ACTUATOR FAULT TOLERANCE EVALUATION OF ...

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Carlos Ocampo-Martínez*, Pedro Guerra*, Vicenç Puig* .... set of compatible predicted states at time k is given by ..... Kieffer, M., L. Jaulin and E. Walter (2002).
ACTUATOR FAULT TOLERANCE EVALUATION OF LINEAR CONSTRAINED MPC Carlos Ocampo-Martínez*, Pedro Guerra*, Vicenç Puig* Marcin Witczak**

*Automatic Control Department Universitat Politècnica de Catalunya (UPC) Rambla de Sant Nebridi, 10 08222 Terrassa (Spain) Email: [email protected] **Institute of Control and Computation Engineering University of Zielona Góra (Poland)

Abstract: This paper presents a computational procedure to evaluate the admissibility of actuator fault configurations when Linear Constrained Model Predictive Control (LCMPC) is used. The admissible solution set for the predictive/optimal control problem including the effect of faults (either through reconfiguration or accommodation) is determined using a zonotope-based algorithm. Finally, the proposed method is tested on a real application consisting on a piece of the Barcelona sewer network. Keywords: Fault tolerance, model predictive control, zonotopes, set computation, sewer networks.

1. INTRODUCTION

lutions set that satisfies the constraints (feasible solutions) and second, the optimal solution determination.

The problem of admissibility evaluation in order to evaluate the tolerance of a certain actuator fault configuration considering a linear predictive/optimal control law with constraints is studied in this paper. This problem has been already treated in the literature for the case of LQR problem without constraints (Staroswiecki, 2003), due to the existence of analytical solution. However, constraints (on states and control signals) are always present in real industrial control problems and could be easily handled using Linear Constrained Model Predictive Control (LCMPC). But in general, an analytical solution for these kind of control laws does not exist, which makes difficult to do this type of analysis. The method proposed in this paper is not of analytical but of computational nature. It follows the idea proposed by (Lydoire and Poignet, 2004) in which the calculation of the control law for a predictive/optimal controller with constraints can be divided in two steps: first, the calculation of so-

Faults in actuators will cause changes in the set of feasible solutions since constraints on the control signals have varied. This causes that the set of admissible solutions for the control objective could be empty. Therefore, the admissibility of the control law facing the actuator faults can be determined knowing the feasible solutions set. One of the aims of this paper is to provide methods to compute this set and then evaluate the admissibility of the control law. The feasible solutions set for the problem of LCMPC is equivalent to the compatible set of inputs and states (Kerrigan, 2000). To find this set, a constraints satisfaction problem could be formulated (Lydoire and Poignet, 2004). However, this problem is computationally demanding and should be solved approximately in a iterative way in time bounding set by its interval hull. Nevertheless, when it is solved in this way, an interval simulation problem is implicitly

solved appearing typical difficulties associated with it (as wrapping effect, among others) (Puig et al., 2003). In order to avoid such problems, the region of possible states should be approximated using more complex domains than intervals, such as subpavings (Kieffer et al., 2002), ellipsoids (Polyak et al., 2004), zonotopes (Kühn, 1998; Alamo et al., 2005), among others. In this paper, a zonotope-based method to evaluate the admissibility of fault actuator configurations is proposed and discussed. The paper structure is the following: in Section 2, fault tolerant linear predictive control concepts and definitions are presented. In Section 3, the generic method for admissibility evaluation based on set computations is presented and discussed. Section 4 describes its implementation using zonotope-based set computations. Section 6 exposes a case of study based on a real application of linear predictive control over a sewage system. This particular system is used to show the effectiveness of the proposed method. Finally, in Section 7, the conclusions and further work are presented.

2. FAULT TOLERANT LCMPC PROBLEM

The solution of a control problem consists on finding a control law in a given set of control laws U such that the controlled system achieves the control objectives O while its behavior satisfies a set of constraints C. Thus, the solution of the problem is completely defined by the triple hO, C, Ui; see (Blanke et al., 2003). Let us denote the sequence variables over the time horizon N N x ˜ = (xk )0 , = (x0 , x1 , ..., xN ), N −1

= (u0 , u1 , ..., uN −1 ).

u ˜

(1)

subject to

  xk+1 = Axk + Buk k ∈ [0, N − 1] ⊂ N C : uk ∈ U  xk ∈ X k ∈ [0, N ] ⊂ N

2.2 Preliminary definitions Definition 2.1. (Feasible solution set). The feasible solution set is given by n o N −1 Ω= x ˜, u ˜ | (xk+1 = Axk + Buk )0 The subset Ω gives the input and state sets compatible with system constraints which originate the set of predictive states. Definition 2.2. (Feasible control objectives set). The feasible control objectives set is given by

and corresponds to the set of all values of J obtained from feasible solutions. In the case of a fault, Ω converts on Ωf and JΩ converts on JΩf . Definition 2.3. (Admissible solution set). Given the following subsets: Ωf (defined as the feasible solution set of a actuator fault configuration) and JA , defined as the admissible control objective set. The solution admissible set is given by A = {˜ x, u ˜ ∈ Ωf | J(˜ x, u ˜ ) ∈ JA }

Thus, in the case of a linear constrained predictive control law, the triple hO, C, Ui is defined by O : min J(˜ x, u ˜)

i=0

where φ is a function that constraints the final state value over N and Φ is a function of states and inputs.

JΩ = {J(˜ x, u ˜) ∈ R | (˜ x, u ˜) ∈ Ω}

2.1 Formulation of the control problem

u ˜ = (uk )0

the previous optimization problem. The initial states x0 are updated from measurements or state estimation. The objective function J is defined, in general form, as N −1 X J(˜ x, u ˜) = φ(xN ) + Φ(xi , ui ) (5)

(2)

and corresponds to the feasible solution subset that produces control objectives in JA . If A = ∅, then the considered actuator fault configuration is not tolerant. Definition 2.4. (Predicted states set). Given the set of states at time k-1, the set of predicted states at time k is defined as Xkp = {xk = Axk−1 + Buk−1 | xk−1 ∈ Xk−1 ,

where

uk−1 ∈ U } ∆

(3)



(4)

U = {u ∈ Rm | umin ≤ u ≤ umax } X = {x ∈ Rn | xmin ≤ x ≤ xmax } ,

and N denotes the time horizon. The control law belongs to the set U and it is obtained using the receding horizon philosophy (Maciejowski, 2002). This technique consists on taking only the first value from the sequence u ˜ computed at each time instant by solving

and corresponds to the set of states at time k originated by the system dynamic. Definition 2.5. (Compatible predicted states set). The set of compatible predicted states at time k is given by Xkc = {xk | xk ∈ Xkp ∩ X } and corresponds to the set of predicted states compatible with system constraints.

Definition 4.2. (Zonotope). Given a vector p ∈ Rn and a matrix H ∈ Rn×m , the set represented as:

Definition 2.6. (Compatible inputs set). The set of compatible inputs at time k-1 is given by c Uk−1 = {uk−1 ∈ U | (xk = Axk−1 + Buk−1 ) ∈ Xkc , ª c xk−1 ∈ Xk−1

and corresponds to the set of inputs that produces the compatible predicted states.

3. ADMISSIBILITY EVALUATION USING SET COMPUTATION 3.1 Admissibility Evaluation and Set Computation The admissibility evaluation using a set computation approach starts obtaining the feasible solution set Ω given a set of initial states X0 ⊆ X , the system dynamic and the system operating constraints over N . This procedure is described in the Algorithm 1. Algorithm 1 Computation of Ω 1: Xk ⇐ X0 2: Ω0 ⇐ X0 3: for k = 1 to N do 4: Uk−1 ⇐ U 5: Compute Xkp from Xk−1 and Uk−1 6: Compute Xkc = X ∩ Xkp c 7: Compute Uk−1 from Xkc c c 8: Ωk = Xk × Uk−1 Xk ⇐ Xkc 9: 10: end for N S 11: Ω = Ωk

X = p ⊕ HBm = {p + Hz : z ∈ Bm } is called a zonotope of order m and corresponds to the Minkowski sum of the segments defined by the columns of matrix H. In this expression, Bm is an unitary box composed by m unitary intervals.

4.2 Computation of Xkp The computation of the prediction set Xkp can be viewed as the direct image evaluation of AXkc + BUkc . This image can be bounded using zonotopes as follows. Given two zonotopes Xkc = pxk ⊕ Hxk Br1 and Ukc = puk ⊕ Huk Br2 , then e Xk+1 = pk+1 ⊕ Hk+1 Brn

(7)

pk+1 = Apxk + Bpuk

(8)

Hk+1 = [AHxk BHuk ]

(9)

where:

It is important to notice that this method of computing the set of predicted states increases the number of e segments generating the zonotope Xk+1 . In order to control the domain complexity, a reduction step is thus implemented. The method proposed in (Combastel, 2003) can be used to reduce the zonotope complexity.

k=0

At the same time that Ω is computed, the feasible control objectives set (Definition 2.2) can be obtained. Given a fault actuator configuration, JΩf can be obtained. Thus, if some admissible control objective set JA is given, the admissibility of that fault actuator configuration could be determined computing the solution admissible set A as A = JΩf ∩ JA

(6)

If the set A = ∅, the fault actuator configuration is not admissible. Otherwise, that configuration would have a certain admissibility degree according to the system designer.

4. IMPLEMENTATION USING ZONOTOPES 4.1 Background on zonotopes Definition 4.1. (Minkowski Sum). The Minkowski sum of two sets X and Y is defined as X ⊕ Y = {x+y : x ∈ X , y ∈ Y}

4.3 Computation of Xkc In the intersection set step, it is required characterizing the set Xkc . This set is the intersection of two sets: the previously bounded set Xkp and the set that restricts the states for all k over N (see Equation (4)). Property 4.1. Given two zonotopes X1 = p1 ⊕H1 Br1 and X2 = p2 ⊕ H2 Br2 and matrix E, let us define: b pb(E) = Ep1 + (I − E)p2 and H(E) = [EH1 (I − b E)H2 ]. Then, X1 ∩ X2 ⊆ X (E) and Xb(E) = pb(E) ⊕ r1 +r2 b H(E)B To reduce the size of the intersection zonotope Xb(E), a convex optimization problem is solved. If H1i and H2j (with i=1,· · · ,m1 , j=1,· · · ,m2 ) are the columns of matrices H 1 and H2 , the function to be minimized is: f (E) =

m1 X

(EH1i )T (EH1i )

i=1

+

m2 X j=1

(H2j − EH2j )T (H2j − EH2j ) (10)

·

c 4.4 Computation of Uk−1 c Uk−1 1

The set can be computed from the Minkowski c difference between the zonotopes Xkc and Xk−1 considering that the matrix B has full row rank (Mayne and Schroeder, 1997). This difference can be expressed as: c c Uk−1 = B + (Xkc ∼ AXk−1 )

(11)

where B + is the pseudoinverse of matrix B. In the case where matrix B has not full row rank, its singular value decomposition (SVD) should be used to compute the matrix B + (Theilliol et al., 2002) · B+ = Φ

¸ Σ−1 0 ΨT 0 0

(12)

¸ · ¸ 11 0 xk+1 = x + u 01 k 1 k £ ¤ yk = 1 0 xk

(15) (16)

with the following constraints for states and control signals: x1 ∈ [−15, 15], x2 ∈ [−6, 6] and u ∈ [−1, 1]. A MPC controller is used to control this system satisfying the associated state and control constraints using the following objective function:

J = xTN P xN +

N −1 X

¡

xTi Qxi + Rx2i

¢

(17)

i=0

To calculate this difference, the interval hull of these two zonotopes is used. The smallest centered interval vector containing the given zonotope X = p ⊕ HBr is called the interval hull and can be computed as: Xb = p ⊕ rs(H)Br where: rs(H)ii =

p X

|Hij |

(13)

(14)

j=1

where rs stands for "row sum" and rs(H) is a diagonal matrix. Then applying interval arithmetic (Moore, 1966) between the two interval enclosures of both zonotopes the difference is obtained.

5. MOTIVATING EXAMPLE First of all, an example to motivate the usefulness and interest of the proposed method is presented. The presence of contraints in MPC makes very difficult to proceed with the fault-tolerance analysis as proposed by (Staroswiecki, 2003). There, the analysis is possible because the expression of how fault affects the objective function is available using LQR theory. However, in constrained MPC this expression is not available, although an explicit expression for the controller could be derived(Bemporad et al., 2002). This motivates the usefulness of the proposed approach.

5.1 Description Let us consider the double integrator system proposed by (Bemporad et al., 2002), whose equivalent discretetime state-space description using the Euler discretisation rule is 1

The Minkowski (or Pontryagin) difference between two given sets Π, Ψ ⊂ Rn is defined as Π ∼ Ψ , {π ∈ Rn | π + ψ ∈ Π, ∀ψ ∈ Ψ}

where N = 2, terminal weight P determined using Ricatti equation with Q = diag([1 0]) and R = 0.01. According to (Bemporad et al., 2002), a statefeedback explicit control law for the MPC controller, piece-wise affine with respect to the states, can be derived: uk = KP W A (xk )xk . Using the MATLAB Hybrid Toolbox (Bemporad, 2003), the expression of KP W A for the proposed example can be determined and represented graphically (see Figure 1). Using this law, the closed-loop state trajectory can be computed and represented (see again Figure 1). Notice that depending on the region of the state space, a different gain for the state feedback is applied. Using the method proposed in this paper, the feasible sets for states x1 and x2 are computed and represented in Figure 2 and 3 (continuous line), respectively. It can be noticed that the closed-loop state trajectories applying the MPC controller are inside the corresponding feasible sets as expected. Now, a fault in the actuator is introduced. This fault corresponds to a reduction of the operating range of the actuator such that u ∈ [−0.75, 0.75]. Computing in this situation the feasible set for states x1 and x2 using again the method proposed, it can be noticed that the closed-loop state trajectories applying the MPC controller in the nonfaulty situation (Figure 2 and 3 (continuous line with stars)) are outside the corresponding feasibles sets for the faulty situation (Figure 2 and 3 (continuous line with circles)). This means that the performance of the MPC controller will be worse that in the case of the non-faulty actuator since MPC trajectories for the faulty situation are not reachable. This means that if MPC trajectories in the non-faulty situation were inside the corresponding feasible sets, the performance of the MPC controller would not be affected by the fault, i.e., would be fault-tolerant. This example shows how easily can be evaluated the tolerance of a control law with respect to a fault using the method proposed. Moreover, the degradation in the performance can also be evaluated with this method, as it will be shown in the following application example.

to virtual/real tank volumes and three input signals corresponding to the control inflows to the command gates (Ocampo et al., 2005). The system constraints are:

Polyhedral partition − 7 regions 6 1 2 3 4 5 6 7

4

θ2

2

• Bounding constraints: refers to physical restrictions.

0

−2

x1 (k) ∈ [0, +∞] x2 (k) ∈ [0, 35000] x3 (k) ∈ [0, +∞]

−4

−6 −15

−10

−5

0 θ1

5

10

15

Figure 1. Explicit MPC law and closed-loop trajectory

10

x1

5

−5

−10

−15 5

10

15

20 k

25

30

35

40

Figure 2. Feasible set corresponding to x1 6

4

2

0

−2

−4

−6 0

5

10 k

15

(20)

where qxi (k) = βi xi (k) (Ocampo et al., 2005). For this application, it is supposed that vector dk (rain) is known at each time instant k, what means known perturbation. This causes that the obtained results have only an interest of design of the tolerant control system. It is desired to evaluate the admissibility of different actuator fault configurations not only in reconfiguration but also in accommodation. Configuration admissibility is defined from a control objectives degradation with respect to nominal (without fault) configuration for a given rain scenario. The selected rain scenario was created from real rain gauge data obtained within the city of Barcelona on 14 September, 1999. This day, severe flooding occurred as a consequence of the rain storm. The actuator faults are no simultaneous and they are present from the beginning of the scenario. Their models are described as change of operating limits (accommodation) or operative annulation (reconfiguration).

0

0

(19)

• Mass conservation constraints d1 (k) = u1 (k) + Q1 (k) qx1 (k) = u2 (k) + Q2 (k) qx2 (k) ≥ u3 (k)

15

x2

u1 (k) ∈ [0, 11] u2 (k) ∈ [0, 25] u3 (k) ∈ [0, 7]

20

Figure 3. Feasible set corresponding to x2 6. CASE OF STUDY

6.2 Control Objective and Admissibility Criterion

6.1 System Description Consider the system corresponding to a piece of Barcelona sewer network described by the discretetime state equations xk+1 = Axk + Buk + Bp dk

(18)

where

The control objective is defined as pollution (water volume that goes to the sea through collector F sea). From system variables, the constraints that defines the control objective are given by: J = Vsea = ∆t

N X

Fsea (k)

(21)

k=0

 A=





1 − ∆tβ1 0 0  0 1 0 ∆tβ1 0 1 − ∆tβ3

 1 0 0 B = ∆t  0 1 −1  −1 −1 1



 0 α2 0 Bp = ∆t  0 0 0  1 0 α3

with sampling time ∆t = 300s and system parameters α2 = 0.5715, α3 = 0.0783, β1 = 5.8 × 10−4 and β3 = 1.0 × 10−3 which are estimated from real data. The system has three state variables corresponding

where Fsea (k) = max(0, qx3 (k) − Qmax ) is the wa3 ter flow to sea. The admissibility criterion is based on a direct comparison between a resultant minimum final value of min volume from the corresponding envelope Vobj (N ) and the same value for degraded nominal system connom (N ) is done (that is due to pollution figuration Vobj corresponds to the accumulated masses over´a given ³ nom min (N ) where (N ) = ψ Vobj scenario). That is Vobj ψ is the relation of degradation and subscript obj denotes the control objective. For this application, if nom min (N ), then the evaluated system (N ) > 8Vobj Vobj configuration is not admissible. In the reconfiguration

case, actuators are considered completely closed or completely open due to the fault, what would change the admissibility of the obtained actuator fault configurations. Table 1 resumes the possible fault cases and their admissibility. Only faults described by actuators completely closed are simulated, that is ui ∈ [0, 0] and Qi ∈ [0, +∞]. On the other hand, in reconfiguration case that faults produces the reduction of the actuators operating range (for example from 0-100% to 0-50%). Thus, the number of fault admissible configurations varies as it is shown in Table 2, where if the admissibility criterion is maintained as in the reconfiguration case, more configurations could be admitted. Two accommodation ranges are presented. This table does not consider accommodation for u3 due to the system insensibility to this actuator shown in Table 1. Table 1. Admissibility of fault configurations for pollution - Reconfiguration Fault Location

Min. Volume [m3 ]

Admissibility Status

No fault Fault in u1 Fault in u2 Fault in u3

1050 8800 52200 1050

— No Admissible No Admissible Admissible

Table 2. Admissibility of fault configurations - Accommodation Fault Location

Operation range

Min. Volume [m3 ]

Admissibility Status

No fault Fault in U1 Fault in U1 Fault in U2 Fault in U2

— 0-20% 0-50% 0-20% 0-50%

1050 5200 2300 34000 15700

— Admissible Admissible No Admissible No Admissible

7. CONCLUSIONS AND FURTHER WORK This paper proposes a method for admissibility evaluation of fault configurations using zonotope-based set computations. This procedure implies the propagation of the feasible solution set at each time instant taking into account the physical system constraints. The results provide the limits of system performance considering all the feasible solutions and how they are degraded after fault occurrence. This allows to evaluate the admissibility of a given actuator fault configuration using a degradation criterion established beforehand. The proposed method has been successfully applied on a linear predictive control system of the Barcelona sewer network. As further work, this approach will be extended to the corresponding nonlinear model, the study of actuator fault sequences will be considered as well and the inclusion of uncertainty in the model and/or in the perturbations (rain) will be also considered.

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