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Constrained tracking control for nonlinear systems
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Research article
Constrained tracking control for nonlinear systems Fatemeh Khani a, Mohammad Haeri b,n a b
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Advanced Control Systems Lab, Electrical Engineering Department, Sharif University of Technology, Tehran 11155-4363, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 21 July 2016 Received in revised form 25 April 2017 Accepted 4 June 2017
This paper proposes a tracking control strategy for nonlinear systems without needing a prior knowledge of the reference trajectory. The proposed method consists of a set of local controllers with appropriate overlaps in their stability regions and an on-line switching strategy which implements these controllers and uses some augmented intermediate controllers to ensure steering the system states to the desired set points without needing to redesign the controller for each value of set point changes. The proposed approach provides smooth transient responses despite switching among the local controllers. It should be mentioned that the stability regions of the proposed controllers could be estimated off-line for a range of set-point changes. The efficiencies of the proposed algorithm are illustrated via two example simulations. & 2017 Published by Elsevier Ltd. on behalf of ISA.
Keywords: Set point tracking Robust model predictive controller Gain scheduling Domain of attraction
1. Introduction Model predictive control (MPC) is one of the well-known control strategies in the process industry due to its capability of optimally controlling multivariable systems with constraints [1,2]. Although a large number of industrial processes are inherently nonlinear, most of the MPC techniques implemented on the real processes are based on linear models. To enhance the performance, a practical control system with large operating regions needs to deal with nonlinearity explicitly. However, a nonlinear MPC could act inadequately because of the computational complexity generally exists in on-line implementation of the controller. As a result, some alternative approaches to consider the nonlinearity in the controller design have been proposed in the literature [3,4]. Robust model predictive control (RMPC) is a convenient approach to consider the modeling error in the controller design. Meanwhile, nonlinear systems could be represented by linear models along with structured or unstructured uncertainties. Thus, a robust control strategy could be implemented instead of a nonlinear one in order to reduce the computational complexity. Therefore, an RMPC strategy could be applied in nonlinear and uncertain cases concurrently. The RMPC techniques mainly are based on minimization of the worst-case objective function incorporating a set of robustness constraints. In this case, [5] introduced a systematic solution for RMPC problem based on linear n
Corresponding author. E-mail address: [email protected] (M. Haeri).
matrix inequality (LMI) for systems with polytopic and structured feedback uncertainties which could estimate the stability region of the controller. This method was modified next to decrease computational time [6,7], reduce conservativeness [8–12], or simplify the representation of the uncertainties [13–16]. Moreover, [17] proposed a scheduled RMPC algorithm that enlarges the operating region of the controller introduced in [5] efficiently and later on [18] proposed a method that improves its transient response. Most of the RMPC methods have been formulated for regulation problems that is steering the system to a fixed point. In practice however, it is often required to track a changing set point. When the set point changes, the stabilizing design of the RMPC may not be valid anymore and/or feasibility of the controller may be lost and the controller fails to track the set point. This issue requires redesigning of the RMPC for each value of the changing set point. In order to solve the tracking problem, [19] introduced RMPC based on a dual-model paradigm which guarantees the offset free tracking of set point changes. Ref. [20] proposed a tracking method which guarantees an H1 norm bound with an optimized linear quadratic performance. Refs. [21–23] presented tracking RMPC approaches by using the notion of tubes. In [24], tracking scheduled RMPC is proposed for linear time invariant systems such that a set of controllers should be designed off-line based on a predefined target. Ref. [25] introduced a robust tracking MPC based on derivation of an invariant set which provides both a large stabilizable set and the closed loop performance. Ref. [26] proposed a nonlinear MPC method augmented with a disturbance observer (DOB). The DOB estimates internal and external disturbances and then in addition to feedback MPC, a feedforward
http://dx.doi.org/10.1016/j.isatra.2017.06.004 0019-0578/& 2017 Published by Elsevier Ltd. on behalf of ISA.
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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2
controller is implemented to compensate the disturbances. In [27], an offset-free MPC approach based on prediction accuracy enhancement via a DOB was developed for a general disturbed system and the computational burden of the proposed method was reduced as compared with the existing offset-free MPCs. Advantages of RMPC such as giving the region of stability and considering input and state constraints in optimization problem make this controller attractive but it has limitations in applications on systems with large operating regions and changing set-points. This paper is devoted to solve the problem by a novel scheduling RMPC scheme. In the off-line part of the proposed method, a set of local controllers is designed that have appropriate overlaps in their stability regions. An on-line part implements a novel switching strategy which uses the designed controllers in off-line part and also some augmented intermediate controllers that are designed on-line. Some features of the suggested method are listed below – The proposed controller could be applied for constrained nonlinear systems. – A prior knowledge of the reference trajectory is not required. – The asymptotic stability of the closed-loop system is guaranteed under set-point changes without needing to change the control scheme for each value of set-points. – The stability region of the controller for a determined range of set-point changes could be approximated off-line. – Implementing augmented intermediate controllers in on-line part could prevent the spikes appearing at the moment of switching between adjacent local controllers in the scheduling scheme. The paper is organized as follows. In Section 2 some mathematical preliminaries are described. Section 3 presents the proposed set point tracking RMPC strategy. In order to illustrate the effectiveness of the proposed method two examples are presented in Section 4. Finally, Section 5 provides the concluding remarks.
where WT W is a positive definite matrix and represents the Lipschitz coefficient. Assumption 1. The pair (A , B ) is stabilizable by state feedback control law. The objective function in MPC which is minimized to optimize performance of the closed-loop system is defined as follows ∞
J ( k) =
∑ i=0
{(
}
T
x k + ik ) Qx( k + ik ) + u(k + i | k )T Ru(k + i | k ) ,
(6)
where Q > 0, R > 0, x( k + ik ) is state at time k + i predicted based on the measurements at time k and u( k + ik ) is control move at time k + i computed by minimizing J ( k ) at time k . To solve the given optimization problem for the nonlinear system in (1) via LMI, first one should replace equality in (6) by an inequality which is done by defining an upper bound for J (k ). Consider a quadratic function V ( x ) = xT Px with P > 0 and V ( 0) = 0 satisfies the following inequality at sampling time k
V ( x(k + i+1k )) − V ( x(k + ik ))
≤
T
−x( k + ik ) Qx( k + ik ) −u(k + i | k )T Ru(k + i | k ).
(7)
By summing both sides of (7) from i = 0 to i = ∞ one could find that T
T
x( ∞k ) Px( ∞k ) − x( kk ) Px( kk ) ≤−J (k ).
(8)
For the asymptotic stability of the closed-loop system, x( ∞k ) must be zero and thus to have an asymptotic stability it follows that
(
)
J (k ) ≤V x( kk ) ≤ γ ,
(9)
where γ is a positive scalar that can be an upper bound for (6). As a result, the RMPC problem is defined as follows Theorem 1. Consider system (1) subject to input and state constraints as (2) and (3). Let x( kk ) be the measured state x(k ) at sample
2. Mathematical preliminaries
time k . Then, the state feedback matrix F (k ) in the control law The proposed regulator in [13] and some useful lemmas and remarks which are used in the remainder of the paper are presented in this section. Let consider a nonlinear discrete-time system represented by
x( k+1) = f (x( k ), u(k )),
X > 0 and Y are obtained from the solution of the following optimization problem with variables γ , ξ , X , Y , M , N , Z = [X;Y ].
min γ, ξ, X, Y , M, N γ
uj (k ) ≤ uj,max , j = 1, 2, …,m,
(2)
xj (k ) ≤ xj,max , j = 1, 2, …,n, n
(3)
(10)
subject to
⎡ T⎤ x( kk ) ⎥ ⎢ I ≥0, X >0, ⎢ x kk X ⎥⎦ ⎣ ( )
(11)
m
where x(k )∈R and u(k )∈R are the state and input vectors of the system respectively, f ( . , . ) is a Lipschitz and C1 function where f ( 0,0) = 0. To design an RMPC based on [13], at first the nonlinear system in (1) is rewritten in uncertain linear representation form
x( k+1) = Ax( k ) + Bu( k ) + f ̃ x( k ), u( k ) ,
(
)
(4)
where A = ∂f /∂x (0,0) , B = ∂f /∂u (0,0). Since f is a Lipschitz nonf ̃ x( k ), u(k ) = f x( k ), u(k ) − Ax( k ) − Bu(k ) is linearity then bounded as
(
)
(
)
T T f ̃ x( k ), u( k ) f ̃ x( k ), u( k ) ≤ ⎡⎣ xT ( k )uT (k )⎤⎦WT W ⎡⎣ xT ( k )uT (k )⎤⎦ ,
) (
the objective function J ( k ) at instant k is given by F = YX−1, where
(1)
and subject to
(
u( k + ik ) = F ( k )x( k + ik ) that minimizes the upper bound V (x( kk )) of
)
⎡ X ⎢ ⎢ 1+ε (AX + BY ) ⎢ ⎢ 1 + 1/ε (WZ ) ⎢ ⎢ Q1/2X ⎢ ⎢⎣ R1/2Y
X − ξI ≥ 0,
1+ε (AX + BY )T X 0 0 0
⎤ 1 + 1/ε (WZ )T (Q1/2X )T (R1/2Y )T ⎥ ⎥ 0 0 0 ⎥ ⎥ ≥0, ξI 0 0 ⎥ ⎥ 0 γI 0 ⎥ 0 0 γI ⎥⎦ (12)
(13)
(5)
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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⎡M *⎤ 2 ⎢ T ⎥ ≥ 0 with Mjj ≤ ur,max , j = 1, 2, …, m, ⎣Y X⎦
(14)
2 N − X ≥ 0 with Njj ≤ x r,max , j = 1, 2, …, n.
(15)
Proof. To prove (10)–(13) and to find how to select ε , see [13]. To prove (14) and (15) see [17]. Remark 1. The mentioned algorithm in this section is based on having equilibrium at the origin i.e. f ( 0,0) = 0. When this is not eq
the case, that is (x eq,u ) ≠ (0,0), at first the shifted nonlinear model
x ̅ ( k + 1) = f x( k ),u( k ) −f x eq , ueq = f ̅ x ̅ ( k ), u̅ ( k ) ,
(
is defined as
) (
)
(
)
eq
where x ̅ = x−x and u̅ = u−ueq . This change of variables should be ≤ uj,max , considered in the constraints as uj(k + i | k ) = u̅j( k + ik) + ueq j
i ≥ 0,
j = 1, 2,⋯,m and
xj( k + ik ) = xj̅ (k + i | k ) + x eq ≤ xj,max , j
i ≥ 0, j = 1, 2,⋯,n as well. Then the described RMPC algorithm is followed to determine u̅ . Finally the control input u = u̅ + ueq is applied to the actual system. Lemma 1. Any feasible solution to the optimization problem defined in Theorem 1 at time k1 is also a feasible solution for all time k>k1. Thus if the optimization problem in Theorem 1 is feasible at time k1 then it is feasible for all times k > k1. Proof. See the proof of Lemma 1 in [5]. Lemma 2. (invariant set) Consider a closed-loop system composed of system (1) and an RMPC designed based on Theorem 1. Suppose control law u(k ) = YX−1x(k ) is obtained from solving the optimization problem defined in Theorem 1 for initial state x0 . Then the subset
ψ=
{ x∈R
n
}
| xT X−1x ≤ 1 of the state space Rn is an asymptotically
stable invariant ellipsoid. Proof. The only LMI in Theorem 1 that depends on the system state is (11) where if it is satisfied for initial state x0 then it is satisfied for any x∈ψ and RMPC in Theorem 1 is guaranteed to be feasible. As a result, the obtained control law for initial state x0 is a feasible solution (not necessary optimal one) for any x∈ψ . Therefore, we could apply this control law for any x∈ψ such that the other constraints in Theorem 1 are satisfied and thereby ensuring that
x( k + i + 1)X−1x( k + i + 1)
i ≥ 0.
Thus, x( k + i) ∈ ψ for i ≥ 1 and x( k + i)→0 as i → ∞, establishing ψ is an asymptotically stable invariant set. Lemma 3. For nonlinear system (1), an ellipsoid region of stability of the RMPC in Theorem 1 is given by =
{ x∈R
n
}
xT −1x ≤ 1 , where
is obtained by solving the following problem.
=
max logdet(X )
(16)
γ , ξ, X , Y , M, N
subject to (12)–(15). Proof. When RMPC in Theorem 1 is applied to a state of the system in (1), the only LMI that depends on the system state is (11) which is automatically satisfied for all states within the ellipsoid ψ =
{ x∈R
n
}
xT X−1x ≤ 1 . Thus, ψ defines a feasible region of
RMPC. The defined problem in Lemma 3 maximizes the volume of this ellipsoid which is proportional to det(X )0.5. Therefore, the maximum region of feasibility for RMPC ( ) could be obtained by solving the defined problem. As a result, considering system (1),
3
for any x0 ∈ , RMPC in Theorem 1 is guaranteed to be feasible and using Lemma 2 establishes that is an asymptotically stable invariant set and RMPC defined in Theorem 1 stabilizes the closed-loop system asymptotically with an estimated region of stability . □
3. Set point tracking RMPC scheme Designing the RMPC for the regulation problem and estimating its domain of attraction were described in the previous section. A similar idea is utilized here to introduce an efficient algorithm for the set point tracking problem. Definition 1. Consider constraints on state and input of system (1) as x ∈ X and u ∈ U , where X and U are compact sets. Determine the steady state response of the system at any ueq∈U as.
x eq = f (x eq , ueq),
(17)
where x eq∈X . The point x eq is called an equilibrium point of system (1). The connection of all equilibria forms an equilibrium surface. Each selected curve on the equilibrium surface represents an equilibrium curve. For each curve one could consider two directions which are named equilibrium paths. In this work equilibrium points over one path (or more paths) belonging to the equilibrium surface are considered as the set points which should be tracked by the closed-loop system. In this regard, first a set of RMPCs with appropriate overlaps in their regions of stability are designed for a specified range in each path. Then based on these off-line computations, an efficient on-line strategy is implemented such that the tracking of any set point in the specified range is guaranteed. The proposed algorithm has the same off-line computations with [17] but a novel on-line switching strategy is implemented here which enables the algorithm to track set point changes efficiently. 3.1. Control description The proposed control approach consists of an off-line computation part and an on-line switching strategy that are described in the following subsections. 3.1.1. Set point tracking RMPC: off-line part To construct the off-line part of the controller, two steps should be performed. First, two paths for each equilibrium curve is considered. Then, a set of RMPCs is designed with respect to each path and appropriately selected middle equilibrium points. In order to demonstrate the proposed method, suppose that Fig. 1(a) exhibits the state space of a nonlinear second order system in the form of (1) where dashed line depicts its equilibrium curve. To design an RMPC which tracks each set point over the equilibrium curve between points a and b , two paths with opposite directions corresponding to this equilibrium curve are considered. Considering Fig. 1(b), Path #1 is assumed in direction of increasing x2 and Fig. 1(c) shows Path #2 which is selected in the direction of decreasing x2. Then a set of RMPCs with respect to each path is designed for appropriately selected middle equilibrium points. These equilibria are chosen such that their related RMPCs have proper overlaps in their regions of stability. Considering Fig. 1(b) to construct the RMPC set for Path #1, point a is eq . The considered as the first middle equilibrium and named x1,0 maximum region of stability for its corresponding RMPC is obtained based on Lemma 3 and is shown by 1. The second middle eq is chosen inside 1 and on the equilibrium path. equilibrium x2,0 eq The approximated stability region of RMPC related to x2,0 is shown
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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Fig. 1. Graphical representation of the proposed tracking scheme.
by 2. The other equilibria are determined in the same manner. This process continues until the last middle equilibrium passes point b . The final step is to augment some equilibria (shown as x ieq ,j and call augmented equilibrium) between sequential middle eq eq equilibria x ieq ,0 and x i − 1,0 , i ≥ 2 such that by increasing j , x i, j takes eq distance from x ieq ,0 and gets close to x i − 1,0 which are shown as
x2,eqj,j = 1, 2 and x3,eqj,j = 1, 2 in Fig. 1(b). These augmented points are considered in order to improve the transient response and guarantee the closed-loop stability. The procedure of how to select these points is explained in the next sections. Assumption 2. It is assumed that each augmented equilibrium points, x ieq , j , is located in the region of stability of RMPC related to the next augmented equilibrium, x ieq , j + 1. This assumption is required to develop the on-line strategy in Section 3.1.2 and will be replaced by a milder one (Assumption 3) in Algorithm 1. Path #2 is selected in the direction of decreasing x2. The corresponding RMPC stability regions are shown in Fig. 1(c) which are designed in the same manner discussed for Path #1 while the first eq middle point, x1,0 , is selected at b . So far the off-line part of the proposed controller has been accomplished which consists of two paths corresponding to one equilibrium curve and a set of RMPCs related to the middle equilibrium points for each path and also some augmented equilibria that considered between each two adjacent middle equilibria. 3.1.2. Set point tracking RMPC: on-line part q q For any initial state x( 0) ∈ (⋃i =1 0 i)Path #1 ∩ (⋃i =2 0 i)Path #2, where q1 and q2 are the number of middle equilibria in Path #1 and Path #2 respectively, the on-line part should determine the appropriate path and its corresponding RMPC set considering the location of the initial state with respect to desired set point. Then it should choose the correct controller at each instant to steer the state to the set point. In this context, whenever a new tracking problem starts (set point variations occur), the region of initial state ( x(0)) and set point ( x sp ) should be specified initially. Definition 2. st and sp are i and j with smallest i and j respectively such that x(0) ∈ i and x sp ∈ j . To clarify this definition, consider the example shown in Fig. 1 (b). x(0) is located only inside 3 therefore st = 3 but x sp belongs to both 1 and 2 thus according to Definition 2, the smallest index i.e. sp = 1 is chosen.
The on-line switching strategy could be expressed as two cases that are explained as follows. Case st ≠ sp : For x( 0) ∈ i,i = st or once x( k ) enters i , at first the feedback matrix, F ( k ), is determined from the RMPC related to (Controller #( i, 0)). According to Lemma 3 this controller x ieq ,0 guarantees feasibility of the solution and according to Lemma 1 it would be feasible for the next sampling times as well though it will not be an optimal solution. At the next sampling time, RMPC based on the first augmented equilibrium, x ieq (Controller #( i, 1)) is ,1 designed which may not be feasible (Assumption 2 implies that it gets feasible after some time instances). In the case of infeasibility, the previous computed control action is applied until Controller #( i, 1) leads to a feasible solution. Then, this feedback is implemented for the next sampling times until the RMPC based on the next augmented equilibrium, x ieq ,2 (Controller #( i, 2)) gets feasible. Other augmented Controllers #( i, j ) are implemented in a same manner sequentially. At each sample time it is checked whether x( k ) has entered i −1 or not and this continues until x( k ) enters Ssp + 1. In this region, the same procedure as for the previous regions is performed except there would be no check for entrance into the next region. That is, at first Controller #( sp + 1,0) and then eq xsp #( sp + 1, j ) corresponding to + 1, j (j = 1,2, …) are implemented. Meanwhile it is checked whether eq sp xsp or not. When the answer is yes, then the + 1, j + 1 has passed x
the next Controllers
computed feedback matrix related to Controller #( sp + 1, j ) is employed until the RMPC regarding x sp gets feasible. From now on, the controller related to x sp is implemented at remaining sampling times which ensures the convergence to x sp . To explain the method by an example, let assume that x(0) and x sp shown in Fig. 1(a) represent an initial state and a set point respectively. Since x sp is placed upper than x(0) then Path #1 which is in direction of increasing x2 is chosen. Considering Path #1 in Fig. 1(b), x(0) is placed in 3 therefore, RMPC designed based on eq (Controller #(3,0)) is implemented at the first sample time. At x3,0 eq the next sample time Controller #(3,1) related to x3,1 (the first augmented equilibrium in 3) is designed but it may not be feasible. In this case, the previous solution is implemented and the process is repeated until Controller #(3,1) leads to a feasible solution. For the next sampling times the same controller continues to work until the RMPC based on the next augmented equilibrium gets feasible. During each sample time it is checked whether x(k ) has entered 2 or not. Since x sp is placed in 1, then when x(k ) enters 2, switching to the next region is stopped and according to the explained procedure, at first Controller #(2,0) is implemented until Controller #(2,1) leads to a feasible solution. This continues
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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Remark 2. According to Definition 1 the equilibrium surface is determined by the steady state relation of the system and then equilibrium curves is selected on this surface. Due to the relation exists among elements of the state vector on the equilibrium curve, by having one of them, the others and the corresponding inputs could be determined. Therefore, for specifying the augmented equilibria x ieq , j ( j = 1, 2, ⋯) for each region i , a move step αi corresponding to one of the elements in the state vector could be employed. Thus, these points could be computed on-line without needing to save them in a lookup table.
Table 1 Middle equilibria and move steps in Example 1.
i
Path #1
(x 1 2 3
eq , i,0
uieq ,0
Path #2
)
([19 96.1875]T , 1.7368)
αi
(x
0
([8 40.5]T , 1.1270)
eq , i,0
uieq ,0
)
αi 0
([15 75.9375] , 1.5432)
0.18
T
([12 60.75] , 1.3803)
0.12
([8 40.5]T , 1.1270)
0.18
([19 96.1875]T , 1.7368)
0.17
T
5
To clarify the point of Remark 2, consider the equilibrium surface and its related input for a nonlinear second order system in the form of (1) which are determined by solving (17) where x eq = ⎡⎣ x1eq x2eq ⎤⎦. Then select a curve on this equilibrium surface. Considering the relations of this equilibrium curve, for a specified x1eq , x2eq and ueq could be computed. Thus for the mentioned ex⎡ x eq x eq ⎤, in region , ample, the augmented equilibria, x ieq 2 i, j ⎦ i , j = ⎣ 1i, j could be obtained on-line by using a move step αi regarding the first element of the state vector as
until the designed RMPC based on x sp becomes feasible. This controller continues to work and guarantees steering state of the system to the set point. When st ≠ sp , considering the direction of the path, x sp is certainly located after the first middle equilibrium ( x ieq ) ,0,i = st where the corresponding controller has the guaranteed feasibility. Therefore, the given procedure ensures the convergence to the set point. However, when st = sp , x sp is located before x ieq ,0,i = st the described strategy will not work. Case st = sp : To establish the stability in this case, at first state is derived to the previous region (st + 1) and once this is taken place, a new tracking problem is defined with current value of state as a new initial state. This modification leads the condition to st ≠ sp case and therefore, the procedure discussed for the case st ≠ sp could be implemented. In this regard, the opposite path and its first middle equilibrium are considered as a pseudo path and a pseudo set point momentarily to guide the state to the previous region. For example consider x( 0) and x sp in Fig. 1(c).
x1ieq = x1ieq + jαi, j ≥1, ,j , j −1
(18)
and consequently the second state
( x2ieq ) ,j
and its related input (uieq ,j )
could be determined by solving (17). and x ieq , should Remark 3. Two adjacent middle equilibria x ieq ,0 + 1,0 not be chosen arbitrarily close. Otherwise i + 1 encompasses i . Remark 4. The larger number of middle equilibria x ieq ,0 caused in a larger lookup table which needs more storage space and longer search time.
Since x( 0) is above x sp , Path #2 should be considered but it is observed that st = sp = 1. Therefore, states should be carried to 2 as a first step. To do so, the opposite path (Path #1) and the eq ) are considered and the problem pseudo set point on this path ( x1,0
The stability regions of the controllers associated with the augmented equilibria are not estimated off-line and they are not considered in the off-line part either. Since they are constructed and implemented on-line, the number and location of the augmented equilibria are not faced with limitations like those of the middle ones mentioned in Remarks 3 and 4. Note that the augmented equilibria are employed to improve the transient response and also provide a guaranteed stability for the tracking problem. In the gain scheduled RMPC approach, using the larger number of intermediate controllers caused to controllers’ switching
is solved with these temporary assumptions until x( k ) gets out from 1 in Path #2. Once it is occurred, the current value of the state is considered as a new x( 0) where st = 2 and then the control procedure for st ≠ sp is followed to track the middle set point x sp .
120
#1
80
xsp3
#3
h2
#2 xsp1
#2
40
#3
xsp2
#1
x(0)
10 0
(a)
20 h1
(b)
40 0
20 h1
40 0
(c)
20 h1
40
Fig. 2. Phase plots in Example 1: (a) stability region of the RMPC set in Path #1, (b) stability region of the RMPC set in Path #2, (c) state trajectory for the tracking problem.
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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6
h1
25 15
h2
5 100 60
u
20 2.4 1.2 0 0
500
1000
Time, sec
1500
Fig. 3. State responses and control action in Example 1 (solid line), set point (dashed line).
Table 2 Middle equilibria and move steps in Example 2.
i
Path #1
(x 1
eq , i,0
uieq ,0
Path #2
)
([371.1264 0.83]T , 308.2529)
αi
(x
0
([423.9139 0.25]T , 297.8278)
eq , i,0
uieq ,0
)
αi 0
([378.7332 0.76] , 309.4664)
0.023
([418.3429 0 .3] , 296.6857)
0.167
([387.7159 0.66]T , 307.4319)
0.033
([411.5618 0.37]T , 297.1235)
0.023
4
([398.9704 0.5203]T , 302)
0.028
([403.8359 0.46]T , 299.6718)
0.025
5
([411.5618 0.37]T , 297.1235)
0.03
([393.4233 0.59]T , 304.8456)
0.037
6
T
([418.3429 .3] , 296.6857)
0.023
([380.6562 0.74] , 309.3125)
0.037
7
([423.9139 0.25]T , 297.8278)
0.012
([371.1264 0.83]T , 308.2529)
0.025
2 3
T
without moving the state trajectory close to their intermediate equilibria which could improve the transition performance. Moreover, RMPC is a model based controller which uses the linearized model at an equilibrium point to construct a feedback matrix (Theorem 1). The scheduled RMPC implements a set of RMPCs associated with some intermediate equilibria while choosing two adjacent equilibria more close causes to less differences in their linearized model and consequently in less differences in their related feedback coefficients. This of course could reduce the jumps during the switches between two consecutive controllers. According to the above discussions, selecting intermediate equilibria very close could enhance transient response. However, according to Remarks 3 and 4 one should not choose two adjacent middle equilibria arbitrarily close. Thus adding augmented equilibria such those proposed in this paper could be an efficient approach in this regard. The stability regions of each path are determined based on its middle equilibria. Therefore, when the controllers are considered without augmented equilibria, one only guarantees the state convergence to the middle equilibria while a set point could be located anywhere on the specified domain over the equilibrium path. Therefore, in this paper an on-line algorithm is proposed to implement a strategy using augmented points to enable the
T
T
convergence to each admissible set point. The following section provides a systematic procedure to design the proposed controller. 3.2. Control algorithm The proposed set point tracking RMPC approach in Section 3.1 could be summarized by the following algorithm. Algorithm 1. For system (1), in order to track each set point between points a and b over a specified equilibrium curve, the set point tracking RMPC is designed as the following steps. Off-line For a specified equilibrium curve, two paths are defined: Path #1 in direction of b to a and Path #2 in direction of a to b. A set of controllers is designed for each path separately. The following three steps are performed for Path #1. In the case of Path #2 the procedure would be the same and just a should be replaced by b. eq 1. Set i = 1 and x1,0 = a.
2. For
(x
eq i,0 ,
)
uieq ,0 , rewrite the system equations as (4) considering
Remark 1. Then obtain the maximum region of stability, i ,
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7
1 #7
#1
0.8
xsp2
#2 #6 #3
0.6
#5
CA
#4
xsp1
#4
0.4
#5
#3 #6 #7
0.2 (a)
0 360
x(0)
(b)
400 T
xsp3
#2 #1
(c)
440
400 T
360
400 T
440
Fig. 4. Phase plots in Example 2: (a) stability region of the RMPC set in Path #1, (b) stability region of the RMPC set in Path #2, (c) state trajectory for the tracking problem.
T
430 400
CA
370 0.9 0.5
0.1 350 Tc
300 250 0
2
4
Time, min
6
8
10
Fig. 5. State responses and control action in Example 2 using Algorithm 1 (solid line), set point (dashed line).
based on Lemma 3 where i =
(x
eq i,0 ,
{ x∈R | x ( ) n
−1
T
i
}
x ≤ 1 . Store
)
uieq ,0 , i , Wi , εi , αi in a lookup table. The Lipschitz weighting
matrix Wi and the coefficient εi are the chosen parameters which are used to obtain i in Lemma 3 and αi is a selected move step for this region ( α0 is set as 0). eq 3. Select x ieq + 1,0 , ui + 1,0 on the equilibrium curve, in opposite direction of the path which satisfying x ieq + 1,0 ∈ i . Let i = i + 1 and
(
)
passes b then set λ which is the value of go to Step 2, until x ieq ,0 admissible time for infeasibility regarding each augmented equilibrium point in the on-line part.
On-line 1. Set k = 0. Check the set point x sp . Detect the appropriate path regarding the location of x(0) and x sp and load the related lookup table. 2. Detect st and sp using Definition 2. If st ≠ sp , set i = st and j = 0 and go to Step 4. If st = sp , save x msp = x sp and m = st , then take opposite path eq from the one detected in Step 1. Set x sp = x1,0 on this path and set f lag = 1 and go to Step 1. 3. If f lag = 1, check if x(k ) gets out from m then set flag = 0 and
x sp = x msp . Go to Step 1.
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else if f lag = 0 and i > sp+1, check if x(k )∈ i enters the next region i − 1. In this case, set i = i − 1 and j = 0. eq 4. Design RMPC based on Theorem 1 for x eq , ueq = x ieq , j , u i, j (Controller #(i, j )). 5. If Step 4 leads to a feasible solution for feedback matrix, set
(
)
(
)
count = 0 and F ( k ) = F ( k ) then implement it in control input i, j
(
)
eq . u( k ) = F (k ) x( k ) − x ieq , j + u i, j
description in Section 3.1.2 for cases st ≠ sp and st = sp . 4. Simulation results In this section, two examples are considered to show the efficiency of the proposed approach where the LMI control toolbox [28] in the MATLAB environment was used to compute the solution to the optimization problem and cvx toolbox [29] was implemented to evaluate the maximum region of stability in Lemma 3.
otherwise set count = count + 1 and F ( k ) = F (k − 1) and im-
(
)
eq . plement it in control input as u( k ) = F (k ) x( k ) − x ieq , j − 1 + u i, j − 1
6. If count = 0, set j = j + 1 then find
(x
eq i, j ,
)
by using αi (conuieq ,j
sidering Remark 2) and go to Step 7. else if 0 < count ≤ λ go to Step 3.
(
eq else if count > λ , set αi = 0.5αi and count = 0 then find x ieq , j , u i, j
)
by using αi (considering Remark 2) and go to Step 3. eq eq 7. If i > sp+1, check if x ieq , j is located between x i,0 and x i − 1,0 go to
Step 3 else set j = j − 1 and go to Step 6. sp go to Step 4 else set else if i = sp+1, check if x ieq , j do not pass x
(x
eq i, j ,
) (
sp sp and go to Step 4. uieq ,j = x , u
)
The proposed method in Section 3.1 has been structured as Algorithm 1 described above. The off-line part of this algorithm describes how to build up the off-line sections of the proposed controller (as explained in Section 3.1.1) and the on-line part determines the appropriate on-line strategy which should be implemented at each sample time to guarantee the closed-loop stability (as described in Section 3.1.2). In the proposed algorithm there is a possibility to tune move step αi if it is initialized improperly. In Step 6 it is checked whether infeasibility in Step 4 occurred more than λ times ( count > λ ) which implies that the sequential augmented points are located far from each other or equivalently αi is chosen large. In this case move step αi is substituted by 0.5αi . Then the related augmented point where the corresponding controller gets infeasible is replaced by a closer one that increases the feasibility chance. Therefore, Assumption 2 is reduced to Assumption 3 which is equivalent to Assumption 1. Assumption 3. RMPC related to each augmented equilibrium point has a non-empty region of stability. It should be noted that selecting small λ in Step 6 causes in multiple dividing of αi and decreasing its value unnecessarily while choosing the large value for λ allows occurrence of several infeasibilities in Step 4 in the case where initial αi is improperly selected.
4.1. Example 1. Consider a two tank system in [13] with the following dynamics
⎧ ρS ḣ = − ρA 2gh + u, ⎪ 1 1 1 1 ⎨ ̇ = ρA 2gh − ρA 2gh , ⎪ ρ S h ⎩ 2 2 1 1 2 2
(19)
where ρ = 0. 001 kg/cm3, g = 980 cm/s2, A2 = 4 cm2, A1 = 9 cm2 ,
S1 = 2500 cm2 , S2 = 1600 cm2, 1 ≤h1 ≤50 cm , 10 ≤h2 ≤120 cm , and 0 ≤u ≤2. 5 kg/s. The equilibrium surface and its related inputs are given in (20) and (21) respectively which are obtained from the steady state solution of (19). These relations are employed to find the augmented equilibria in Step 6 in Algorithm 1 as explained in Remark 2.
h1eq −
A22 A12
h2eq = 0,
ueq = ρA1 2gh1 .
(20) (21)
To design a controller based on Algorithm 1 to track any set point on equilibrium curve in (20) between a = [19 96.1875]T and
b = [8 40.5]T , two path regarding (20) are considered, Path #1 in direction of increasing h2 and Path #2 in the opposite direction. For each path a set of RMPCs is designed according to mentioned steps T in the off-line part of Algorithm 1. Let x = ⎡⎣ h1 h2 ⎤⎦ . The chosen
middle equilibria regarding Path #1 and Path #2 and their corresponding move steps regarding the first state are listed in Table 1. The estimated regions of stability are shown in Fig. 2(a) and (b). The sampling time is 1 s and the control parameters are set as R = 0.1, Q = I , W = diag(⎣⎡ 0.001 0.001 0 ⎦⎤), ε = 0.01, and λ = 5. In this study, the control act starts from x( 0) = [10 20 ]T and first set point is set as x sp1 = [11 56.6875]T . There is a set point change from x sp1 to x sp2 = [ 9 45.5625]T at t ¼ 500 s and then from x sp2 to
x ieq ,0
x sp3 = [8 91.125]T at t ¼ 1000 s. The state trajectory of the control system is shown in Fig. 2(c).
and x ieq ” in Step 7 implies that in each region i , the associated − 1,0
At t = 0 , the set point, x sp1, is upper than x( 0) therefore, the
augmented equilibrium x ieq , j should not exceed the next middle
controllers in Path #1 are implemented where st = 3 and sp = 2.
Remark 5. The mentioned condition: “if
x ieq ,j
is located between
equilibrium x ieq . − 1,0
At t ¼ 500 s, the set point changes to x sp2 while x sp1 is an initial state for this new tracking action ( x( 0) = x sp1). Now set point is
Remark 6. In the extended scheduled RMPC, x(k )∈ i could exceed the predefined stability region but the condition x ieq , j ∈ i −1 ensures that the trajectory returns back to this region.
located under initial state and it means Path #2 should be chosen but st = sp = 2. Therefore, according to on-line part of Algorithm 1, the opposite path (Path #1) and its related controllers are implemented until x(k ) gets out 2 in Path #2 as it is obvious in Fig. 2 (c). Then the related controllers in Path #2 are used which results in convergence to x sp2 . At t ¼ 1000 s, controllers in Path #2 are used because of the location of set point ( x sp3) and initial state
Theorem 2. Consider nonlinear system in (1) along with one of its equilibrium curves. The corresponding set point tracking RMPC which is designed based on Algorithm 1 has an estimated region of stability q q . defined by ⋃i =1 0 i ∩ ⋃i =2 0 i
( x( 0) = x sp2). The time responses of state and control actions are
Proof. The proof could be easily obtained by considering the
shown in Fig. 3.
(
)
Path #1
(
)
Path #2
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specified range on the equilibrium curve. Moreover, the proposed algorithm could provide an admissible transient performance in spite of switching between the local controllers because of using the augmented equilibria which could be constructed on-line without needing to save them in the look-up table.
4.2. Example 2. Consider the following CSTR system [17]
⎧ − E q UA ∆H ⎪ Ṫ = Tf − T − k 0e RT CA + ( Tc − T ), ⎪ V VρCp ρ Cp ⎨ ⎪ − E q CAf − CA − k 0e RT CA, ⎪ CȦ = ⎩ V
(
)
(
)
(22)
where T is the reactor temperature, CA is the concentration of substance A in the reactor, and Tc is the temperature of the coolant stream. The parameters are q = 100 l/min , V = 100 l, CAf = 1 mol/l,
Tf = 400 K ,
ρ = 103 g/l,
Cp = 1 J/( g K),
k0 = 4.71×108 min−1,
E /R = 8000 K , UA = 105 J/( min K), ∆H = − 2×105 J/( min K), 0 ≤CA ≤1 mol/l, 250 K ≤T ≤ 500 K , and 250 K ≤ Tc≤ 500 K . The equilibrium curve of the system in (22) and its related input are given by (23) and (24). These equations can be employed to construct the augmented equilibria in Algorithm 1 by using move step regarding one of the states.
T eq = −
E R
⎛ q ⎞ ln ⎜ C − CAeq ⎟ ⎝ k 0VC Aeq Af ⎠
(
Tceq =
,
)
(23)
⎞ VρCp ⎛ q ∆H − E ⎜⎜ − Tf − T eq + k 0e RT eq CAeq⎟⎟ + T eq. UA ⎝ V ρ Cp ⎠
(24)
(
)
9
To design a controller based on Algorithm 1 to track each set point on the equilibrium curve in (23) between
a = [ 371.1264 0.83]T and b = [ 423.9139 0.25]T , Path #1 in direction of increasing CA and Path #2 in the opposite direction are T assumed. Consider x = ⎡⎣ T CA ⎤⎦ and u = Tc . The selected middle
equilibria for each path and move steps regarding CA are given in Table 2. Two paths and their related RMPC stability regions are shown in Fig. 4(a) and (b). In this example a series of set point are selected as x sp1 = [ 398.1979 0.53]T , x sp2 = [ 374.5904 0.8]T , and x sp3= [ 423.9139 0.25]T where changes happen at multiple of 3.3 min. The tracking procedure starts from x( 0) = [ 410 0.1] which locates lower than x sp1 therefore, Path #1 is selected for the first tracking attempt. For the second and third set points, Path #1 and Path #2 are chosen respectively. State trajectory for this tracking problem is shown in Fig. 4(c). The time response of the states and control action are depicted in Fig. 5 which shows that the closed-loop system could track the set point changes efficiently. Moreover, the control signal jumps just once the set point changes occur and the proposed strategy could smooth the control action during switches between local controllers. In the simulations of this example the sample time was 0.33 min and the controller parameters were chosen as Q = diag([1/400 1/0.5]), R = 0.1×1/300, W = diag(⎡⎣ 0.001 0.01 0 ⎤⎦), ε = 0.01, and λ = 5 for all regions. 5. Conclusion In this paper, unlike the most existing RMPC algorithms, a scheduled RMPC scheme is introduced for the set point tracking problem. The proposed algorithm consists of the off-line and online parts. In the off-line part of the proposed approach some equilibrium points are determined based on stability regions of their corresponding RMPCs. These regions should have proper overlaps. Then the on-line part of the algorithm uses the information prepared in the off-line part and implements a novel switching strategy which guarantees tracking any set point in the
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Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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Research article
Constrained tracking control for nonlinear systems Fatemeh Khani a, Mohammad Haeri b,n a b
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Advanced Control Systems Lab, Electrical Engineering Department, Sharif University of Technology, Tehran 11155-4363, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 21 July 2016 Received in revised form 25 April 2017 Accepted 4 June 2017
This paper proposes a tracking control strategy for nonlinear systems without needing a prior knowledge of the reference trajectory. The proposed method consists of a set of local controllers with appropriate overlaps in their stability regions and an on-line switching strategy which implements these controllers and uses some augmented intermediate controllers to ensure steering the system states to the desired set points without needing to redesign the controller for each value of set point changes. The proposed approach provides smooth transient responses despite switching among the local controllers. It should be mentioned that the stability regions of the proposed controllers could be estimated off-line for a range of set-point changes. The efficiencies of the proposed algorithm are illustrated via two example simulations. & 2017 Published by Elsevier Ltd. on behalf of ISA.
Keywords: Set point tracking Robust model predictive controller Gain scheduling Domain of attraction
1. Introduction Model predictive control (MPC) is one of the well-known control strategies in the process industry due to its capability of optimally controlling multivariable systems with constraints [1,2]. Although a large number of industrial processes are inherently nonlinear, most of the MPC techniques implemented on the real processes are based on linear models. To enhance the performance, a practical control system with large operating regions needs to deal with nonlinearity explicitly. However, a nonlinear MPC could act inadequately because of the computational complexity generally exists in on-line implementation of the controller. As a result, some alternative approaches to consider the nonlinearity in the controller design have been proposed in the literature [3,4]. Robust model predictive control (RMPC) is a convenient approach to consider the modeling error in the controller design. Meanwhile, nonlinear systems could be represented by linear models along with structured or unstructured uncertainties. Thus, a robust control strategy could be implemented instead of a nonlinear one in order to reduce the computational complexity. Therefore, an RMPC strategy could be applied in nonlinear and uncertain cases concurrently. The RMPC techniques mainly are based on minimization of the worst-case objective function incorporating a set of robustness constraints. In this case, [5] introduced a systematic solution for RMPC problem based on linear n
Corresponding author. E-mail address: [email protected] (M. Haeri).
matrix inequality (LMI) for systems with polytopic and structured feedback uncertainties which could estimate the stability region of the controller. This method was modified next to decrease computational time [6,7], reduce conservativeness [8–12], or simplify the representation of the uncertainties [13–16]. Moreover, [17] proposed a scheduled RMPC algorithm that enlarges the operating region of the controller introduced in [5] efficiently and later on [18] proposed a method that improves its transient response. Most of the RMPC methods have been formulated for regulation problems that is steering the system to a fixed point. In practice however, it is often required to track a changing set point. When the set point changes, the stabilizing design of the RMPC may not be valid anymore and/or feasibility of the controller may be lost and the controller fails to track the set point. This issue requires redesigning of the RMPC for each value of the changing set point. In order to solve the tracking problem, [19] introduced RMPC based on a dual-model paradigm which guarantees the offset free tracking of set point changes. Ref. [20] proposed a tracking method which guarantees an H1 norm bound with an optimized linear quadratic performance. Refs. [21–23] presented tracking RMPC approaches by using the notion of tubes. In [24], tracking scheduled RMPC is proposed for linear time invariant systems such that a set of controllers should be designed off-line based on a predefined target. Ref. [25] introduced a robust tracking MPC based on derivation of an invariant set which provides both a large stabilizable set and the closed loop performance. Ref. [26] proposed a nonlinear MPC method augmented with a disturbance observer (DOB). The DOB estimates internal and external disturbances and then in addition to feedback MPC, a feedforward
http://dx.doi.org/10.1016/j.isatra.2017.06.004 0019-0578/& 2017 Published by Elsevier Ltd. on behalf of ISA.
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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2
controller is implemented to compensate the disturbances. In [27], an offset-free MPC approach based on prediction accuracy enhancement via a DOB was developed for a general disturbed system and the computational burden of the proposed method was reduced as compared with the existing offset-free MPCs. Advantages of RMPC such as giving the region of stability and considering input and state constraints in optimization problem make this controller attractive but it has limitations in applications on systems with large operating regions and changing set-points. This paper is devoted to solve the problem by a novel scheduling RMPC scheme. In the off-line part of the proposed method, a set of local controllers is designed that have appropriate overlaps in their stability regions. An on-line part implements a novel switching strategy which uses the designed controllers in off-line part and also some augmented intermediate controllers that are designed on-line. Some features of the suggested method are listed below – The proposed controller could be applied for constrained nonlinear systems. – A prior knowledge of the reference trajectory is not required. – The asymptotic stability of the closed-loop system is guaranteed under set-point changes without needing to change the control scheme for each value of set-points. – The stability region of the controller for a determined range of set-point changes could be approximated off-line. – Implementing augmented intermediate controllers in on-line part could prevent the spikes appearing at the moment of switching between adjacent local controllers in the scheduling scheme. The paper is organized as follows. In Section 2 some mathematical preliminaries are described. Section 3 presents the proposed set point tracking RMPC strategy. In order to illustrate the effectiveness of the proposed method two examples are presented in Section 4. Finally, Section 5 provides the concluding remarks.
where WT W is a positive definite matrix and represents the Lipschitz coefficient. Assumption 1. The pair (A , B ) is stabilizable by state feedback control law. The objective function in MPC which is minimized to optimize performance of the closed-loop system is defined as follows ∞
J ( k) =
∑ i=0
{(
}
T
x k + ik ) Qx( k + ik ) + u(k + i | k )T Ru(k + i | k ) ,
(6)
where Q > 0, R > 0, x( k + ik ) is state at time k + i predicted based on the measurements at time k and u( k + ik ) is control move at time k + i computed by minimizing J ( k ) at time k . To solve the given optimization problem for the nonlinear system in (1) via LMI, first one should replace equality in (6) by an inequality which is done by defining an upper bound for J (k ). Consider a quadratic function V ( x ) = xT Px with P > 0 and V ( 0) = 0 satisfies the following inequality at sampling time k
V ( x(k + i+1k )) − V ( x(k + ik ))
≤
T
−x( k + ik ) Qx( k + ik ) −u(k + i | k )T Ru(k + i | k ).
(7)
By summing both sides of (7) from i = 0 to i = ∞ one could find that T
T
x( ∞k ) Px( ∞k ) − x( kk ) Px( kk ) ≤−J (k ).
(8)
For the asymptotic stability of the closed-loop system, x( ∞k ) must be zero and thus to have an asymptotic stability it follows that
(
)
J (k ) ≤V x( kk ) ≤ γ ,
(9)
where γ is a positive scalar that can be an upper bound for (6). As a result, the RMPC problem is defined as follows Theorem 1. Consider system (1) subject to input and state constraints as (2) and (3). Let x( kk ) be the measured state x(k ) at sample
2. Mathematical preliminaries
time k . Then, the state feedback matrix F (k ) in the control law The proposed regulator in [13] and some useful lemmas and remarks which are used in the remainder of the paper are presented in this section. Let consider a nonlinear discrete-time system represented by
x( k+1) = f (x( k ), u(k )),
X > 0 and Y are obtained from the solution of the following optimization problem with variables γ , ξ , X , Y , M , N , Z = [X;Y ].
min γ, ξ, X, Y , M, N γ
uj (k ) ≤ uj,max , j = 1, 2, …,m,
(2)
xj (k ) ≤ xj,max , j = 1, 2, …,n, n
(3)
(10)
subject to
⎡ T⎤ x( kk ) ⎥ ⎢ I ≥0, X >0, ⎢ x kk X ⎥⎦ ⎣ ( )
(11)
m
where x(k )∈R and u(k )∈R are the state and input vectors of the system respectively, f ( . , . ) is a Lipschitz and C1 function where f ( 0,0) = 0. To design an RMPC based on [13], at first the nonlinear system in (1) is rewritten in uncertain linear representation form
x( k+1) = Ax( k ) + Bu( k ) + f ̃ x( k ), u( k ) ,
(
)
(4)
where A = ∂f /∂x (0,0) , B = ∂f /∂u (0,0). Since f is a Lipschitz nonf ̃ x( k ), u(k ) = f x( k ), u(k ) − Ax( k ) − Bu(k ) is linearity then bounded as
(
)
(
)
T T f ̃ x( k ), u( k ) f ̃ x( k ), u( k ) ≤ ⎡⎣ xT ( k )uT (k )⎤⎦WT W ⎡⎣ xT ( k )uT (k )⎤⎦ ,
) (
the objective function J ( k ) at instant k is given by F = YX−1, where
(1)
and subject to
(
u( k + ik ) = F ( k )x( k + ik ) that minimizes the upper bound V (x( kk )) of
)
⎡ X ⎢ ⎢ 1+ε (AX + BY ) ⎢ ⎢ 1 + 1/ε (WZ ) ⎢ ⎢ Q1/2X ⎢ ⎢⎣ R1/2Y
X − ξI ≥ 0,
1+ε (AX + BY )T X 0 0 0
⎤ 1 + 1/ε (WZ )T (Q1/2X )T (R1/2Y )T ⎥ ⎥ 0 0 0 ⎥ ⎥ ≥0, ξI 0 0 ⎥ ⎥ 0 γI 0 ⎥ 0 0 γI ⎥⎦ (12)
(13)
(5)
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⎡M *⎤ 2 ⎢ T ⎥ ≥ 0 with Mjj ≤ ur,max , j = 1, 2, …, m, ⎣Y X⎦
(14)
2 N − X ≥ 0 with Njj ≤ x r,max , j = 1, 2, …, n.
(15)
Proof. To prove (10)–(13) and to find how to select ε , see [13]. To prove (14) and (15) see [17]. Remark 1. The mentioned algorithm in this section is based on having equilibrium at the origin i.e. f ( 0,0) = 0. When this is not eq
the case, that is (x eq,u ) ≠ (0,0), at first the shifted nonlinear model
x ̅ ( k + 1) = f x( k ),u( k ) −f x eq , ueq = f ̅ x ̅ ( k ), u̅ ( k ) ,
(
is defined as
) (
)
(
)
eq
where x ̅ = x−x and u̅ = u−ueq . This change of variables should be ≤ uj,max , considered in the constraints as uj(k + i | k ) = u̅j( k + ik) + ueq j
i ≥ 0,
j = 1, 2,⋯,m and
xj( k + ik ) = xj̅ (k + i | k ) + x eq ≤ xj,max , j
i ≥ 0, j = 1, 2,⋯,n as well. Then the described RMPC algorithm is followed to determine u̅ . Finally the control input u = u̅ + ueq is applied to the actual system. Lemma 1. Any feasible solution to the optimization problem defined in Theorem 1 at time k1 is also a feasible solution for all time k>k1. Thus if the optimization problem in Theorem 1 is feasible at time k1 then it is feasible for all times k > k1. Proof. See the proof of Lemma 1 in [5]. Lemma 2. (invariant set) Consider a closed-loop system composed of system (1) and an RMPC designed based on Theorem 1. Suppose control law u(k ) = YX−1x(k ) is obtained from solving the optimization problem defined in Theorem 1 for initial state x0 . Then the subset
ψ=
{ x∈R
n
}
| xT X−1x ≤ 1 of the state space Rn is an asymptotically
stable invariant ellipsoid. Proof. The only LMI in Theorem 1 that depends on the system state is (11) where if it is satisfied for initial state x0 then it is satisfied for any x∈ψ and RMPC in Theorem 1 is guaranteed to be feasible. As a result, the obtained control law for initial state x0 is a feasible solution (not necessary optimal one) for any x∈ψ . Therefore, we could apply this control law for any x∈ψ such that the other constraints in Theorem 1 are satisfied and thereby ensuring that
x( k + i + 1)X−1x( k + i + 1)
i ≥ 0.
Thus, x( k + i) ∈ ψ for i ≥ 1 and x( k + i)→0 as i → ∞, establishing ψ is an asymptotically stable invariant set. Lemma 3. For nonlinear system (1), an ellipsoid region of stability of the RMPC in Theorem 1 is given by =
{ x∈R
n
}
xT −1x ≤ 1 , where
is obtained by solving the following problem.
=
max logdet(X )
(16)
γ , ξ, X , Y , M, N
subject to (12)–(15). Proof. When RMPC in Theorem 1 is applied to a state of the system in (1), the only LMI that depends on the system state is (11) which is automatically satisfied for all states within the ellipsoid ψ =
{ x∈R
n
}
xT X−1x ≤ 1 . Thus, ψ defines a feasible region of
RMPC. The defined problem in Lemma 3 maximizes the volume of this ellipsoid which is proportional to det(X )0.5. Therefore, the maximum region of feasibility for RMPC ( ) could be obtained by solving the defined problem. As a result, considering system (1),
3
for any x0 ∈ , RMPC in Theorem 1 is guaranteed to be feasible and using Lemma 2 establishes that is an asymptotically stable invariant set and RMPC defined in Theorem 1 stabilizes the closed-loop system asymptotically with an estimated region of stability . □
3. Set point tracking RMPC scheme Designing the RMPC for the regulation problem and estimating its domain of attraction were described in the previous section. A similar idea is utilized here to introduce an efficient algorithm for the set point tracking problem. Definition 1. Consider constraints on state and input of system (1) as x ∈ X and u ∈ U , where X and U are compact sets. Determine the steady state response of the system at any ueq∈U as.
x eq = f (x eq , ueq),
(17)
where x eq∈X . The point x eq is called an equilibrium point of system (1). The connection of all equilibria forms an equilibrium surface. Each selected curve on the equilibrium surface represents an equilibrium curve. For each curve one could consider two directions which are named equilibrium paths. In this work equilibrium points over one path (or more paths) belonging to the equilibrium surface are considered as the set points which should be tracked by the closed-loop system. In this regard, first a set of RMPCs with appropriate overlaps in their regions of stability are designed for a specified range in each path. Then based on these off-line computations, an efficient on-line strategy is implemented such that the tracking of any set point in the specified range is guaranteed. The proposed algorithm has the same off-line computations with [17] but a novel on-line switching strategy is implemented here which enables the algorithm to track set point changes efficiently. 3.1. Control description The proposed control approach consists of an off-line computation part and an on-line switching strategy that are described in the following subsections. 3.1.1. Set point tracking RMPC: off-line part To construct the off-line part of the controller, two steps should be performed. First, two paths for each equilibrium curve is considered. Then, a set of RMPCs is designed with respect to each path and appropriately selected middle equilibrium points. In order to demonstrate the proposed method, suppose that Fig. 1(a) exhibits the state space of a nonlinear second order system in the form of (1) where dashed line depicts its equilibrium curve. To design an RMPC which tracks each set point over the equilibrium curve between points a and b , two paths with opposite directions corresponding to this equilibrium curve are considered. Considering Fig. 1(b), Path #1 is assumed in direction of increasing x2 and Fig. 1(c) shows Path #2 which is selected in the direction of decreasing x2. Then a set of RMPCs with respect to each path is designed for appropriately selected middle equilibrium points. These equilibria are chosen such that their related RMPCs have proper overlaps in their regions of stability. Considering Fig. 1(b) to construct the RMPC set for Path #1, point a is eq . The considered as the first middle equilibrium and named x1,0 maximum region of stability for its corresponding RMPC is obtained based on Lemma 3 and is shown by 1. The second middle eq is chosen inside 1 and on the equilibrium path. equilibrium x2,0 eq The approximated stability region of RMPC related to x2,0 is shown
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Fig. 1. Graphical representation of the proposed tracking scheme.
by 2. The other equilibria are determined in the same manner. This process continues until the last middle equilibrium passes point b . The final step is to augment some equilibria (shown as x ieq ,j and call augmented equilibrium) between sequential middle eq eq equilibria x ieq ,0 and x i − 1,0 , i ≥ 2 such that by increasing j , x i, j takes eq distance from x ieq ,0 and gets close to x i − 1,0 which are shown as
x2,eqj,j = 1, 2 and x3,eqj,j = 1, 2 in Fig. 1(b). These augmented points are considered in order to improve the transient response and guarantee the closed-loop stability. The procedure of how to select these points is explained in the next sections. Assumption 2. It is assumed that each augmented equilibrium points, x ieq , j , is located in the region of stability of RMPC related to the next augmented equilibrium, x ieq , j + 1. This assumption is required to develop the on-line strategy in Section 3.1.2 and will be replaced by a milder one (Assumption 3) in Algorithm 1. Path #2 is selected in the direction of decreasing x2. The corresponding RMPC stability regions are shown in Fig. 1(c) which are designed in the same manner discussed for Path #1 while the first eq middle point, x1,0 , is selected at b . So far the off-line part of the proposed controller has been accomplished which consists of two paths corresponding to one equilibrium curve and a set of RMPCs related to the middle equilibrium points for each path and also some augmented equilibria that considered between each two adjacent middle equilibria. 3.1.2. Set point tracking RMPC: on-line part q q For any initial state x( 0) ∈ (⋃i =1 0 i)Path #1 ∩ (⋃i =2 0 i)Path #2, where q1 and q2 are the number of middle equilibria in Path #1 and Path #2 respectively, the on-line part should determine the appropriate path and its corresponding RMPC set considering the location of the initial state with respect to desired set point. Then it should choose the correct controller at each instant to steer the state to the set point. In this context, whenever a new tracking problem starts (set point variations occur), the region of initial state ( x(0)) and set point ( x sp ) should be specified initially. Definition 2. st and sp are i and j with smallest i and j respectively such that x(0) ∈ i and x sp ∈ j . To clarify this definition, consider the example shown in Fig. 1 (b). x(0) is located only inside 3 therefore st = 3 but x sp belongs to both 1 and 2 thus according to Definition 2, the smallest index i.e. sp = 1 is chosen.
The on-line switching strategy could be expressed as two cases that are explained as follows. Case st ≠ sp : For x( 0) ∈ i,i = st or once x( k ) enters i , at first the feedback matrix, F ( k ), is determined from the RMPC related to (Controller #( i, 0)). According to Lemma 3 this controller x ieq ,0 guarantees feasibility of the solution and according to Lemma 1 it would be feasible for the next sampling times as well though it will not be an optimal solution. At the next sampling time, RMPC based on the first augmented equilibrium, x ieq (Controller #( i, 1)) is ,1 designed which may not be feasible (Assumption 2 implies that it gets feasible after some time instances). In the case of infeasibility, the previous computed control action is applied until Controller #( i, 1) leads to a feasible solution. Then, this feedback is implemented for the next sampling times until the RMPC based on the next augmented equilibrium, x ieq ,2 (Controller #( i, 2)) gets feasible. Other augmented Controllers #( i, j ) are implemented in a same manner sequentially. At each sample time it is checked whether x( k ) has entered i −1 or not and this continues until x( k ) enters Ssp + 1. In this region, the same procedure as for the previous regions is performed except there would be no check for entrance into the next region. That is, at first Controller #( sp + 1,0) and then eq xsp #( sp + 1, j ) corresponding to + 1, j (j = 1,2, …) are implemented. Meanwhile it is checked whether eq sp xsp or not. When the answer is yes, then the + 1, j + 1 has passed x
the next Controllers
computed feedback matrix related to Controller #( sp + 1, j ) is employed until the RMPC regarding x sp gets feasible. From now on, the controller related to x sp is implemented at remaining sampling times which ensures the convergence to x sp . To explain the method by an example, let assume that x(0) and x sp shown in Fig. 1(a) represent an initial state and a set point respectively. Since x sp is placed upper than x(0) then Path #1 which is in direction of increasing x2 is chosen. Considering Path #1 in Fig. 1(b), x(0) is placed in 3 therefore, RMPC designed based on eq (Controller #(3,0)) is implemented at the first sample time. At x3,0 eq the next sample time Controller #(3,1) related to x3,1 (the first augmented equilibrium in 3) is designed but it may not be feasible. In this case, the previous solution is implemented and the process is repeated until Controller #(3,1) leads to a feasible solution. For the next sampling times the same controller continues to work until the RMPC based on the next augmented equilibrium gets feasible. During each sample time it is checked whether x(k ) has entered 2 or not. Since x sp is placed in 1, then when x(k ) enters 2, switching to the next region is stopped and according to the explained procedure, at first Controller #(2,0) is implemented until Controller #(2,1) leads to a feasible solution. This continues
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Remark 2. According to Definition 1 the equilibrium surface is determined by the steady state relation of the system and then equilibrium curves is selected on this surface. Due to the relation exists among elements of the state vector on the equilibrium curve, by having one of them, the others and the corresponding inputs could be determined. Therefore, for specifying the augmented equilibria x ieq , j ( j = 1, 2, ⋯) for each region i , a move step αi corresponding to one of the elements in the state vector could be employed. Thus, these points could be computed on-line without needing to save them in a lookup table.
Table 1 Middle equilibria and move steps in Example 1.
i
Path #1
(x 1 2 3
eq , i,0
uieq ,0
Path #2
)
([19 96.1875]T , 1.7368)
αi
(x
0
([8 40.5]T , 1.1270)
eq , i,0
uieq ,0
)
αi 0
([15 75.9375] , 1.5432)
0.18
T
([12 60.75] , 1.3803)
0.12
([8 40.5]T , 1.1270)
0.18
([19 96.1875]T , 1.7368)
0.17
T
5
To clarify the point of Remark 2, consider the equilibrium surface and its related input for a nonlinear second order system in the form of (1) which are determined by solving (17) where x eq = ⎡⎣ x1eq x2eq ⎤⎦. Then select a curve on this equilibrium surface. Considering the relations of this equilibrium curve, for a specified x1eq , x2eq and ueq could be computed. Thus for the mentioned ex⎡ x eq x eq ⎤, in region , ample, the augmented equilibria, x ieq 2 i, j ⎦ i , j = ⎣ 1i, j could be obtained on-line by using a move step αi regarding the first element of the state vector as
until the designed RMPC based on x sp becomes feasible. This controller continues to work and guarantees steering state of the system to the set point. When st ≠ sp , considering the direction of the path, x sp is certainly located after the first middle equilibrium ( x ieq ) ,0,i = st where the corresponding controller has the guaranteed feasibility. Therefore, the given procedure ensures the convergence to the set point. However, when st = sp , x sp is located before x ieq ,0,i = st the described strategy will not work. Case st = sp : To establish the stability in this case, at first state is derived to the previous region (st + 1) and once this is taken place, a new tracking problem is defined with current value of state as a new initial state. This modification leads the condition to st ≠ sp case and therefore, the procedure discussed for the case st ≠ sp could be implemented. In this regard, the opposite path and its first middle equilibrium are considered as a pseudo path and a pseudo set point momentarily to guide the state to the previous region. For example consider x( 0) and x sp in Fig. 1(c).
x1ieq = x1ieq + jαi, j ≥1, ,j , j −1
(18)
and consequently the second state
( x2ieq ) ,j
and its related input (uieq ,j )
could be determined by solving (17). and x ieq , should Remark 3. Two adjacent middle equilibria x ieq ,0 + 1,0 not be chosen arbitrarily close. Otherwise i + 1 encompasses i . Remark 4. The larger number of middle equilibria x ieq ,0 caused in a larger lookup table which needs more storage space and longer search time.
Since x( 0) is above x sp , Path #2 should be considered but it is observed that st = sp = 1. Therefore, states should be carried to 2 as a first step. To do so, the opposite path (Path #1) and the eq ) are considered and the problem pseudo set point on this path ( x1,0
The stability regions of the controllers associated with the augmented equilibria are not estimated off-line and they are not considered in the off-line part either. Since they are constructed and implemented on-line, the number and location of the augmented equilibria are not faced with limitations like those of the middle ones mentioned in Remarks 3 and 4. Note that the augmented equilibria are employed to improve the transient response and also provide a guaranteed stability for the tracking problem. In the gain scheduled RMPC approach, using the larger number of intermediate controllers caused to controllers’ switching
is solved with these temporary assumptions until x( k ) gets out from 1 in Path #2. Once it is occurred, the current value of the state is considered as a new x( 0) where st = 2 and then the control procedure for st ≠ sp is followed to track the middle set point x sp .
120
#1
80
xsp3
#3
h2
#2 xsp1
#2
40
#3
xsp2
#1
x(0)
10 0
(a)
20 h1
(b)
40 0
20 h1
40 0
(c)
20 h1
40
Fig. 2. Phase plots in Example 1: (a) stability region of the RMPC set in Path #1, (b) stability region of the RMPC set in Path #2, (c) state trajectory for the tracking problem.
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6
h1
25 15
h2
5 100 60
u
20 2.4 1.2 0 0
500
1000
Time, sec
1500
Fig. 3. State responses and control action in Example 1 (solid line), set point (dashed line).
Table 2 Middle equilibria and move steps in Example 2.
i
Path #1
(x 1
eq , i,0
uieq ,0
Path #2
)
([371.1264 0.83]T , 308.2529)
αi
(x
0
([423.9139 0.25]T , 297.8278)
eq , i,0
uieq ,0
)
αi 0
([378.7332 0.76] , 309.4664)
0.023
([418.3429 0 .3] , 296.6857)
0.167
([387.7159 0.66]T , 307.4319)
0.033
([411.5618 0.37]T , 297.1235)
0.023
4
([398.9704 0.5203]T , 302)
0.028
([403.8359 0.46]T , 299.6718)
0.025
5
([411.5618 0.37]T , 297.1235)
0.03
([393.4233 0.59]T , 304.8456)
0.037
6
T
([418.3429 .3] , 296.6857)
0.023
([380.6562 0.74] , 309.3125)
0.037
7
([423.9139 0.25]T , 297.8278)
0.012
([371.1264 0.83]T , 308.2529)
0.025
2 3
T
without moving the state trajectory close to their intermediate equilibria which could improve the transition performance. Moreover, RMPC is a model based controller which uses the linearized model at an equilibrium point to construct a feedback matrix (Theorem 1). The scheduled RMPC implements a set of RMPCs associated with some intermediate equilibria while choosing two adjacent equilibria more close causes to less differences in their linearized model and consequently in less differences in their related feedback coefficients. This of course could reduce the jumps during the switches between two consecutive controllers. According to the above discussions, selecting intermediate equilibria very close could enhance transient response. However, according to Remarks 3 and 4 one should not choose two adjacent middle equilibria arbitrarily close. Thus adding augmented equilibria such those proposed in this paper could be an efficient approach in this regard. The stability regions of each path are determined based on its middle equilibria. Therefore, when the controllers are considered without augmented equilibria, one only guarantees the state convergence to the middle equilibria while a set point could be located anywhere on the specified domain over the equilibrium path. Therefore, in this paper an on-line algorithm is proposed to implement a strategy using augmented points to enable the
T
T
convergence to each admissible set point. The following section provides a systematic procedure to design the proposed controller. 3.2. Control algorithm The proposed set point tracking RMPC approach in Section 3.1 could be summarized by the following algorithm. Algorithm 1. For system (1), in order to track each set point between points a and b over a specified equilibrium curve, the set point tracking RMPC is designed as the following steps. Off-line For a specified equilibrium curve, two paths are defined: Path #1 in direction of b to a and Path #2 in direction of a to b. A set of controllers is designed for each path separately. The following three steps are performed for Path #1. In the case of Path #2 the procedure would be the same and just a should be replaced by b. eq 1. Set i = 1 and x1,0 = a.
2. For
(x
eq i,0 ,
)
uieq ,0 , rewrite the system equations as (4) considering
Remark 1. Then obtain the maximum region of stability, i ,
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7
1 #7
#1
0.8
xsp2
#2 #6 #3
0.6
#5
CA
#4
xsp1
#4
0.4
#5
#3 #6 #7
0.2 (a)
0 360
x(0)
(b)
400 T
xsp3
#2 #1
(c)
440
400 T
360
400 T
440
Fig. 4. Phase plots in Example 2: (a) stability region of the RMPC set in Path #1, (b) stability region of the RMPC set in Path #2, (c) state trajectory for the tracking problem.
T
430 400
CA
370 0.9 0.5
0.1 350 Tc
300 250 0
2
4
Time, min
6
8
10
Fig. 5. State responses and control action in Example 2 using Algorithm 1 (solid line), set point (dashed line).
based on Lemma 3 where i =
(x
eq i,0 ,
{ x∈R | x ( ) n
−1
T
i
}
x ≤ 1 . Store
)
uieq ,0 , i , Wi , εi , αi in a lookup table. The Lipschitz weighting
matrix Wi and the coefficient εi are the chosen parameters which are used to obtain i in Lemma 3 and αi is a selected move step for this region ( α0 is set as 0). eq 3. Select x ieq + 1,0 , ui + 1,0 on the equilibrium curve, in opposite direction of the path which satisfying x ieq + 1,0 ∈ i . Let i = i + 1 and
(
)
passes b then set λ which is the value of go to Step 2, until x ieq ,0 admissible time for infeasibility regarding each augmented equilibrium point in the on-line part.
On-line 1. Set k = 0. Check the set point x sp . Detect the appropriate path regarding the location of x(0) and x sp and load the related lookup table. 2. Detect st and sp using Definition 2. If st ≠ sp , set i = st and j = 0 and go to Step 4. If st = sp , save x msp = x sp and m = st , then take opposite path eq from the one detected in Step 1. Set x sp = x1,0 on this path and set f lag = 1 and go to Step 1. 3. If f lag = 1, check if x(k ) gets out from m then set flag = 0 and
x sp = x msp . Go to Step 1.
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8
else if f lag = 0 and i > sp+1, check if x(k )∈ i enters the next region i − 1. In this case, set i = i − 1 and j = 0. eq 4. Design RMPC based on Theorem 1 for x eq , ueq = x ieq , j , u i, j (Controller #(i, j )). 5. If Step 4 leads to a feasible solution for feedback matrix, set
(
)
(
)
count = 0 and F ( k ) = F ( k ) then implement it in control input i, j
(
)
eq . u( k ) = F (k ) x( k ) − x ieq , j + u i, j
description in Section 3.1.2 for cases st ≠ sp and st = sp . 4. Simulation results In this section, two examples are considered to show the efficiency of the proposed approach where the LMI control toolbox [28] in the MATLAB environment was used to compute the solution to the optimization problem and cvx toolbox [29] was implemented to evaluate the maximum region of stability in Lemma 3.
otherwise set count = count + 1 and F ( k ) = F (k − 1) and im-
(
)
eq . plement it in control input as u( k ) = F (k ) x( k ) − x ieq , j − 1 + u i, j − 1
6. If count = 0, set j = j + 1 then find
(x
eq i, j ,
)
by using αi (conuieq ,j
sidering Remark 2) and go to Step 7. else if 0 < count ≤ λ go to Step 3.
(
eq else if count > λ , set αi = 0.5αi and count = 0 then find x ieq , j , u i, j
)
by using αi (considering Remark 2) and go to Step 3. eq eq 7. If i > sp+1, check if x ieq , j is located between x i,0 and x i − 1,0 go to
Step 3 else set j = j − 1 and go to Step 6. sp go to Step 4 else set else if i = sp+1, check if x ieq , j do not pass x
(x
eq i, j ,
) (
sp sp and go to Step 4. uieq ,j = x , u
)
The proposed method in Section 3.1 has been structured as Algorithm 1 described above. The off-line part of this algorithm describes how to build up the off-line sections of the proposed controller (as explained in Section 3.1.1) and the on-line part determines the appropriate on-line strategy which should be implemented at each sample time to guarantee the closed-loop stability (as described in Section 3.1.2). In the proposed algorithm there is a possibility to tune move step αi if it is initialized improperly. In Step 6 it is checked whether infeasibility in Step 4 occurred more than λ times ( count > λ ) which implies that the sequential augmented points are located far from each other or equivalently αi is chosen large. In this case move step αi is substituted by 0.5αi . Then the related augmented point where the corresponding controller gets infeasible is replaced by a closer one that increases the feasibility chance. Therefore, Assumption 2 is reduced to Assumption 3 which is equivalent to Assumption 1. Assumption 3. RMPC related to each augmented equilibrium point has a non-empty region of stability. It should be noted that selecting small λ in Step 6 causes in multiple dividing of αi and decreasing its value unnecessarily while choosing the large value for λ allows occurrence of several infeasibilities in Step 4 in the case where initial αi is improperly selected.
4.1. Example 1. Consider a two tank system in [13] with the following dynamics
⎧ ρS ḣ = − ρA 2gh + u, ⎪ 1 1 1 1 ⎨ ̇ = ρA 2gh − ρA 2gh , ⎪ ρ S h ⎩ 2 2 1 1 2 2
(19)
where ρ = 0. 001 kg/cm3, g = 980 cm/s2, A2 = 4 cm2, A1 = 9 cm2 ,
S1 = 2500 cm2 , S2 = 1600 cm2, 1 ≤h1 ≤50 cm , 10 ≤h2 ≤120 cm , and 0 ≤u ≤2. 5 kg/s. The equilibrium surface and its related inputs are given in (20) and (21) respectively which are obtained from the steady state solution of (19). These relations are employed to find the augmented equilibria in Step 6 in Algorithm 1 as explained in Remark 2.
h1eq −
A22 A12
h2eq = 0,
ueq = ρA1 2gh1 .
(20) (21)
To design a controller based on Algorithm 1 to track any set point on equilibrium curve in (20) between a = [19 96.1875]T and
b = [8 40.5]T , two path regarding (20) are considered, Path #1 in direction of increasing h2 and Path #2 in the opposite direction. For each path a set of RMPCs is designed according to mentioned steps T in the off-line part of Algorithm 1. Let x = ⎡⎣ h1 h2 ⎤⎦ . The chosen
middle equilibria regarding Path #1 and Path #2 and their corresponding move steps regarding the first state are listed in Table 1. The estimated regions of stability are shown in Fig. 2(a) and (b). The sampling time is 1 s and the control parameters are set as R = 0.1, Q = I , W = diag(⎣⎡ 0.001 0.001 0 ⎦⎤), ε = 0.01, and λ = 5. In this study, the control act starts from x( 0) = [10 20 ]T and first set point is set as x sp1 = [11 56.6875]T . There is a set point change from x sp1 to x sp2 = [ 9 45.5625]T at t ¼ 500 s and then from x sp2 to
x ieq ,0
x sp3 = [8 91.125]T at t ¼ 1000 s. The state trajectory of the control system is shown in Fig. 2(c).
and x ieq ” in Step 7 implies that in each region i , the associated − 1,0
At t = 0 , the set point, x sp1, is upper than x( 0) therefore, the
augmented equilibrium x ieq , j should not exceed the next middle
controllers in Path #1 are implemented where st = 3 and sp = 2.
Remark 5. The mentioned condition: “if
x ieq ,j
is located between
equilibrium x ieq . − 1,0
At t ¼ 500 s, the set point changes to x sp2 while x sp1 is an initial state for this new tracking action ( x( 0) = x sp1). Now set point is
Remark 6. In the extended scheduled RMPC, x(k )∈ i could exceed the predefined stability region but the condition x ieq , j ∈ i −1 ensures that the trajectory returns back to this region.
located under initial state and it means Path #2 should be chosen but st = sp = 2. Therefore, according to on-line part of Algorithm 1, the opposite path (Path #1) and its related controllers are implemented until x(k ) gets out 2 in Path #2 as it is obvious in Fig. 2 (c). Then the related controllers in Path #2 are used which results in convergence to x sp2 . At t ¼ 1000 s, controllers in Path #2 are used because of the location of set point ( x sp3) and initial state
Theorem 2. Consider nonlinear system in (1) along with one of its equilibrium curves. The corresponding set point tracking RMPC which is designed based on Algorithm 1 has an estimated region of stability q q . defined by ⋃i =1 0 i ∩ ⋃i =2 0 i
( x( 0) = x sp2). The time responses of state and control actions are
Proof. The proof could be easily obtained by considering the
shown in Fig. 3.
(
)
Path #1
(
)
Path #2
Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i
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specified range on the equilibrium curve. Moreover, the proposed algorithm could provide an admissible transient performance in spite of switching between the local controllers because of using the augmented equilibria which could be constructed on-line without needing to save them in the look-up table.
4.2. Example 2. Consider the following CSTR system [17]
⎧ − E q UA ∆H ⎪ Ṫ = Tf − T − k 0e RT CA + ( Tc − T ), ⎪ V VρCp ρ Cp ⎨ ⎪ − E q CAf − CA − k 0e RT CA, ⎪ CȦ = ⎩ V
(
)
(
)
(22)
where T is the reactor temperature, CA is the concentration of substance A in the reactor, and Tc is the temperature of the coolant stream. The parameters are q = 100 l/min , V = 100 l, CAf = 1 mol/l,
Tf = 400 K ,
ρ = 103 g/l,
Cp = 1 J/( g K),
k0 = 4.71×108 min−1,
E /R = 8000 K , UA = 105 J/( min K), ∆H = − 2×105 J/( min K), 0 ≤CA ≤1 mol/l, 250 K ≤T ≤ 500 K , and 250 K ≤ Tc≤ 500 K . The equilibrium curve of the system in (22) and its related input are given by (23) and (24). These equations can be employed to construct the augmented equilibria in Algorithm 1 by using move step regarding one of the states.
T eq = −
E R
⎛ q ⎞ ln ⎜ C − CAeq ⎟ ⎝ k 0VC Aeq Af ⎠
(
Tceq =
,
)
(23)
⎞ VρCp ⎛ q ∆H − E ⎜⎜ − Tf − T eq + k 0e RT eq CAeq⎟⎟ + T eq. UA ⎝ V ρ Cp ⎠
(24)
(
)
9
To design a controller based on Algorithm 1 to track each set point on the equilibrium curve in (23) between
a = [ 371.1264 0.83]T and b = [ 423.9139 0.25]T , Path #1 in direction of increasing CA and Path #2 in the opposite direction are T assumed. Consider x = ⎡⎣ T CA ⎤⎦ and u = Tc . The selected middle
equilibria for each path and move steps regarding CA are given in Table 2. Two paths and their related RMPC stability regions are shown in Fig. 4(a) and (b). In this example a series of set point are selected as x sp1 = [ 398.1979 0.53]T , x sp2 = [ 374.5904 0.8]T , and x sp3= [ 423.9139 0.25]T where changes happen at multiple of 3.3 min. The tracking procedure starts from x( 0) = [ 410 0.1] which locates lower than x sp1 therefore, Path #1 is selected for the first tracking attempt. For the second and third set points, Path #1 and Path #2 are chosen respectively. State trajectory for this tracking problem is shown in Fig. 4(c). The time response of the states and control action are depicted in Fig. 5 which shows that the closed-loop system could track the set point changes efficiently. Moreover, the control signal jumps just once the set point changes occur and the proposed strategy could smooth the control action during switches between local controllers. In the simulations of this example the sample time was 0.33 min and the controller parameters were chosen as Q = diag([1/400 1/0.5]), R = 0.1×1/300, W = diag(⎡⎣ 0.001 0.01 0 ⎤⎦), ε = 0.01, and λ = 5 for all regions. 5. Conclusion In this paper, unlike the most existing RMPC algorithms, a scheduled RMPC scheme is introduced for the set point tracking problem. The proposed algorithm consists of the off-line and online parts. In the off-line part of the proposed approach some equilibrium points are determined based on stability regions of their corresponding RMPCs. These regions should have proper overlaps. Then the on-line part of the algorithm uses the information prepared in the off-line part and implements a novel switching strategy which guarantees tracking any set point in the
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Please cite this article as: Khani F, Haeri M. Constrained tracking control for nonlinear systems. ISA Transactions (2017), http://dx.doi. org/10.1016/j.isatra.2017.06.004i