Quasi-LPV Model Predictive Recongurable Control for Constrained Nonlinear Systems

Quasi-LPV Model Predictive Reconfigurable Control for Constrained Nonlinear Systems Lamia Ben Hamouda1,2 , Ouadie Bennouna2 , Mounir Ayadi1 and Nicolas Langlois2 Abstract— This paper deals with the control laws reconfiguration of nonlinear systems, by using a Fuzzy-Model-based Predictive Control (FMPC). It should be noted that the studied systems are written in the quasi-linear parametric varying (quasiLPV) form. This FMPC strategy is developed to preserve closed-loop stability in the nominal and actuator faulty case. Fault accommodation by perturbations rejection is presented. This step is done by interpolation-based control to cover the entire area of operation. To allow the process to maintain current performances closed to desired performances, a dynamic optimizer is used. Our contribution comes from the combination of several aspects: fuzzy model, quadratic programming and faults decoupling principle. The linearization around a family of equilibrium points is also studied. The operating points are appropriately configured by a set of variables called premise. Index Terms— nonlinear system, fuzzy model, quasi-linear parametric varying system, accommodation, quadratic programming, model predictive control.

I. INTRODUCTION The T-S fuzzy modelling is based on the decomposition of the nonlinear system dynamic behavior around several operational areas. According to the area where the system is, each sub-model contributes more or less to approximate the overall system behavior, [1], [2] and [3]. Inside an operation area, the system has an homogeneous dynamic behavior. The muli-model (MM) is based on the T-S fuzzy model, known for its universal approximator properties, (it is a polytopic representation). The T-S fuzzy model is based on rules such as, [4]: IF PREMISE THEN CONSEQUENCE. Premises are obtained from linguistic propositions. They allow the evaluation of weighting functions and the consequences correspond to sub-models. A nonlinear system may also be included under the quasi-LPV system definition. These arguments are motivating to choose an approach which preserves the nonlinear modeling. Thus, the study and the analysis of global quasi-LPV modeling (non-stationary linearization) is presented. Afterwards comes the construction 1 Universit´ e Tunis El Manar, Ecole Nationale d’Ing´ enieurs de Tunis, Laboratoire de Recherche en Automatique, BP 37 le Belv´ ed` ere, Tunisie 2 Institut de Recherche en Syst` emes Electroniques Embarqu´ es, Universit´ e de Rouen, BP 10024 Avenue Galil´ ee, France

[email protected], [email protected], [email protected] and [email protected]

of non-stationary linear controllers set, satisfying properties of the specification for each equilibrium point. The aim is to configure a quasi-LPV controller for each sub-model to obtain then a nonlinear controller. For such local linear model an eigenvalue assignment (linear feedback) regulator is designed in order to obtain the desired dynamic of the closed loop system. The principle object of control system is to track the reference and regulate state and output system in the nominal and faulty case. This paper relies on the idea of tolerating various faults to maintain current performances closed to desired performances. The nonlinear predictive controller represents accommodation based on faults decoupling principle. The layout of this paper is as follows: In Section 2, a global non stationary linearization method is given. Then, a FMPC formulations for reconfigurable control is proposed in Section 3. Section 4 shows the simulation results while Section 5 concludes the paper. II. Global non stationary linearization The considered system is based on an interpolation between the local linear models. Suppose a set of N local models describing the dynamic behavior of the nonlinear system in different operation areas [5]. Activation function (weighting) µj (x(t), u(t)) is normalized; it determines the activation degree of the j th associated local model, by providing a gradual transition from this model to local model neighbors. These functions are generally triangular shaped, sigmoidal or Gaussian and satisfy the following properties: N ∑

µj (x(t), u(t)) = 1 and 0 ≤ µj (x(t), u(t)) ≤ 1

(1)

j=1

where x ∈ ℜn stands for the state and u ∈ ℜm denotes the control input. As assumed above, the dynamic behavior of the nonlinear system is characterized by fuzzy model representation, composed of N linear or affine local models. Each local model is obtained by applying stationary linearization, around equilibrium points. Let f and g two nonlinear functions and y ∈ ℜp the output such as, the system state space representation is: { x(t) ˙ = f (x(t), u(t)) (2) y(t) = g(x(t)) A. The quasi-LPV system First, the aim is to transform nonlinear system described by (2) to quasi-LPV system with polytopic form

described by the following: { x(t) ˙ = A(θ)x(t) + B(θ)u(t) y(t) = C(θ)x(t)

where the operation point belongs to the set Γ of equilibrium points defined by: { } Γ = (x, u) ∈ ℜn×p |f (x, u) = 0, g(x) = 0 (7)

(3)

where A ∈ ℜn×n , B ∈ ℜn×m and C ∈ ℜp×n are the state space matrices with variable parameters. with A(θ) = ∇x f (θ(x, u)), B(θ) = ∇u f (θ(x, u)), C(θ) = ∇x g(θ(x, u)) and θ the premise variables around the equilibrium point (x0 , u0 ). For quasi-LPV system, varying parameters of the linear system are known at the current time, but unknown in the future. Then, transform it in the fuzzy model form, which is given by this state space representation:  ∑N   x(t) ˙ = µj (θ) (Aj x(t) + Bj u(t)) j=1 (4) ∑ N   y(t) = µj (θ)Cj x(t)

Equilibrium points of nonlinear system are set by a vector of premise variables, [8]. Equations (5) and (6) give the quasi-LPV system described by (3). •

2nd step: Polytopic convex transformation To obtain a fuzzy model formed by two sets of subLTI model representing the lower and upper bounds, [6] and [9], the polytopic convex transformation is used as described in the following. Lemme 2.1: Consider a function θ(x, u) continuous and bounded on the domain D ⊂ ℜn × ℜm with values in ℜ, where x ∈ ℜn , u ∈ ℜm . Then there exists two functions (i = 1, 2)

j=1

µi : D 7→ [0, 1]

All three Aj , Bj and Cj are constant matrices. The structure fuzzy model is described by the weighting functions µj (θ), [6]. Indeed, the j th linear model describes the system dynamic around the j th operating point. Indeed many studies exist in this attractive area, where researchers deal with various mathematical representation of nonlinear system. It can be written in a quasi-LPV form under the polytopic representation ∑N A0 + j=1 µj (t)Aj used by [7].

(x, u) 7→ µi (x, u)) with µ1 (x, u) + µ2 (x, u) = 1, such as: θ(x, u) = µ1 (x, u)θ¯ + µ2 (x, u)θ− θ¯ = max {θ(x, u)} and − θ = min {θ(x, u)} , x,u∈D

x,u∈D

functions µ1 and µ2 are defined by:

B. Algorithm 1: non stationary linearization µ1 (θ) =

Polytope is obtained with N = 2r peaks, where r is the number of premise variables. This polytopic form is a generalization of affine systems. Variations of the vector θ is represented by a set of M th peak matrices which define the polytope. Under the condition that the matrix M is considered as a matrix of sub-models. Relation with fuzzy N ∑ model structure is obvious, where : M (θ) = µj Mj ,

θ(x, u) − − θ θ¯ − − θ

θ¯ − θ(x, u) θ¯ − − θ Remember that the objective is to obtain a decomposition which is not unique for two equilibrium points. µ2 (θ) = 1 − µ1 (θ) =

j=1

assuming θmin is the lower bound of θ(x, u) and θmax is the upper one. The µj (θ) is the interpolation variable according to θ which depends on system variables (state, input, ...); µj (θ) can be considered as a time-dependent parameter. •

1st step: Nonlinear to quasi-LPV model transformation The Taylor series expansion linearization (limited to one order) of (2) around the operating point (x0 , u0 ) gives the following gap system: { δ x(t) ˙ = ∇x f (x0 , u0 )δx(t) + ∇u f (x0 , u0 )δu(t) δy(t) = ∇x g(x0 , u0 )δx(t) (5) with    δx(t) = x(t) − x0 , δu(t) = u(t) − u0 (6)   δy(t) = y(t) − y0

•

3rd step: Choose the θj corresponding to the intervals premise variables and then rewrite Aj (θ), Bj (θ), Cj (θ) and µj (θ) to obtain the Fuzzy Model. III. Fuzzy-based Model Predictive Reconfigurable Control formulations

Model Predictive Control (MPC) requires an optimization at every sampling instance to reach desired set points and subject to constraints on control input and system state. The main objective of control system is to accommodate faults and changes in underlying system dynamics while achieving tracking and regulation of performance variable. Without any specific knowledge of faults, the MPC is able to accommodate, by learning and prediction, faults that have not been anticipated. The ability to incorporate input and state constraints directly in the control algorithm is a unique characteristic of MPC, [10], [11] and [12].

A. MPC-based strategies reconfiguration The principle of MPC is based on the repetitive minimization of cost function applied to (3): J(k) =

Hp ∑

∥ x(k + l) − xd (k + l |k ) ∥2Q

l=1

+

H∑ u −1

∥ u(k + l |k )

(8) ∥2R

l=0

subject to contraints x(k + 1) = Aj x(k) + Bj u(k), umin 6 u(l) 6 umax , ∆umin 6 ∆u(l) 6 ∆umax , where k 6 l 6 k + Hu − 1 and ∆u = u(k) − u(k − 1), xmin 6 x(l) 6 xmax , where k + 1 6 l 6 k + Hp . where xd is the state target trajectory, Q and R are weights independent of time k and Hp and Hu are prediction and control horizons, respectively. The MPC algorithm drives the predicted state over the prediction horizon, towards the target trajectory and yields a sequence of future control inputs. It is also assumed that the dynamic system defined by the model (Aj , Bj ) is controllable. The controllability condition is required to ensure that the FMPC optimization solved at each step is feasible in the nominal case. This optimization can be formulated as a quadratic programming (QP) problem. To ensure a well-posed optimization problem, the constraints defined on control inputs and states must be consistent and convex. B. Stabilized predictions An application of the MPC to an unstable system can lead to severe numerical problems in evaluating the prediction equations. As demonstrated by [13]. An effective way to prevent this case is to pre-stabilize the prediction equations. The standard MPC prediction equation assumes that the system is operating in openloop and control signals are computed as deviations from a nominal value. A straightforward strategy is to assume state feedback as a baseline controller to which MPC control signals are added. The pre-stabilization is also an efficient tool to guarantee nominal closed-loop stability using the MPC controller, [14], [15] and [16]. Pre-stabilization also reduces conservation in optimal control moves by counteracting to the adverse effects of disturbances and model uncertainties, [17]. Assuming a base-line stabilizing state feedback controller, consider N state feedback gains ku1 , ku2 , . . . , kuN as    −kuj x(k + i |k ) + qij , j uF M P C (k + i |k ) = i = 0, . . . , Hu − 1 (9)   −kuj x(k + i |k ), i > Hu where j = 1, ..., N , qj the j th predicted control input calculated by the FMPC and ujF M P C is the control law

signal generated by the j th controller. As the baseline feedback controller remains active after Hu , it must also satisfy related control constraints. Hence, the FMPC control moves must bring the state at the end of horizon within a terminal region, such that the baseline control law remains valid thereafter. To maximize this terminal set, the feedback gain can chosen to be one that merely pushes the unstable modes inside the unit circle. The cost of the baseline control is then added to MPC cost ¯ x(k + Hu |k ), where Q ¯ function as x(k + Hu |k )T Q is the terminal penalty matrix computed by solving an appropriate Lyapunov equation. C. The T-S fuzzy-based MPC proposed The fuzzy model approach is based on a set of models j = 1, · · · , N describing the system in different operating conditions (in the nominal and faulty case, ). The j th state representation is described by the follow: { x(k + 1) = Aj x(k) + Bj uj (k)) (10) y(k) = Cj x(k) The strategy is to design a fuzzy model with local controller weighted by activation functions depending on various parameters such as the state and the control vectors. The fuzzy model control law applied to the nonlinear system is as follow: ∑N u(k) = µj (θ)uj (k) (11) j=1

The quasi-LPV controller input is developed to control a nonlinear system via the interpolation control laws uj which are designed from the local controllers around N different operating points. Stability studies on such systems exist in [6], [18] and recently in [19], but the presence of faults is not taken into account. D. Algorithm 2: faults accommodation by perturbations rejection In the faulty case, the nonlinear system described by (2) becomes: { x(k + 1) = f (x(k), u(k)) + F w(k) (12) y(k) = g(x(k)) where w ∈ ℜw the persist disturbances and F represents the additive actuator fault matrix. The j th linear model described by (10) becomes:: { x(k + 1) = Aj x(k) + Bj uj (k) + Fj w(k) (13) y(k) = Cj x(k) The considered faults accommodation method is based on the following basic equation: uj (k) = ujF M P C (k) + ujF (k)

(14)

where ujF M P C the control law defined by (9) and ujF represents additive control compensating faults, [8] and [20]. ujF (k) must solve the following equality: Bj ujF (k) + Fj w(k) = 0, with Bj = Fj

(15)

Fig. 1. Interpolation based control using decoupled multiple model

System dynamic equations, fault information from actuator, at instance k, and constraints are formulated as a quadratic programming (QP) problem, [21]. System (13) with the control law (14) under the condition (15) is used to cancel faults effect in closed loop, [22] and [23], as follows:  x(k + 1) = Aj x(k) + Bj uj (k) + Fj w(k)       = Aj x(k) + Bj (ujF M P C (k) + ujF (k)) + Fj w(k)   = Aj − Bj kuj x(k) + Bj qj (k) + Bj ujF (k) + Fj w(k) {z } | {z } |    =0  =Ajk     = Ajk x(k) + Bj qj (k), where j = 1, ..., N where qj (k) represents the j th predicted control. The schema of the control law based on interpolation apply on the system is shown in Fig. 1 and the interpolation based control is written like following: ∑N u(k) = µj (θ)[ujF M P C (k) + ujF (k)] (16) j=1

IV. Simulation Results Consider the nonlinear system described under the following differential equations form, [24]:    x˙ 1 (t) = −x1 (t) + u(t) x˙ 2 (t) = x1 (t) − |x2 (t)| x2 (t) − 10 (17)   y(t) = x2 (t) From the Taylor series development around the operating point (x10 , x20 , u0 , y0 ), it comes : [ ]  [ ] 1  −1 0  δ x(t) ˙ = δx(t) + δu(t) 1 −2 |x20 | 0 (18)  [ ]  δy(t) = 0 1 δx(t)

where δx1 (t) = x1 (t) − x10 , δx2 (t) = x2 (t) − x20 , δu(t) = u(t) − u0 and δy(t) = y(t) − y0 . The linearized SISO system dynamics can be represented by the following quasi-LPV differential equations: { x(t) ˙ = A(θ)x(t) + Bu(t) (19) y(t) = Cx(t) [ ] [ ] [ ] 1 −1 0 with A(θ) = , B= ,C= 0 1 1 −2θ 0 and θ = |x2 |. The vector θ is supposed to vary arbitrarily in the interval [0 10], [24]. Remind that equilibrium points of nonlinear system are set by a vector of premise variables. In this case, we have only one premise variable, so two sub-models with polytopic form are obtained: M (θ) = ∑2 j=1 µj Mj = µ1 M1 + µ2 M2 , where θ1 = 0.707 and θ2 = 0.35. The fuzzy model state space representation is written as :  ∑2   x(t) ˙ = µj (θ) (Aj x(t) + Bj u(t)) j=1 (20) ∑ 2   y(t) = µj (θ)Cj x(t) j=1

with

[

] [ ] −1 0 −1 0 A1 = , A2 = ; 1 −2θ1 1 −2θ2 [ ]T [ ] B1 = B2 = 1 0 ; C1 = C2 = 0 1 ;

The objective is to design a constrained FMPC for the state regulation problem, with the weighting matrices given in Table I. The Linear-quadratic controller gains [ ] 0.4965 0.1198 k = [k k ] are: k = and k u1 u2 u2 = [u ] u1 0.4965 0.1198 . The tuning parameters used in the FMPC and the persistent actuator fault matrices are given in Table I. The outputs responses and control inputs from initial T condition x0 = (0.32, 0.35) , y0 = 0.35 and u0 = 0, can be seen in the following figures. TABLE I controller tuning parameters Sample time T e Prediction horizon Hp Control horizon Hu Input constraints Output constraints Input weights R Output weights Q Fault matrices

0.5 s 8 Te 6 Te −2 6 uk 6 25 −0.358 6 ∆uk 6 2 −3.02 6 yk 6 3.02 , ∀k > 0 0.1 1 [ ]T F1 = F2 = 1 0

From the results of Fig. 2, the response time at + − 5% with the MPC (16.35 s) is lower than the FMPC one (25.65 s). However, the FMPC response does not include oscillation. It is more stable thanks to the stabilizing effect. The desired trajectory is tracked with acceptable performances and small tracking error with a slight

30

0.5 0

25

−0.5

20

−1

15 −1.5

10 −2

MPC FMPC u1

5 −2.5 −3 0

FMPC 2

Reference MPC FMPC 10

20

30 time(s)

40

50

uFMPC

0 0

10

20

60

25

30 time (s)

40

50

60

Fig. 4. Scenario fault (top), output (middle) and control (bottom) signals vs.time actuator linear-time-varying fault

20

1

15

µ

1

µ

0.9

2

0.8

10

0.7 MPC FMPC u1 FMPC u2 FMPC

0 0

0.6

µ (θ)

5

10

20

30 time (s)

40

50

0.5 0.4

60

0.3 0.2 0.1

Fig. 2. Output (top) and control (bottom) signals vs.time in the nominal operating

0 0

10

20

30 time (s)

40

50

60

14

Fig. 5. Variation of the activation functions according to the premise variable vs.time in the nominal operating

12

10 1

µ

8

2

µ1

0.9

6

0.8 0.7

4

µ (θ)

0.6

2

0 0

0.5 0.4

10

20

30 time (s)

40

50

60 0.3 0.2 0.1

2

0 0

1.5

10

20

30 time (s)

40

50

60

1 0.5 0

Fig. 6. Variation of the activation functions according to the premise variable vs.time: case linked to the first scenario fault

−0.5 −1 −1.5

1

−2

−3 0

10

20

30 time (s)

40

50

µ

1

Reference MPC FMPC

−2.5

µ2

0.9 0.8 60

0.7

µ (θ)

0.6 0.5 0.4

25

0.3 20

0.2 0.1

15

0 0 10 MPC FMPC u1 FMPC

5

u2

0 0

FMPC

10

20

30 time (s)

40

50

60

Fig. 3. Scenario fault (top), output (middle) and control (bottom) signals vs.time in the faulty case with the amplitude w=13 at 40 s 20 18 16 14 12 10 8 6 4 2 0 0

10

20

30 time (s)

40

50

10

20

30 time (s)

40

50

60

3

2

1

0

−1

−2 Reference MPC FMPC

−3 0

60

10

20

30 time (s)

40

50

60

Fig. 7. Variation of the activation functions according to the premise variable vs.time: case linked to the second scenario fault

advantage to the FMPC. The bottom of Fig. 2 shows the control effort produced by the MPC and the FMPC. The control effort is increased, where the fuzzy-based model predictive controller is applied to the nonlinear system. An actuator linear-time-varying fault is generated from t = 25 s, scenarios faults are given by each top of Fig. 3 and 4. The middle of Fig. 3 and 4 demonstrates that the use of algorithm 2 accommodates additive actuator faults by perturbations rejection. The FMPC is also improving the capability of the closed-loop system convergence. It tolerates persistent faults and maintains desired performances. This accommodation describes decoupling principle of disturbances in the faulty case. However, the outputs of Fig. 3 and 4 illustrate the limits of the MPC. Contrary to the FMPC, the output cannot track the desired trajectory. The figures show deteriorated behavior

that the MPC try to reduce but still unable to completely eliminate after t = 50 s. Figure 5, 6 and 7 illustrate the evolution of the activation functions according to the premise variable. Comparing simulations with theoretical results, the fault cancelation formulas (14) and (15) are satisfied. The interpolation based control formulas (11) and (16) are checked with experimental values. For the first faulty case, at time t = 35 s (k = 70), we obtain: u1F M P C = 18, u2F M P C = 13, µ1 = 0.1515, µ2 = 0.8485 and u = 20.7575. From (16), it is possible to check the theoretical and practical results: µ1 u1F + µ2 u2F = u − µ1 u1F M P C − µ2 u2F M P C = 20.7575 − 2.9081 − 10.8494 = 7, with u1F = u2F = 7. From (14), we obtain the following: u1 = 18 + 7 = 25 and u2 = 13 + 7 = 20. Equation (16) is verified like follows: u = µ1 u1 + µ2 u2 = 20.7575. The considered faults accommodation method based on (14) is also verified. The performances of the nonlinear system have been considerably improved by using FMPC. According to results, we conclude that the performances of the FMPC strategy are satisfying and allowing a normal behavior of the system even in the occurrence of actuator faults. The reconfigurable FMPC for nonlinear system preserves stability conditions and relbiability, in the nominal and faulty case. V. CONCLUSIONS The FMPC is proposed to get more exact input control law for nonlinear system. The main contribution of this research is the development of a reconfigurable control for nonlinear processes. The method proposed to obtain a convex hull is a very conservative embedding procedure. We deduce that the FMPC maintains acceptable tracking performances according to an acceptable dynamic. The controller accommodates the faults tested in the simulations for any operating point of the process. For a quasiLPV system with both constraints and disturbances, a design approach of dynamic FMPC was presented. The receding horizon implementation of this FMPC strategy guarantees quadratic convergence of the output to a neighborhood of the family of equilibrium points. ACKNOWLEDGMENT The authors gratefully thank ”R´egion Haute Normandie”, FEDER and Ecole Doctorale Sciences et Techniques de l’Ing´enieur de Tunis for financially supporting this work within the framework of the VIRTUOSE project. References [1] D. Filev, “Fuzzy modeling of complex systems,” International Journal of Approximate Reasoning, vol. 5, no. 3, pp. 281–290, 1991. [2] F.J. Uppal, R.J Patton and M. Witczak, “A neuro-fuzzy multiple-model observer approach to robust fault diagnosis based on the Damadics benchmark problem,” Control Engineering Practice J., vol. 14, no. 6, pp. 699–717, 2006. [3] R. Orjuela, B. Marx, D. Maquin and J. Ragot, “State estimation for nonlinear systems using decoupled multiple model,” International Journal of Modelling Identification and Control, vol. 4, no. 1, pp. 59–67, 2008.

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