 in G2 ðzÞ, observed in Figs. 9 and 12. For this system, an interesting aspect may be depicted from Tables 3 and 4: the PSO average values for Kd and δ are next to that manually tuned for QP, what means that with some effort a good pair may be found by simulations, but this is what this proposal tries to avoid. This trade-off is even more clear in Figs. 15 and 18, where Dual3 provides the best robustness, but with some overshoot in the nondisturbed case. For the hardware and software described above, the average computation time for QP is 0.1 s and for the Dual approach is 0.3 s, being reasonable for
Kd QP
1 0.5
−0.2
0
Kd
Kd
Kd Dual1
−0.15
−0.25
−0.5 −0.3
−1 −1.5
0
50
100
−0.35
150
0
50
time (s) Kd Dual2
−0.1
0.5
−0.2
0
−0.3
−0.5
−0.4
−1
−0.5
0
50
150
100
150
100
150
100
150
Kd Dual3
1
Kd
Kd
0
100
time (s)
100
−1.5
150
0
50
time (s)
time (s) Fig. 16. G3 ðzÞ: nondisturbed, Kd parameter.
Delta QP
5.5
Delta Dual1
3
5
Delta
Delta
2.5 4.5
2 4 3.5
0
50
100
1.5
150
0
50
time (s)
time (s)
Delta Dual2
4
2.5
Delta
Delta
3 2 1 0
Delta Dual3
3
2 1.5 1
0
50
100
150
0.5
0
time (s)
50
time (s) Fig. 17. G3 ðzÞ: nondisturbed, δ parameter.
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QP−SMGPC x Dual−SMGPC
2
Process Variable
13
0 QP SetPoint Dual1 Dual2 Dual3
−2 −4 0
50
100
150
time (s) Control Signal 6 QP Dual1 Dual2 Dual3
4
u
2 0 −2 −4 0
50
100
150
time (s) Fig. 18. G3 ðzÞ: disturbed.
Kd QP
1 0.5
−0.2
0
Kd
Kd
Kd Dual1
−0.15
−0.25
−0.5 −0.3
−1 −1.5
0
50
100
−0.35
150
0
50
time (s) Kd Dual2
0
150
100
150
Kd Dual3
−0.2
−0.1
−0.25
−0.2
Kd
Kd
100
time (s)
−0.3
−0.3 −0.35
−0.4 −0.5
0
50
100
−0.4
150
0
time (s)
50
time (s) Fig. 19. G3 ðzÞ: disturbed, Kd parameter.
the chosen sampling time. If compiled programming languages were used, these computation times would be even lower.
4. Conclusions and perspectives A new model independent scheme based on Particle Swarm Optimization has been developed for automatic tuning of the Sliding Mode Generalized Predictive Controller discontinuous control law component parameters, providing robustness and making the design process easier. Ease of tuning is of great interest in practice. Simulations results and performance indexes for a
nonminimum phase and time delayed systems, commonly found in process control industry, were presented, showing a better tracking without significant control effort and computational time increase. Therefore, future works will explore:
The effect of the search space limits for ðK d ; δÞ. PSO convergence aspects and its relation with the closed loop stability.
The use of different regression schemes (linear, nonlinear) to
get fixed parameters pairs from PSO data as an option for QP manual tuning. Test of other cost functions structures and their comparison with the quadratic form.
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14
Delta SQP
5.5
Delta Dual1
3
5
Delta
Delta
2.5 4.5
2 4
3.5
0
50
100
1.5
150
0
50
time (s) Delta Dual2
4
100
150
time (s) Delta Dual3
2
3
Delta
Delta
1.5 2
1 1
0
0
50
100
150
0.5
0
50
100
150
time (s)
time (s) Fig. 20. G3 ðzÞ: disturbed, δ parameter.
A Pareto's view of the problem [36], based on a multi objective
perspective. Practical implementation and studies to further embedding in Digital Signal Processors (DSPs) and Field Programmable Gate Arrays (FPGAs).
Acknowledgments First author would like to thank CAPES-Brazil and the Science without Borders program for the financial support, through process no. 2485-13-3. References [1] Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans ASME 1942;64:759–68. [2] Al-Ghazzawi A, Ali E, Nouh A, Zafiriou E. On-line tuning strategy for model predictive controllers. J Process Control 2001;11:265–84. [3] Kennedy J, Eberhart R. Particle swarm optimization. In: IEEE International conference on neural networks. Perth, Australia; 1995. p. 1942–8. [4] Oliveira P, Boaventura-Cunha J, Coelho J. Design of PID controllers using particle swarm algorithm. In: IASTED international conference modeling, identification and control. Innsbruck, Austria; 2002. p. 263–8. [5] Camacho EF, Alba CB. Model predictive control. second edition. London: Springer; 2007. [6] Darby M, Nikolaou M. MPC: current practice and challenges. Control Eng Pract 2012;20:328–42. [7] Abu-Ayyad M, Dubay R. Real-time comparison of a number of predictive controllers. ISA Trans 2007;46:411–8. [8] Clarke DW, Mohtadi C, Tuffs PS. Generalized predictive control—Part i. The basic algorithm. Automatica 1987;23(2):137–48. [9] Clarke DW, Mohtadi C, Tuffs PS. Generalized predictive control—Part ii. Extension and interpretations. Automatica 1987;23(2):149–60.
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A swarm intelligence-based tuning method for the sliding mode generalized predictive control J.B. Oliveira n,1, J. Boaventura-Cunha, P.B. Moura Oliveira, H. Freire INESC TEC - INESC Technology and Science (formerly INESC Porto, UTAD pole) Department of Engineering, School of Sciences and Technology 5001-811 Vila Real, Portugal
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2013 Received in revised form 9 June 2014 Accepted 11 June 2014 This paper was recommended for publication by Dr. R. Dubay
This work presents an automatic tuning method for the discontinuous component of the Sliding Mode Generalized Predictive Controller (SMGPC) subject to constraints. The strategy employs Particle Swarm Optimization (PSO) to minimize a second aggregated cost function. The continuous component is obtained by the standard procedure, by Quadratic Programming (QP), thus yielding an online dual optimization scheme. Simulations and performance indexes for common process models in industry, such as nonminimum phase and time delayed systems, result in a better performance, improving robustness and tracking accuracy. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model Predictive Control Sliding Mode Particle Swarm Optimization Soft computing Robustness
1. Introduction Controllers play a central role in industrial plants, since they are designed to regulate process variables in accordance with some performance criteria. A low energy consumption is also of practical interest, thus arising the well known trade-off: tracking or regulation accuracy versus energy consumption. To cope with these issues, controller parameters must be appropriately adjusted and research in this subject keeps its relevance since the work by [1], regarding Proportional Integral Derivative (PID) controller. This tuning method allows getting an easier initial set for PID parameters. When some intelligence is incorporated into the closed loop, one has an online or automatic tuning method [2]. Metaheuristic algorithms, such as Particle Swarm Optimization (PSO) are quite feasible in such cases. PSO is a natural inspired computation technique introduced by [3] and it is characterized for its simplicity and high efficiency in searching global optimal solutions in problem spaces. This feature attracted the attention of control
n
Corresponding author: Tel.: þ 55 84 33424836. E-mail addresses: [email protected] (J.B. Oliveira), [email protected] (J. Boaventura-Cunha), [email protected] (P.B. Moura Oliveira), [email protected] (H. Freire). 1 Permanent address: Agricultural School of Jundiai - Federal University of Rio Grande do Norte 59280-000 Macaiba, Brazil.
engineers in the sense of a simple way of searching optimal or semi-optimal tuning of controller's parameters [4]. Besides PID, known for its simplicity due to just three parameters to tune, Model Predictive Controllers (MPC) are interesting for linear, nonlinear, time-delayed and nonminimum phase systems [5]. It offers a straightforward design method to anticipate future control actions within some time horizon (control horizon), in order to track a future behavior (in some prediction horizon), predicted by an explicit model. The most common model forms in the various MPC products rely on convolution (Finite Step Response FSR and Finite Impulse Response FIR) models, such as Dynamic Matrix Control (DMC), but recent controllers suggest a trend toward state space formulations which provides flexibility in representing stable, unstable, integrating and unmeasured disturbances, just as the Controller Auto-Regressive Integrated MovingAverage (CARIMA) model in the Generalized Predictive Control (GPC) [6,7]. According to [8,9], the objective function of GPC is very similar to that of DMC, with the fundamental difference of using a Diophantine equation and CARIMA model to formulate the dynamic matrix. Abu-Ayyad and Dubay [7] showed that GPC and Extended Predictive Control (EPC) can handle the system matrix ill-conditionality better than other MPC methods and, therefore, it still motivates the development and applications of GPC, as in [6]. The direct treatment of practical constraints such as actuator and output limits is carried out by the minimization or maximization of some objectives, expressed in its simpler form as an aggregated
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Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
2
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quadratic cost function. For this optimization scenario, Quadratic Programming (QP) based on active-set algorithm is usual and PSO can be also applied to tune its parameters, as proposed by [10–13]. Considering its wide application in environments subject to disturbances, robustness is a necessary feature and must be taken into account. An attempt to aggregate robustness into GPC by combining GPC with Sliding Mode Control (SMC) was firstly reported in [14]. SMC is a nonlinear control scheme known to be robust to model uncertainties, disturbances and unmodeled dynamics, being quite suitable for industrial environments. Since the considerations by [15], the research on SMC theory and its applications have been of increasing interest, providing an engineering look at SMC. Key aspects were clarified, such as the chattering phenomena, both in continuous and discrete time. The key idea consists in choosing a state variables function (sliding surface) in which all trajectories must reach in finite time (reaching phase) and, once reached, cannot escape, sliding to the desired final value (sliding phase). A control law is then designed to force the trajectories towards this surface (corrective action) and, moreover, to keep them thereafter (equivalent control [16]). This control law must be discontinuous or, at least, it must contain a discontinuous component. In classical SMC, a possibility is for instance the tuning of sliding surface parameters, as obtained by [17,18]. A GPC based on PSO was compared with the traditional QP-based GPC in a greenhouse experiment, giving better results without great increase of computational burden [19]. The well succeeded melting of Sliding Mode Predictive Controllers (SMPC) motivated other works and applications [20–30]. Most of the work referenced is concerned with common process control problems, such as delayed and nonminimum phase systems, often represented by a First Order Plus Dead Time (FOPDT) transfer function. For this type of continuous-time models, Camacho and Smith [20] proposed a set of tuning equations for the initial values of the discontinuous component of the control law, as a function of the characteristic parameters of the FOPDT model. When other model structures are considered, including discrete and higher order systems, these equations are no longer valid and computational intelligent approaches are interesting in order to help online tuning of the controller. In [6] it is stated that, for the MPC controllers used in process industries today, the tuning emphasis is on disturbance rejection and suggest as trends and research directions the development of improved disturbance estimators and robust controllers, by means of randomized algorithms which would rely on extensive offline simulation. Tuning is therefore commonly based on offline simulation and the actual performance of the online controller. It is typically carried out using the nominal model and via trial and error try do determine steady state behavior, providing initial tuning values for the parameters. Such simulations need to consider expected model errors and incorporate the characteristics of unmeasured (stochastic) disturbances obtained, for instance, from the actual controller. In this sense, a tuning strategy incorporated into the control loop can provide adaptability and robustness, without significant increase of the computational load, considering the hardware technology currently available. In [2], it also corroborates that while tuning guidelines for initial tuning values may be found in the literature, these rules are not general and do not learn from the controller operation and system response. Following some design steps of the SMPC presented in [27], here named Sliding Mode Generalized Predictive Controller (SMGPC), this paper keeps QP active-set for the optimization of the continuous component of the control law responsible for the sliding phase, but proposes PSO as an optimization tool for selecting optimal parameters for the discontinuous component of the control law, thus yielding a dual optimization scheme (henceforth named Dual-SMGPC), applicable to a wider class of systems. Moreover, now
its parameters are adaptive, providing robustness during reaching phase. In order to test the way PSO can find the optimum solutions, besides the common approach of getting the results from the initial population, two other variations are compared: restarting the population after some iterations while keeping a member in the next population, and a totally random new population after restarting, to avoid possible local minimum. In traditional SMGPC, these parameters are kept constant and calculated offline, normally through simulations. Simulations on common process models will be presented and the results compared with SMGPC without PSO, with a fixed pair ðK d ; δÞ. The remaining of this paper is organized as follows: Section 2 states the GPC and both optimization problems (QP and PSO), discussing adjustments criteria; Section 3 presents and comment simulations results; Section 4 provides some conclusions and encourage future works.
2. Controller design The SMGPC presented here is based on a Controller AutoRegressive Integrated Moving-Average model (CARIMA), considered linear around each operating point and described as
ΔAðq 1 ÞyðkÞ ¼ Bðq 1 ÞΔuðk d 1Þ þ ξðkÞ;
ð1Þ
where d is the delay from input to output (here considered as a multiple of the sampling time), u is the input signal, q 1 is the backward-shift operator, Δ : 1 q 1 and ξ is the zero mean white noise. A and B are polynomials in q 1 defined as Aðq 1 Þ ¼ 1 þ a1 q 1 þa2 q 2 þ … þ ana q na ;
ð2Þ
Bðq 1 Þ ¼ b0 þ b1 q 1 þ b2 q 2 þ …þ anb q nb :
ð3Þ
According to the SMC theory [16], the first step to design the controller is to define a sliding surface, S(t), along which the process can slide to find its desired final value. Very often, S(t) is chosen in such a way that represents a desired system dynamics and/or control objective. For instance, S(t) could be the tracking error eo ¼ y w, with w being some reference signal. The problem of tracking a reference value can be reduced to keeping S(t) at zero. From [25,27], the j-step ahead prediction of S(k) with information until the actual instant t¼ k is given by ^ þjÞ ¼ P s ðq 1 Þðyðk ^ þjÞ wðk þ jÞÞ þ Q s ðq 1 ÞΔuðk þ j 1 dÞ: Sðk
ð4Þ
Polynomials P s ðq 1 Þ, Q s ðq 1 Þ have degree np and nq respectively, and allow to design the desired dynamics in the sliding condition. A common adjustment is choosing P s ðq 1 Þ and Q s ðq 1 Þ as a first order system: Q s ðq 1 Þ ð1 αÞq 1 ¼ ; 1 αq 1 P s ðq 1 Þ
ð5Þ
with 0 o α r 1, since all roots of P s ðq 1 Þ must be inside the unit circle [27]. As α-0 the dynamic is faster. The cost function aggregates two simultaneous objectives: Ny
Nu
j ¼ N1
j¼1
^ þ jÞ2 þ ∑ λ½Δuðk þj 1Þ2 J C ¼ ∑ ½Sðk
ð6Þ
where λ is set constant and N 1 N y is the period of time in which one desires the output tracks the reference signal and Nu is the control horizon. For these parameters, [27] suggested some intuitive relations, which can be used as initial values. Other online tuning strategies for these specific parameters are available in the literature [2].
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Likewise GPC, to minimize (6) the output prediction within the interval j ¼ ½N 1 ; N y is formed by two parts: ^ þ jÞ ¼ yf þ yl ; yðk
ð7Þ
where yf is the forced response (considering the initial conditions null but subject to future control actions) and yl is the free response (natural system response from the initial conditions with no future control actions).
3
qs0 ¼ 0; qs1 ¼ 1 α; np ¼ 1; nq ¼ 1, 2 3 2 ps0 qs0 0 … 0 6p 7 6q p … 0 6 s1 7 6 s0 7; Q s ¼ 6 s1 Ps ¼ 6 6 ⋮ 6 ⋮ ⋮ ⋱ ⋮ 7 4 4 5 0 psnp … ps0 0
0 qs0
… …
⋮
⋱
qsnq
…
3 0 0 7 7 7; ⋮ 7 5 qs0
with order ðN y N 1 þ 1Þ ðN y N1 þ 1Þ and ðNy N 1 þ 1Þ N u , respectively. Substituting (17) into (6), the cost function becomes J C ¼ 12 ΔuTc GΔuc þ ΔuTc b þf 0 ;
2.1. Forced response, yf
ð19Þ
ð8Þ
with G ¼ 2ðλ1 Z Z þ λ2 IÞ, Z ¼ P s H þ Q s , f 0 ¼ λ1 ðP s ðyl wÞÞ ðP s ðyl wÞÞ and b ¼ 2λ1 Z T P s ðyl wÞ. A necessary and sufficient condition for discrete time reaching ^ þ 1Þj r SðkÞ ^ motion is jSðk [27] and a control law satisfying this condition does guarantee that all trajectories will enter and remain within at least a non-increasing domain. In order to guarantee the reaching condition of (17), a constraint is added into the optimization problem of (19), namely:
ð9Þ
SðkÞ1 þ P s ðyl wÞ r ðP s H þ Q s ÞΔuc r SðkÞ1 P s ðyl wÞ;
T
The forced response may be calculated from the step response of the parametric model (1): yf ¼ H Δu; where, 2
H ðNy N1 þ 1ÞNu
h0 6 h 6 1 ¼6 6 ⋮ 4 hNy 1
0
…
0
h0 ⋮
... ⋱
0 ⋮
hNy 2
…
hNy Nu
3 7 7 7; 7 5
ΔuðNu 1Þ ¼ ½ΔuðkÞ Δuðk þ 1Þ … Δuðk þ N u 1ÞT : j1
i¼1
i¼0
hj ¼ ∑ aj hj i þ ∑ bi ;
hj ¼ 0; 8 ðj iÞ o 0:
ð10Þ
ð11Þ
Although yf and yl might be calculated by solving a Diophantine equation [27], as usual, here the available parametric model is used, thus simplifying the design. The concept of free response states that the future control signal increments after t ¼ k 1 are zero, then Δuðk þ jÞ ¼ 0; j Z 0. Another assumption is that ^ þ jÞ; j r0, since yl takes into account the initial yl ðk þjÞ ¼ yðk conditions. Considering these conditions and after some manipulation in (1), one has for the general case: T
ð12Þ
A~ l ¼ ½ a~ 1 a~ 2 … a~ na þ 1 ; T ;
ð13Þ
Y l ¼ ½yl ðk þ jÞ yl ðk þj 1Þ … yl ðk þ j naÞT ;
ð14Þ
Bl ¼ ½b0 b1 … bnb T ;
ð15Þ
Δul ¼ ½Δuðk d þ jÞ Δuðk d þ j 1Þ … Δuðk d þ j nbÞT ;
ð16Þ
with
and a~ 1 ¼ a1 1; a~ 2 ¼ a2 a1 ; …; a~ na ¼ ana ana 1 ; a~ na þ 1 ¼ ana . 2.3. Prediction of the sliding surface SMGPC control law, ΔuSMGPC ðkÞ, is the combination of two additive parts: a continuous part Δuc ðkÞ ¼ ΔuðkÞ (10) developed like a GPC by the minimization of (6) using QP, which is responsible for keeping the process variable on the reference value, and a discontinuous part Δud ðkÞ to be detailed further, responsible for guiding the system to the sliding surface. To calculate Δuc ðkÞ, (7) is substituted into (4) and, after putting it into matrix form, one has ^ SðkÞ ¼ ðP s H þQ s ÞΔuc ðkÞ þ P s ðyl ðkÞ wðkÞÞ;
ð20Þ
2.4. Discontinuous control signal component The discontinuous part Δud is given by
2.2. Free response, yl
yl ðk þj þ 1Þ ¼ A~ l Y l þ BTl Δul ;
T
where 1Nu 1 is a vector whose entries are ones.
The parameters that compound H may be obtained from: j
ð18Þ
ð17Þ
with the matrices Ps and Qs composed from (5), ps0 ¼ 1; ps1 ¼ α;
^ þ jÞ Sðk
Δud ðk þ jÞ ¼ K d ^ ; jSðk þ jÞj þ δ
δ Z 0;
ð21Þ
where Kd is a gain responsible for the velocity of the reaching mode but also increases chattering, reduced by an appropriate δ. The pair ðK d ; δÞ directly affects system performance and then are commonly selected through simulations. For continuous FOPDT models, [20] proposed some initial tuning equations. However, this work proposes a new methodology based on PSO for a wider class of systems which is model independent, providing an adaptive online scheme for tuning these parameters. PSO is a popular population based metaheuristic search algorithm which is inspired by the mechanism of biological swarm social behavior, such as bird flocking and fish schooling, which can be used to search for optima [3]. In this case, besides (19), a second quadratic aggregated bi-objective cost function must be minimized by PSO, as follows: J D ¼ λ3 ½ðyl wÞT ðyl wÞ þ λ4 ½ΔuTd Δud ;
ð22Þ
where
λ3 ðkÞ ¼ signð sin ð2π k=FÞÞ;
λ4 ðkÞ ¼ 1 λ3 ðkÞ;
ð23Þ
with F is a user-defined adaptation frequency. This strategy named Bang-Bang Weighted Aggregation (BWA) [31,32] is based on sign function and force the algorithm to promote movements towards the optimal solutions. The weights change periodically depending on F. For example, if F¼ 200, the weights behave as shown in Fig. 1. One entity of the Np individuals is named particle and its position in the swarm represents a possible solution for the problem. Therefore, a pair ðK d ; δÞ represents a particle in a 2-dimensional search space and modifies its movement according to its own experience and its neighboring particle experience. In classical PSO, only two equations are used and updated in each iteration: position update (X) and velocity update (V). For the ith particle of the swarm (of size Np), one has for this case the vectors X i ¼ ðxi1 ; xi2 Þ and V i ¼ ðvi1 ; vi2 Þ. The previously best visited position of the ith particle is denoted by P i ¼ ðpi1 ; pi2 Þ, and g is the index of the best particle of the swarm (global best). The movement equations are vid ðtÞ ¼ ωðtÞvid ðt 1Þ þ c1 r 1 ðpid ðtÞ xid ðtÞÞ þ c2 r 2 ðpgd ðtÞ xid ðtÞÞ;
ð24Þ
Please cite this article as: Oliveira JB, et al. A swarm intelligence-based tuning method for the sliding mode generalized predictive control. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.06.007i
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4
λ3 BWA 1.5 1 0.5 0 −0.5 −1 −1.5
0
50
100
150
100
150
time (s) λ4 BWA
2.5 2 1.5 1 0.5 0 −0.5
0
50
time (s) Fig. 1. Bang-Bang weighted aggregation for λ3 and λ4.
Table 3 Performance Indexes G2 ðzÞ – average Values for 21 runs – nondisturbed case. Performance Index
QP-SMGPC
Dual1
Dual2
Dual3
Eu σu IAE ISE ITAE
141.3 0.2083 13.2 10.6 105.6 0.4000
142.9 0.1989 11.8 9.8 82.7 0.5236
144.0 0.1956 11.7 9.5 87.0 0.5249
143.6 0.1954 11.4 9.5 77.7 0.4462
– 3.0000 –
0.0218 3.2345 0.1590
0.2740 3.2036 1.5937
0 2.2965 0
Kd σK d δ σδ Fig. 2. Block diagram – dual SMGPC.
Table 1 Performance Indexes G1 ðzÞ – average values for 21 runs – nondisturbed case. Performance Index
QP-SMGPC ð103 Þ
Dual1 ð103 Þ
Dual2 ð103 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
1.24 0.0002 0.0292 0.0197 0.5918 0.0001
1.26 0.0002 0.0153 0.0116 0.1507 0.0005
1.27 0.0001 0.0126 0.0094 0.1159 0.0005
1.26 0.0002 0.0144 0.0110 0.1342 0.0004
– 0.0010 –
0 0.0032 0.0001
0.0003 0.0033 0.0016
0 0.0023 0
Kd σKd δ σδ
Table 2 Performance Indexes G1 ðzÞ – average values for 21 runs – disturbed case.
Table 4 Performance Indexes G2 ðzÞ – average values for 21 runs – disturbed case. Performance Index
QP-SMGPC ð103 Þ
Dual1 ð103 Þ
Dual2 ð103 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
0.32 0.0013 0.1231 0.3436 9.6530 0.0004
0.28 0.0011 0.0977 0.2387 7.3 0.0005
0.27 0.0010 0.0910 0.1966 6.7 0.0005
0.28 0.0011 0.0952 0.2271 7.1 0.0004
– 0.0030 –
0 0.0032 0.0012
0.0003 0.0032 0.0016
0 0.0023 0
Kd σK d δ σδ
Table 5 Performance indexes G3 ðzÞ – average values for 21 runs – nondisturbed case.
Performance Index
QP-SMGPC ð104 Þ
Dual1 ð104 Þ
Dual2 ð104 Þ
Dual3 ð104 Þ
Performance Index
QP-SMGPC
Dual1
Dual2
Dual3
Eu σu IAE ISE ITAE
0.26 0.0002 0.0389 0.0802 8.0653 0.00001
0.26 0.0002 0.0141 0.0187 3.0694 0.0001
0.27 0.0003 0.0122 0.0154 2.3814 0.0001
0.26 0.0003 0.0131 0.0173 2.7276 0.00004
Eu σu IAE ISE ITAE
172.6 0.3462 24.37 20.07 342.6 0.4000
176.0 0.3385 23.13 19.03 319.6 0.2719
177.6 0.3422 19.02 19.0 368.0 0.2771
180.6 0.3315 23.39 17.87 388.4 0.3113
– 0.0001 –
0 0.0003 0
0 0.0003 0.0002
0 0.0002 0
0.0197 2.2440 0.1639
0.1296 2.2328 0.9863
0 1.6433 0
Kd σKd δ σδ
Kd σK d δ σδ
– 4.500 –
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xid ðtÞ ¼ xid ðt 1Þ þ vid ðtÞ;
Table 6 Performance Indexes G3 ðzÞ – average values for 21 runs – disturbed case. Performance Index
QP-SMGPC ð104 Þ
Dual1 ð104 Þ
Dual2 ð104 Þ
Dual3 ð103 Þ
Eu σu IAE ISE ITAE
0.0217 0.0002 0.0176 0.0419 1.55 0.00004
0.0229 0.0002 0.0167 0.0388 1.45 0
0.0225 0.0002 0.0172 0.0409 1.51 0
0.2688 0.0018 0.1069 0.2209 7.8 0.0003
– 0.00045 –
0 0.0002 0.0001
0 0.0002 0
0 0.0011 0.0005
Kd σKd δ σδ
ð25Þ
where d¼1,2, i ¼ 1; …; N p . c1 ; c2 are constants, called respectively cognitive and social parameters, r 1 ; r 2 are random numbers obtained from a uniform distribution in the set ½0; 1, and ω is the inertia weight, balancing the effect of initial velocity in the exploration and exploitation process. After studying 15 different strategies for selecting ω, Bansal et al. [33] indicated that a fixed ω or a linear decaying ω would give good results. In this paper, both approaches are tested. The limits for the search space (therefore, for the controller parameters) are expressed by Xmax and the maximum velocity Vmax, normally associated with Xmax, controls the global exploration of the particles in the swarm.
QP−SMGPC x Dual−SMGPC
1.5
Process Variable
5
1 QP SetPoint Dual1 Dual2 Dual3
0.5 0 −0.5
0
50
100
150
time (s) Control Signal 2
u
1.5 1
QP Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Fig. 3. G1 ðzÞ: nondisturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
Kd Dual1
0.65
Kd
Kd
1.5
0.4 0
50
100
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 4. G1 ðzÞ: nondisturbed, Kd parameter.
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6
13. Solve quadratic optimization problem for the continuous component subject to constraints (19,20) 14. // Discontinuous control component evaluation, Δu d 15. // PSO Loop, to obtain Kd and δ 16. Define nIter Reinitialize 17. t¼ 0 18. Initialize swarmX(t) 19. While(!(Iter)) 20. if(Rem(t; nIterReinitialize) ¼0) Reinitialize Swarm// Dual 1 and Dual 2 21. evaluate the discontinuous weighting cost
Therefore, the complete control signal increment for the SMGPC at t¼ k is
ΔuSMGPC ðkÞ ¼ Δuc ðkÞ þ Δud ðkÞ:
ð26Þ
It is noteworthy that at each instant k, although Nu components are calculated for Δuc and Δud , only the first component of each (uc(k), ud(k)) is considered, neglecting the others components, obeying the receding horizon principle. Then, the actual control signal sent to the process is given by uSMGPC ðkÞ ¼ uSMGPC ðk 1Þ þ uc ðkÞ þ ud ðkÞ:
ð27Þ
A generalized block diagram for the proposed Dual SMGPC may be seen in Fig. 2.
22. 23.
2.5. Control algorithm - summary
24. 25. 26. 27. 28.
In order to facilitate the simulation and implementation of the proposed technique, the aforementioned steps may be summarized by the following algorithm with the respective equations numbers: 1. 2. 3. 4.
Initialize SMGPC parameters: α, N1, Ny, Nu, F Define system polynomials and delay: A, B, d (2,3) Define polynomials Qs and Ps with a initialized in step 1 (5) Define control signal saturation and slew rate limits
Define PSO search intervals for Kd and δ 6. Define Swarm size, Np, number of iterations, Iter 7. Control Loop begins, k ¼1 8. //Continuous control component evaluation, Δuc 9. Evaluate the forced response component, yf (8)–(11) 10. Evaluate the free response component, yl (12)–(16) 11. Evaluate the output prediction (7) 12. Evaluate the prediction of the sliding surface (17) 5.
29. 30.
3. Simulations, results and discussion All simulations were carried out on an Intel Core i7 2.3 GHz, 4 GB DDR3 RAM, MacOS X 10.7.3, Matlab R-2011 64-bit and represent a time of 150 s. Nondisturbed and disturbed scenarios are tested. The disturbances are a load step change of 0.3 acting between 60 s and 90 s and a parameter variation of 20% equally applied to all discrete model parameters after t¼ 100 s. The initial conditions are set to zero. The following general parameters are set for PSO: c1 ¼ c2 ¼ 1:4962, Np ¼100, Iter ¼80. For comparison
Delta QP
2
3.5
Delta
Delta
Delta Dual1
4
1.5 1 0.5 0
functions: λ3, λ4 (23) evaluate X(t) (22) update individual and global best particles, pi, pg evaluate velocity, v(t) and position x(t) (24,25) update particles velocity and position t ¼ t þ1 End while Evaluate the discontinuous control component, Δud (21) Evaluate the complete control signal uSMGPC (27) End control loop
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
time (s)
50
time (s) Fig. 5. G1 ðzÞ: nondisturbed, δ parameter.
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purposes, besides traditional SMGPC with QP active-set, three PSO variations are considered, changing the way the population is reinitialized. Therefore, four cases are simulated as follows:
7
Dual2: Every 20 epochs, the population is restarted, totally random.
Dual3: Without population restart.
QP active-set: Traditional SMGPC with fixed ðK d ; δÞ defined offline and tuned to avoid overshoot.
Dual1: Every 20 epochs (NIterRenitialize¼20, found appropriate both for Dual1 and Dual 2, by prior experimentation), the population is restarted keeping the average of the present population position and velocity vectors as a new individual in the next randomly generated swarm population.
For the initial population, in order to guarantee reproducibility, all cases used the same seed for the random number generator (Mersenne Twister, seed 0). For the restarts, to provide some variability, different seeds based on the computer clock are used to force new members. Besides, the control signal amplitude is restricted to 7 10 and its slew rate to 7 1.
QP−SMGPC x Dual−SMGPC Process Variable
6 QP SetPoint Dual1 Dual2 Dual3
4 2 0 −2
0
50
100
150
time (s) Control Signal 10 8
u
6 QP Dual1 Dual2 Dual3
4 2 0
0
50
100
150
time (s) Fig. 6. G1 ðzÞ: disturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
Kd Dual1
0.65
Kd
Kd
1.5
100
0.4
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 7. G1 ðzÞ: disturbed, Kd parameter.
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Integral of Square Error (ISE):
3.1. System description and design parameters
k
Three common plant models were chosen for simulation in each scenario (disturbed, nondisturbed) and controlled by the four algorithms listed above. For each system and scenario four plots are presented: set point tracking, control signal magnitude, Kd evolution (for each algorithm) and δ evolution (for each algorithm). Considering the stochastic nature of PSO, each algorithm was run 21 times and the following average performance indexes are used:
ISE ¼ ∑ ðwðjÞ yðjÞÞ2
Integral of Time Multiplied Absolute Error (ITAE): k
ITAE ¼ ∑ jjwðjÞ yðjÞj
k
k
IAE ¼ ∑ jwðjÞ yðjÞj
Eu ¼ ∑ juðjÞj
ð28Þ
j¼0
Delta QP
Delta Dual1
4 3.5
Delta
Delta
ð31Þ
j¼0
1.5 1 0.5 0
ð30Þ
j¼0
Energy Consumption (Eu):
Integral of Absolute Error (IAE):
2
ð29Þ
j¼0
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2.5 2
2
1.5 0
0
50
100
1
150
0
50
time (s)
time (s) Fig. 8. G1 ðzÞ: disturbed, δ parameter.
QP−SMGPC x Dual−SMGPC
Process Variable
1.5 1
QP SetPoint Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Control Signal 1.5
u
1 QP Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Fig. 9. G2 ðzÞ: nondisturbed.
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Standard Deviation – Control Signal: σu ¼
9
Kd Mean:
k 1 ∑ ðuðjÞ uÞ2 ; k 1 j ¼ 1
ð32Þ
Kd ¼
1 k ∑ K ðjÞ kj¼1 d
ð34Þ
Standard Deviation – Kd: u¼
k
1 ∑ uðjÞ kj¼1
Kd QP
1.5 1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
k 1 ∑ ðK d ðjÞ K d Þ2 k1 j ¼ 1
100
0.4
150
0
50
time (s)
0.6
0.5
Kd
Kd
1
0.4
0
0.2
−0.5 0
50
150
100
150
Kd Dual3
1.5
0.8
0
100
time (s)
Kd Dual2
1
ð35Þ
Kd Dual1
0.65
Kd
Kd
σKd ¼
ð33Þ
100
−1
150
0
50
time (s)
time (s) Fig. 10. G2 ðzÞ: nondisturbed, Kd parameter.
Delta QP
4
3.5
Delta
Delta
3.5 3 2.5 2
Delta Dual1
4
3 2.5
0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
time (s)
50
time (s) Fig. 11. G2 ðzÞ: nondisturbed, δ parameter.
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δ Mean: δ¼
with a sampling rate of T ¼ 0:2 s, as simulated by [25], arising a nonminimum phase system described as
k
1 ∑ δðjÞ kj¼1
ð36Þ G1 ðzÞ ¼
Standard Deviation – δ: k 1 ∑ ðδðjÞ δ Þ2 σδ ¼ k1 j ¼ 1
ð37Þ
The first model represents the isothermal Van de Vussen reaction system, physically described in [34]. Here, it is used in its discretized form (Zero Order Hold) around an operating point
ð38Þ
General design parameters: N 1 ¼ 1; N y ¼ 30; N u ¼ 12; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:1; δ ¼ 1. For PSO: 0:05 rK d r 1, 0:5 r δ r 6, ω ¼ 0:4 fixed. The second model represents a FOPDT discrete transfer function, commonly associated with processes in the chemical industry, with a delay of 2 sampling times and time constant 1: G2 ðzÞ ¼
0:6321z 1 2 z 1 0:3679z 1
ð39Þ
QP−SMGPC x Dual−SMGPC
6
Process Variable
0:0939 þ 0:1745z 1 : 1 1:2573z 1 þ 0:3951z 2
QP SetPoint Dual1 Dual2 Dual3
4 2 0
0
50
100
150
time (s) Control Signal 6 QP Dual1 Dual2 Dual3
u
4 2 0
0
50
100
150
time (s) Fig. 12. G2 ðzÞ: disturbed.
Kd QP
1
0.6
0.5
0.55
0
0.5
−0.5
0.45
−1
0
50
Kd Dual1
0.65
Kd
Kd
1.5
100
0.4
150
0
50
time (s) Kd Dual2
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
0
50
150
100
150
Kd Dual3
1.5
Kd
Kd
1
100
time (s)
100
150
−1
0
time (s)
50
time (s) Fig. 13. G2 ðzÞ: disturbed, Kd parameter.
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General design parameters: N 1 ¼ 3; N y ¼ 30; N u ¼ 12; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:4; δ ¼ 3. For PSO: 0:05 r K d r 1, 0:5 r δ r 6, ω ¼ 0:4 fixed. The third model represents also a FOPDT discrete transfer function, but with a higher time delay [22,26] and a controllability ratio bigger than one, known for producing control difficulties [35]: G3 ðzÞ ¼
0:3069z 1 5 z 1 0:6065z 1
ð40Þ
General design parameters: N 1 ¼ 6; N y ¼ 30; N u ¼ 30; α ¼ 0:3; F ¼ 200. For QP: K d ¼ 0:4; δ ¼ 4:5. For PSO: 0:05 rK d r 0:5, 0:5 r δ r 4, 0:4 r ω r5 linear decay.
3.2. Results and discussion When compared to conventional GPC, it is well known that the better robustness properties obtained with SMGPC with fixed, manually tuned Kd and δ, as may be seen in [14,24–26]. In this work, an even better robustness is obtained, since PSO is used to automatic tuning of Kd and δ, in order to minimize tracking error and control effort. For all simulated systems, in both scenarios, the error indexes for PSO approaches are lower, without significant increase in control effort and control slew rate. This fact is shown in Tables 1–6, observing Eu, σu, IAE, ISE and ITAE. The evolution of Kd and δ in Dual1 and Dual2 is similar (4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20) with a lower oscillation amplitude in Dual1 due to the use of some elitism. This term is inherited from evolutionary
Delta QP
4
Delta Dual1
4 3.5
Delta
3.5
Delta
11
3 2.5
3 2.5
2 0
50
100
2
150
0
50
time (s) Delta Dual2
6
100
150
100
150
time (s) Delta Dual3
3.5 3
Delta
Delta
4
2
2.5 2 1.5
0
0
50
100
150
1
0
50
time (s)
time (s) Fig. 14. G2 ðzÞ: disturbed, δ parameter.
QP−SMGPC x Dual−SMGPC
Process Variable
1.5 1
QP SetPoint Dual1 Dual2 Dual3
0.5 0
0
50
100
150
time (s) Control Signal
0
QP Dual1 Dual2 Dual3
u
−0.5 −1 −1.5 −2
0
50
100
150
time (s) Fig. 15. G3 ðzÞ: nondisturbed.
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algorithms and it is related with keeping one or more members of the present population (and thus its characteristics) in the next population. When no population restart is employed, Dual3 with fixed inertia weight for G1 ðzÞ and G2 ðzÞ gets a pair ðK d ; δÞ which satisfies de minimization problem and keeps that value. For the higher time delay in G3 ðzÞ, the varying inertia weight was shown to be better, providing a better disturbance rejection in Figs. 18–20. The choice of which strategy is more suitable depends on the engineering requirements. If some small overshoot is allowed, Dual2 presented the best disturbance rejection for G1 ðzÞ
(Figs. 3 and 6). The same occurs for Dual3 in G2 ðzÞ, observed in Figs. 9 and 12. For this system, an interesting aspect may be depicted from Tables 3 and 4: the PSO average values for Kd and δ are next to that manually tuned for QP, what means that with some effort a good pair may be found by simulations, but this is what this proposal tries to avoid. This trade-off is even more clear in Figs. 15 and 18, where Dual3 provides the best robustness, but with some overshoot in the nondisturbed case. For the hardware and software described above, the average computation time for QP is 0.1 s and for the Dual approach is 0.3 s, being reasonable for
Kd QP
1 0.5
−0.2
0
Kd
Kd
Kd Dual1
−0.15
−0.25
−0.5 −0.3
−1 −1.5
0
50
100
−0.35
150
0
50
time (s) Kd Dual2
−0.1
0.5
−0.2
0
−0.3
−0.5
−0.4
−1
−0.5
0
50
150
100
150
100
150
100
150
Kd Dual3
1
Kd
Kd
0
100
time (s)
100
−1.5
150
0
50
time (s)
time (s) Fig. 16. G3 ðzÞ: nondisturbed, Kd parameter.
Delta QP
5.5
Delta Dual1
3
5
Delta
Delta
2.5 4.5
2 4 3.5
0
50
100
1.5
150
0
50
time (s)
time (s)
Delta Dual2
4
2.5
Delta
Delta
3 2 1 0
Delta Dual3
3
2 1.5 1
0
50
100
150
0.5
0
time (s)
50
time (s) Fig. 17. G3 ðzÞ: nondisturbed, δ parameter.
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QP−SMGPC x Dual−SMGPC
2
Process Variable
13
0 QP SetPoint Dual1 Dual2 Dual3
−2 −4 0
50
100
150
time (s) Control Signal 6 QP Dual1 Dual2 Dual3
4
u
2 0 −2 −4 0
50
100
150
time (s) Fig. 18. G3 ðzÞ: disturbed.
Kd QP
1 0.5
−0.2
0
Kd
Kd
Kd Dual1
−0.15
−0.25
−0.5 −0.3
−1 −1.5
0
50
100
−0.35
150
0
50
time (s) Kd Dual2
0
150
100
150
Kd Dual3
−0.2
−0.1
−0.25
−0.2
Kd
Kd
100
time (s)
−0.3
−0.3 −0.35
−0.4 −0.5
0
50
100
−0.4
150
0
time (s)
50
time (s) Fig. 19. G3 ðzÞ: disturbed, Kd parameter.
the chosen sampling time. If compiled programming languages were used, these computation times would be even lower.
4. Conclusions and perspectives A new model independent scheme based on Particle Swarm Optimization has been developed for automatic tuning of the Sliding Mode Generalized Predictive Controller discontinuous control law component parameters, providing robustness and making the design process easier. Ease of tuning is of great interest in practice. Simulations results and performance indexes for a
nonminimum phase and time delayed systems, commonly found in process control industry, were presented, showing a better tracking without significant control effort and computational time increase. Therefore, future works will explore:
The effect of the search space limits for ðK d ; δÞ. PSO convergence aspects and its relation with the closed loop stability.
The use of different regression schemes (linear, nonlinear) to
get fixed parameters pairs from PSO data as an option for QP manual tuning. Test of other cost functions structures and their comparison with the quadratic form.
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Delta SQP
5.5
Delta Dual1
3
5
Delta
Delta
2.5 4.5
2 4
3.5
0
50
100
1.5
150
0
50
time (s) Delta Dual2
4
100
150
time (s) Delta Dual3
2
3
Delta
Delta
1.5 2
1 1
0
0
50
100
150
0.5
0
50
100
150
time (s)
time (s) Fig. 20. G3 ðzÞ: disturbed, δ parameter.
A Pareto's view of the problem [36], based on a multi objective
perspective. Practical implementation and studies to further embedding in Digital Signal Processors (DSPs) and Field Programmable Gate Arrays (FPGAs).
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