network-induced time-varying delay modeling for compensation

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi Tomul LVIII (LXII), Fasc. 1, 2012 SecŃia AUTOMATICĂ şi CALCULATOARE

NETWORK-INDUCED TIME-VARYING DELAY MODELING FOR COMPENSATION THROUGH PREDICTIVE CONTROL BY

CONSTANTIN F. CĂRUNTU ∗ and CORNELIU LAZĂR “Gheorghe Asachi” Technical University of Iaşi, Faculty of Automatic Control and Computer Engineering

Received: December 9, 2011 Accepted for publication: January 5, 2012

Abstract. The goal of this paper is to provide a control design methodology that can assure the closed-loop performances of a physical plant, while compensating the time-varying delays introduced by the communication network that links the controller with the remote process. Firstly, the error caused by the time-varying delays is modeled as a disturbance and a method of bounding the disturbances is presented. Then, a robust one step ahead predictive controller based on flexible control Lyapunov functions is designed, which explicitly takes into account the bounds of the disturbances caused by time-varying delays and guarantees also the input-to-state stability of the system in a non-conservative way. Moreover, it is shown that by choosing an appropriately Lyapunov function, the MPC algorithm amounts solving a single, low-complexity linear program each sampling instant. The modeling method and the control strategy were tested on a vehicle drivetrain controlled through CAN, with the aim of damping driveline oscillations. Several TrueTime simulations based on realistic scenarios show that the proposed control scheme can handle the performance and physical constraints and the strict limitations on the computational complexity. Key words: networked control systems, time-varying delays, predictive control, TrueTime simulation. 2000 Mathematics Subject Classification: 93C10, 93D05, 00A72. ∗

Corresponding author; e-mail: [email protected]

64

Constantin F. Căruntu and Corneliu Lazăr

1. Introduction The evolution of standalone control systems to networked control systems (NCSs) brought many attractive advantages, but the use of communication networks makes it necessary to deal with the effects of the network-induced delays in the control loop. The delays introduced by the communication network may be time-varying and unpredictable and may deteriorate the closed-loop system performance and even reduce the system’s stability region. Existing constant time-delay stabilization and compensation methodologies may not be directly suitable for controlling a system over a communication network; therefore, to handle network delays in a closed-loop control system over a network, an advanced methodology is required. During the last few years several delay compensation strategies have been proposed: a multiple-delay Smith predictor (Ibeas et al., 2007), a predictive networked controller based on Smith predictor (Velagic, 2008) and even model predictive control (MPC) (Căruntu & Lazăr, 2011). Another recent approach is to consider the communication delay as a disturbance and to use a communication delay observer (CDOB) as in (Natori et al., 2004), which observes the condition of the network and compensates the effect of the variable delays in the control algorithm. An alternative approach to the communication observers (Natori et al., 2004) is to design an advanced technique, as model predictive control, which takes into account the uncertainty of communication delays. Recently, the need for increased driveability and passenger comfort lead to the development of numerous control strategies to damp driveline oscillations: robust pole placement (Stewart et al., 2005), linear quadratic Gaussian control design with loop transfer recovery (LQG/LTR) (Berriri et al., 2008) and model predictive control (MPC) (Rostalski et al., 2007; Căruntu et al., 2011). All of the previous control solutions assume that the sensors, controllers and actuators are directly connected, which is not realistic. Rather, in modern vehicles, the control signals from the controllers and the measurements from the sensors are exchanged using a communication network, e.g., Controller Area Network (CAN), among control system components. This brings up a new challenge on how to deal with the effects of the networkinduced time-varying delays and packet losses in the control loop. As such, the problem considered in this paper is to control a system through a communication network, while compensating the time-varying delays introduced by the network. The proposed solution consists of two steps: firstly, the error caused by the time-varying network-induced delays is considered as a disturbance and a method of finding the bounds of the disturbance is presented. Secondly, a robust one step ahead MPC scheme is designed using the concept of flexible control Lyapunov functions (CLF) (Lazăr, 2009) that explicitly accounts for rejection of disturbances introduced by the time-varying delay in

65

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

the communication network and guarantees also the input-to-state stability of the system in a non-conservative way. Several TrueTime simulation scenarios of a CAN-controlled vehicle drivetrain are used in this paper to illustrate the effectiveness and robustness of the proposed delay modeling and control strategy. Notation and basic definitions. ℝ, ℝ + , ℤ and ℤ + are the real, nonnegative real, integer and non-negative integer numbers. The notations ℤ ≥ c1 and ℤ ( c1 , c2 ] are used to denote the sets {k ∈ ℤ + ∣ k ≥ c1} and {k ∈ ℤ + ∣ c1 < k ≤ c2 },

respectively,

for

some

c1 , c2 ∈ ℤ + .

Let

‖ x‖ ∞ := max i∈ℤ[1,n ] [ x]i , where ⋅ denotes the absolute value. A function

ϕ : ℝ + → ℝ + belongs to class K if it is continuous, strictly increasing and ϕ (0) = 0 . A function ϕ : ℝ + → ℝ + is said to belong to class K∞ if it is of class K and lim s →∞ ϕ ( s ) = ∞ . 2. Network Delay Modeled as Disturbance Consider the standard NCS illustrated in Fig. 1, which is composed of five parts: a communication network, where it is assumed that delay occurs completely randomly, a physical plant, one sensor node (S), one controller node and one actuator node (A). The bounded delays introduced by the communication network, which can be smaller and larger than a sampling period, are represented as τ ca for the delay in the forward channel and as τ sc for the delay in the feedback channel. The physical plant is given by a linear state-space model xɺ (t ) = Ac x(t ) + Bc u (t − τ ca )

(1)

where x(t ) is the system state, u (t ) is the control signal and Ac and Bc are the continuous-time system matrices.

Fig. 1 – Control system with network-induced time delay.

It is easy to observe that Fig. 1 is equivalent to Fig. 2, in which the difference between the actual control signal and the delayed one is regarded as a

66

Constantin F. Căruntu and Corneliu Lazăr

hypothetical disturbance (Natori et al., 2004). Furthermore, the difference between the actual output and the delayed one is regarded as another hypothetical disturbance. Note that, this method is different than the one proposed in (Căruntu & Lazăr, 2011), where some assumptions were made in order to consider that the time-varying delays that appear in the feedback channel can be considered as time delays in the forward channel.

Fig. 2 – Control system with network-induced time delay as disturbance.

Notation: Let ak and bk denote the delay in the forward channel and in the feedback channel, respectively, at time instant k , expressed as a number of sampling periods: ak = τ kca / Ts  , bk = τ ksc / Ts  , where Ts is the sampling period of the system. Moreover, let a and b denote the maximum delays in the forward channel and in the feedback channel, respectively, expressed as a max max number of sampling periods: a = τ ca / Ts  , b = τ sc / Ts  .     2.1. Forward Channel Delays

The discrete-time forward channel disturbance representation is given by ukd = uk − ak − uk

(2)

and the goal is to find a bounded set Wu , in which to include all possible disturbances that can appear due to the time-varying delays introduced in the forward channel, knowing that the input of the remote process is bounded. This method is different from the one presented in (Natori et al., 2004), in which a communication delay observer is used to determine the difference between the computed control signal and the control signal that produced the measured output. Now, consider that the control signal is bounded by lower and upper bounds

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

u min ≤ uk ≤ u max ,

67

(3)

where u min and u max are the minimum and the maximum control signal values that can be given as input to the remote process. Then, the disturbance can be bounded as u min − u max ≤ ukd ≤ u max − u min .

(4)

Furthermore, if the discrete control signal is also restricted by an incremental bound −u ∆ ≤ ∆uk ≤ u ∆ ,

(5)

where ∆uk := uk − uk −1 , for all k ∈ ℤ ≥1 , with u0 some predetermined value and u ∆ is the maximum increase/decrease of the control signal at each sampling time instant k ∈ ℤ ≥1 , the disturbance can be rebounded as − au ∆ ≤ ukd ≤ au ∆ .

(6)

2.2. Feedback Channel Delays

The hypothetical disturbance exerted on the physical plant delays the feedback signal, the discrete-time feedback channel disturbance representation being given by

xkd = xk − ak − bk − xk − ak .

(7)

The goal is to find a bounded set Wx , in which to include all possible disturbances that can appear due to the time-varying delays introduced in the feedback channel, knowing that the input of the remote process is bounded. Consider the physical plant (1) with input disturbance xɺ (t ) = Ac x(t ) + Bc (u (t ) + u d (t )) .

(8)

Discretizing (8) yields

xk +1 = Ad xk + Bd (uk + ukd ) j −1

= Adj xk − j +1 + ∑ Adi Bd (uk − i + ukd−i ) i =0

,

(9)

68

Constantin F. Căruntu and Corneliu Lazăr Ts

where Ad = e AcTs , Bd = ∫ e Ac (Ts −θ ) dθ Bc . 0

Calculating

xk − ak = Ad xk − ak −1 + Bd (uk − ak −1 + ukd− ak −1 ) ,

(10)

xkd = ( I n − Adbk ) xk − ak − bk − ∑ Adi Bd (uk -ak -i -1 + ukd-ak -i -1 ) .

(11)

bk -1

= A xk − ak −bk + + ∑ Adi Bd (uk - ak -i -1 + ukd-ak -i -1 ) bk d

i =0

(7) becomes bk -1 i =0

xk − ak − bk is known at time instant k , all uk −i , i ∈ ℤ ≥1 are known, all u , i ∈ ℤ ≥1 are bounded, ak and bk are bounded by a and b , respectively, so d k −i d k

x can be dynamically bounded at each sampling instant k ∈ ℤ + . Now, (9) becomes xk +1 = Ad xk + Bd uk + Bd ukd + xkd .

(12)

Even though the time-varying delay gives a time-varying disturbance, the sets Wu , which is defined by Bd ukd , and Wx , which is defined by xkd , remain fixed since the delays are substituted by their bounds, so this modeling technique is suitable for the use of the results presented in (Lazăr et al., 2008), in which the disturbances are explicitly taken into account during the design phase of the predictive controller, which will be accomplished in the next section. 3. One Step Ahead Predictive Controller Design In this section, a robust one step ahead MPC scheme that employs a flexible Lyapunov function (Lazăr, 2009) to attain stability and performance, in which the disturbances are explicitly taken into account during the design phase by using input-to-state stability (ISS) concepts (see, e.g. (Jiang & Wang, 2001)), is presented. Furthermore, it is shown that for a particular choice of the Lyapunov function candidate the proposed design leads to a low-complexity linear program (LP) that can be solved efficiently within the required time range, while strictly enforcing the constraints and considering the delays induced by the communication network, via the usage of the disturbance model described in Section 2.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

69

3.1. MPC Setup

Consider the perturbed discrete-time constrained nonlinear system of the form xk +1 = φ ( xk , uk , wk ) := φ ( xk , uk ) + wk := f ( xk ) + g ( xk )uk + wk , k ∈ ℤ + ,

(13)

where xk ∈ X ⊆ ℝ n is the state, uk ∈ U ⊆ ℝ m is the control input and wk ∈ W ⊆ ℝ n is an unknown disturbance at the discrete-time instant k .

φ : ℝ n × ℝ m × ℝ n → ℝ n , φ : ℝ n × ℝ m → ℝ n , f : ℝ n → ℝ n and g : ℝ n → ℝ n×m are arbitrary nonlinear, possibly discontinuous, functions with φ (0,0,0) = 0, φ (0,0) = 0, f (0) = 0 and g (0) = 0 . Naturally, it is assumed that the set of feasible states X , the set of feasible inputs U and the disturbance set W are bounded polyhedra with non-empty interiors containing the origin. Next, let α1 , α 2 , α 3 ∈ K∞ and let σ ∈ K . Definition 1. A function V : ℝ n → ℝ + that satisfies α1 (‖ x ‖) ≤ V ( x ) ≤ α 2 (‖ x ‖), ∀x ∈ ℝ n

(14)

and for which there exists a possibly set-valued control law π : ℝ n ⇉ U such that V (φ ( x, u, w)) − V ( x ) ≤ −α 3 (‖ x ‖) + σ (‖ w ‖), ∀x ∈ X, ∀u ∈ π ( x ), ∀w ∈ W

(15)

is called an input-to-state stability control Lyapunov function (ISS-CLF) in X for system (1) and disturbances in W . ISS theory (Jiang & Wang, 2001) can be used to derive an input-to-state stabilizing predictive control scheme with improved disturbance rejection, as done in (Lazăr et al., 2008), where this property is referred to as optimized ISS. As such, let W be a convex hull of the vertices we , e = 1,…, E , and let λke , k ∈ ℤ + , be optimization variables associated with each vertex we . Let J (λ 1 ,…, λ E , λ ) : ℝ E+ × ℝ + → ℝ + be a strictly convex, radially unbounded function (i.e. J (·) tends to infinity when its arguments tend to infinity) and let J (λ 1 ,…, λ E , λ ) → 0 ⇒ λ e → 0 for all e = 1,…, E and λ → 0, and J (0,…,0, 0) = 0 . Choose off-line a CLF V (·) for system (13) without disturbances and let α 3 ∈ K∞ and x ∈ X be given. At each control sampling instant k ∈ ℤ + the one step ahead ISS MPC controller solves the following problem.

70

Constantin F. Căruntu and Corneliu Lazăr

Problem 1. At time k ∈ ℤ + measure the state xk and minimize the cost J (λ ,…, λkE , λk ) over uk , λk1 ,…, λkE and λk , subject to the constraints 1 k

uk ∈ U, φ ( xk , uk ) ∈ X, λke ≥ 0, λk ≥ 0,

(16a)

V (φ ( xk , uk ,0)) − V ( xk ) + α 3 (‖ xk ‖) ≤ λk , .

(16b)

V (φ ( xk , uk , w )) − V ( xk ) + α 3 (‖ xk ‖) ≤ λ ,

(16c)

e

e k

for all e = 1,…, E . Let π ( xk ) := {uk ∈ ℝ m ∣ ∃λk , λke , e ∈ ℤ [1, E ] s.t. (16) holds}

and

let

φcl ( xk , π ( xk ), wk ) := {φ ( xk , uk , wk )∣ uk ∈ π ( xk )} denote the difference inclusion corresponding to system (13) in closed-loop with the set of feasible solutions obtained by solving Problem 1 at each sampling instant k ∈ ℤ + . Next, the main robust stability result in terms of ISS is stated. This result is an adaptation of the main result in (Lazăr et al., 2008), to fit the relaxation (16b) of Problem 1, i.e., λk = 0 for all k ∈ ℤ + corresponds to the problem considered in (Lazăr et al., 2008). Theorem 1. Let α1 , α 2 , α 3 ∈ K∞ , a continuous and convex CLF V (·) and a cost J (·) be given. Suppose that Problem 1 is feasible for all states x in X and assume that lim k →∞ λk∗ = 0 . Then, the trajectories generated by the difference inclusion xk +1 ∈ φcl ( xk , π ( xk ), wk ), k ∈ ℤ + ,

(17)

with initial state x0 ∈ X converge in finite time to a robustly positively invariant subset of X , in which the difference inclusion is ISS for disturbances in W . The proof of Theorem 1 follows from standard arguments employed in proving input-to-state stability and Lyapunov stability and is therefore omitted here. The interested reader is referred to (Lazăr et al., 2008; Lazăr, 2009) for more details. Advantageous properties of the proposed robust controller are that ISS is guaranteed for any (feasible) solution of the optimization problem, state and input constraints can be explicitly accounted for, and feedback to disturbances is provided actively, on-line. The key of the stability proof is the limiting condition lim k →∞ λk∗ = 0 . In what follows a non-conservative solution for guaranteeing this condition is provided, such that λk can be small or even 0 for some time after which it is allowed to increase again, as long as this does not violate the upper bound.

71

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

Lemma 1. Let ∆ ∈ ℝ + be a fixed constant to be chosen a priori and let

ρ ∈ ℝ [0,1) and M ∈ ℤ >0 . If 1

0 ≤ λk ≤ ρ M (λk∗−1 + ρ

k −1 M

(18)

∆ ), ∀k ∈ ℤ ≥1 ,

then lim k →∞ λk = 0 . Lemma 1 is proven in (Căruntu et al., 2011) and is omitted here for brevity. By augmenting Problem 1 with constraint (18) the property lim k →∞ λk* = 0 is thus guaranteed, which is sufficient for asymptotic stability. Note that, in constraint (18), ρ k −1Ω gives a decreasing ramp and λk∗−1 dictates with what value the ramp is allowed to be violated in order to maintain the feasibility of the closed-loop control system. It is worth to mention that the proposed limiting condition is less conservative than the solution presented in (Lazăr, 2009; Căruntu & Lazăr, 2011), which corresponds to setting Ω = 0 . In the next subsection it is described how the developed one step ahead MPC scheme for the constrained system (13) can be implemented by solving a single LP during each control cycle. 3.2. Implementation Issues

Let ukss and xkss be the corresponding input and state steady-state values, respectively. Next, consider the following cost function to be minimized J 1 ( xk , uk , λk , λk ) := J MPC ( xk , uk ) + J (λk , λk )

:=‖ Px ( f ( xk ) + g ( xk )uk − xkss ) ‖∞ +

,

(19)

+ ‖ Q x ( xk − xkss ) ‖∞ + ‖ R (uk − ukss ) ‖∞ + J (λk , λk )

where the cost on the optimization variables λk := [λk1 ,…, λkE ]T and λk is defined as J (λk , λk ) :=‖ Λλk ‖∞ + | Λλ | , where Λ is a full-column rank matrix of appropriate dimensions and Λ ∈ ℝ > 0 . The cost J (·) is chosen as required in Problem 1 and the matrices Px , Qx and R are known full-column rank matrices of appropriate dimensions. The cost function (19) is subjected to the following constraints:

72

Constantin F. Căruntu and Corneliu Lazăr

u min − ukss ≤ uk ≤ u max − ukss , −u ∆ ≤ ∆uk ≤ u ∆ ,

(20)

x min − xkss ≤ xk +1 ≤ x max − xkss , where x min and x max are appropriate vectors that bound the states of the system. Now consider the following infinity-norm based candidate CLF, i.e., V ( x ) =‖ Px ‖∞ ,

(21)

where P ∈ ℝ p×n is a full-column rank matrix to be determined. This function satisfies (14) with α1 ( s ) =

σ p

s , where σ is the smallest singular value of P ,

and with α 2 ( s ) =‖ P ‖∞ s . Substituting (13) and (21) in (16b) and (16c) yields P ( f ( xk ) + g ( xk )uk − xkss ) − ‖ P( xk − xkss ) ‖∞ + Q ( xk − xkss ) ∞

∞

P ( f ( xk ) + g ( xk )uk + we − xkss ) − ‖ P ( xk − xkss ) ‖∞ + Q ( xk − xkss ) ∞

≤ λk ,

∞

≤ λke ,

(22) (23)

where: P , xk and xkss are known, e = 1,…, E and ‖ Qxk ‖≤ ξ ‖ xk ‖ , for

α 3 ( s ) := ξ x, ∀x ∈ X . Although (19), (22) and (23) appear to be nonlinear in the optimization variables, due to using only horizon 1, the optimization problem can be recast as a linear program via a particular set of equivalent linear inequalities. By definition of the infinity norm, for ‖ x ‖∞ ≤ c to be satisfied, it is necessary and sufficient to require that ±[ x] j ≤ c for all j ∈{1, 2,., n} , yielding a total of 2 p + 2 p( E + 1) linear inequalities in the optimization variables uk , λk1 ,…, λkE

and λk . Solving Problem 1, which includes minimizing the cost function (19), can be reformulated as the following problem. Problem 2. min (εk1 + εk2 + εk3 + εk4 )

uk , λk , λk

subject to (18), (20), (22), (23) and

(24)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

±[ Px ( f ( xk ) + g ( xk )uk − xkss )] j + Qx ( xk − xkss )

∞

≤ εk1 , ∀j ∈ {1, 2,…, n}

73

(25a)

± R (u k − u ) ≤ ε ,

(25b)

±Λλk ≤ εk3 ,

(25c)

Λλk ≤ ε .

(25d)

ss k

2 k

4 k

Notice that Problem 2 is a linear program, since xk , xkss , ukss and λk⊻−1 are known at time k ∈ ℤ + and thus, all constraints are linear in the unknowns uk , λk , λk and ε k1,2,3,4 . Next, the one step ahead MPC algorithm can be summarized as follows: Algorithm 1. At each sampling instant k ∈ ℤ + : Step 1: Measure the current state xk ;

Step 2: Solve the LP Problem 2 and pick any feasible control action u ∗ ( xk ) ; Step 3: Implement uk := u ∗ as control action. An advantageous feature of the proposed scheme is that only a feasible solution of (24) is required in order to assure stability. This implies that the control algorithm can be executed in a shorter time than standard MPC schemes, which typically require optimality to guarantee stability. 4. CAN-Controlled Drivetrain Architecture The structure of a vehicle drivetrain consists, in general, of the following parts: engine, clutch, transmission (gearbox), final drive, driveshafts and wheels. 4.1. CAN-Based Architecture

As it can be seen in Fig. 3 the components of the closed-loop vehicle drivetrain control system, e.g., sensors, controller, actuator, communicate through a communication network, i.e., the engine torque (control signal), and also the engine and wheel speeds are sent through CAN. Many of the messages transmitted on CAN, have real-time constraints associated with them.

Fig. 3 – Drivetrain control architecture.

74

Constantin F. Căruntu and Corneliu Lazăr

The complete network-controlled architecture considered in this paper, which is graphically depicted in Fig. 3, consists of the following operations: − the sensors measure the outputs of the system and send the samples to the controller through CAN; − the transmission controller receives the measurements from the sensors and the desired velocity reference vvr = rwωwr and computes the required torque, while handling the physical constraints and the delays, where rw is the wheel radius and ωwr is the desired wheel speed; − the control signal, i.e., the torque computed by the controller, is sent to the engine controller through CAN; − the engine controller actuates the spark timing and airflow as requested for driveline control. In Fig. 3 the dashed lines represent the direction of the messages sent to and from the controllers. 4.2. Drivetrain Model

A PWA three inertias model, which takes into consideration the clutch flexibility together with the driveshaft flexibility, was derived from the laws of motion (Kiencke & Nielsen, 2005), in which the first inertia corresponds to the engine, the second one includes the inertia of the gearbox and the inertia of the final drive, and the last inertia corresponds to the wheel and vehicle mass, as it is represented in Fig. 4.

Fig. 4 – Three inertias drivetrain model including the clutch.

When studying a clutch in detail, it is seen that its torsional flexibility is a result of an arrangement with smaller stiffness springs in series with springs with higher stiffness, as it can be seen in Fig. 5. (Fig. 5 a illustrates the clutch stiffness characteristic and the spring arrangement of the clutch is presented in Fig. 5 b).

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

75

The reason for this arrangement is vibration insulation. In this paper four working modes for the clutch are introduced, i.e., open, closing, closed and locked, which yields a more accurate model of the clutch. In the open mode, there is no mechanical connection between the engine and the rest of the driveline, so no torque is transmitted towards the wheels. In the closing mode, the smaller springs in the clutch are compressed, which means that the engine torque is gradually transmitted to the driveline. In the closed mode, the stiffer springs in the clutch are compressed and the gradual transmission of torque to the driveline continues. The locked mode corresponds to the phase when the springs in the clutch cannot be compressed any further, i.e., the clutch hits a mechanical stop.

Fig. 5 – Clutch functionality a − stiffness characteristic; b − clutch springs.

Consider as state variables: the torsional angle between engine and transmission, the torsional angle between transmission and wheel, the angular speed of the engine, the angular speed of the transmission and the angular speed of the wheel, and as control input of the system, the engine torque, i.e., x1 = θ e − θt g t , x2 =

θt gf

− θ w , x3 = ωe , x4 = ωt , x5 = ωw , u = Te ,

(26)

where θ e , θt and θ w are angles of the engine, transmission and wheel, respectively, ωe , ωt and ωw are the angular speeds of the engine, transmission and wheel, respectively, and Te is the torque produced by the engine. The following PWA state-space model is obtained: xɺ (t ) = Aci x(t ) + bc u (t ) + f c

if

x(t ) ∈ Ω i ,

(27)

76

Constantin F. Căruntu and Corneliu Lazăr

where x := ( x1 ,…, x5 )⊤ ∈ ℝ 5 and i ∈ I := ℤ [1,4] . Here i denotes the active mode at time t ∈ ℝ + , Aci ∈ ℝ 5×5 , bc ∈ ℝ 5×1 are the system matrices and f c ∈ ℝ 5×1 is the affine term. Notice that although there are 3 angles, only two states are introduced as only the angle difference is relevant. The collection of sets {Ωi ∣ i ∈ I } defines a partition of the state-space X ⊆ ℝ 5 as follows: Ω1 := {x ∈ ℝ5 ∣ x3 ≤ ωeclosing }, Ω 2 := {x ∈ ℝ 5 ∣ x3 > ωeclosing & Ω 3 := {x ∈ ℝ ∣ x3 > ω 5

closing e

| x1 |≤ θ1},

& θ1 ωeclosing & θ 2 t1 , if x(τ ) ∈Ω1 , ∀τ ∈ ℝ [ t1 ,t2 ) and x (t2 ) ∈ Ω 2 , set x1 (t2 ) := 0

(29)

As the engine angle θ e tends to infinity in the open mode, so the state x1 tends to infinity, synchronization of the engine angle and the transmission angle must be attained at the moment the clutch switches from the open mode to the closing mode. Notice that the reset condition does not apply to initial conditions, as the model cannot be initialized in the closing mode. The matrices Aci , bc and f c can be obtained by deriving the whole dynamical model of the drivetrain using the generalized Newton's second law of motion as in (Căruntu et al., 2011). The engine torque (i.e., the control input) is restricted by lower and upper bounds and by a torque rate constraint as follows: 0 ≤ u (t ) ≤ Temax , ∀t ∈ ℝ + , Tem ≤ uɺ (t ) ≤ TeM , ∀t ∈ ℝ + ,

(30)

where Temax is the maximum torque that can be generated by the internal combustion engine and Tem , TeM are torque rate bounds. Furthermore, the engine and wheel speeds are bounded, i.e.,

ωemin ≤ x3 (t ) ≤ ωemax , ∀t ∈ ℝ + ,

(31)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

ωwmin ≤ x5 (t ) ≤ ωwmax , ∀t ∈ ℝ + ,

77

(32)

where ωemin and ωemax are the idle speed and the engine limit speed, respectively, and ωwmin and ωwmax are the minimum and the maximum speed of the wheels. 5. TrueTime Results The CAN closed-loop drivetrain was simulated using TrueTime simulator (Henriksson et al., 2003), which is a very powerful MATLAB-based network simulation toolbox that can effectively simulate real-time networked control systems. 5.1. Simulation Environment

There are two primary Simulink blocks in the TrueTime package: the computer block and the network block, both being easy to customize in order to obtain a practical NCS. The computer block simulates an ECU with a flexible real-time kernel, A/D and D/A converters, a network interface, and external interrupt channels. The user has to write code functions, which define the evolution of the control tasks and the interrupt handlers. The network block, which allows nodes to communicate over the simulated network, is eventdriven, so it executes every time a message enters or leaves the network. To simulate a required network condition, different parameters such as: medium access control (MAC) protocol (Ethernet, CAN, etc.), transmission rate, minimum frame size, etc., can be customized (Henriksson et al., 2003). In the designed NCS simulation platform (Fig. 6), the sensors, controller and actuator are implemented using computer blocks and the CAN communication network is realized using a network block in which the MAC protocol is specified as CSMA/AMP. The sensors and the actuator are connected to the plant through their A/D converters and D/A converter, respectively. The platform also includes an interfering node sending disturbing traffic over the network. 5.2. Numerical Results

The continuous-time PWA model (27)-(29) was implemented in Matlab/Simulink and the proposed one step ahead predictive controller proposed in this paper was applied to damp driveline oscillations. The sampling period of the system was chosen as Ts = 5 ms. The values of the parameters used in simulations, which relate to a medium size passenger car, can be found in (Căruntu et al., 2011).

78

Constantin F. Căruntu and Corneliu Lazăr

Fig. 6 – TrueTime Simulink diagram - complex drivetrain model.

The upper bound of the delays that are induced by CAN was calculated using the methodology described in (Klehmet et al., 2008), resulting that τ max = 3.4Ts = 0.017s < 4Ts , which is used to apply the methodology described in Section 2 to model the effects of the variable time delays as disturbances. Then, the bounds of the disturbances are explicitly taken into account by the robust one step ahead predictive control strategy described in Section 3. The delays induced by CAN are time-varying and uniformly distributed in the interval [0,τ max ] . The control objective is to reach a desired speed reference in a short time, but, at the same time, to increase the passenger comfort by reducing the oscillations that appear in the driveline. The axle wrap is calculated as the difference between the engine speed (divided by the total transmission ratio) and the wheel speed, and it is used as a measure of the driveline oscillations. The predictive controller proposed in this paper uses the following weight matrices of the cost (19): Px = 0.0001I 5 , Qx = 0.002 I 5 , R = 0.25 , Λ = 0.01 and Λ = 1 . The technique presented in (Lazăr, 2006) was used for the off-line computation of the infinity norm based local CLF V ( x ) =‖ Px ‖∞ . For the one step ahead MPC scheme developed in this paper, recursive feasibility implies asymptotic stability. However, recursive feasibility is not a priori guaranteed and hinges mainly on the constraint (18) on the future evolution of λk∗ . For all simulation scenarios case studies, the values ρ = 0.99 , ∆ = 15000 and M = 5 proved to be large enough to guarantee recursive feasibility for the desired operating scenarios. The worst case time needed for computation of the control input for the proposed predictive controller was less than 2 ms on a PC running Matlab TrueTime, which meets the timing constraints.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

79

Different simulations were conducted, to evaluate the vehicle behavior in response to acceleration and tip-in and tip-out maneuvers, which are presented in the following. 5.3. Scenario 1: Acceleration Test

A first simulation test is performed on an acceleration scenario where the vehicle has to accelerate from 0 km/h to 30 km/h and the trajectories are plotted in Fig. 7.

Fig. 7 – Scenario 1: Acceleration test.

The amplitude of the axle wrap is represented in Fig. 8, bottom left. The evolution of the CLF relaxation variable λk∗ (green dashed-line) and the corresponding upper bound defined by (18) (black line) is shown in Fig. 7, top left. It can be observed that λk∗ may decrease or even go to 0, after which it is allowed to increase again, as long as this does not violate the upper bound. However as k → ∞ , λk∗ is forced to converge to 0. In Fig. 8 the clutch mode history was represented for the predictive controller, to show that in the transient the closed-loop system frequently switches between the operating modes.

80

Constantin F. Căruntu and Corneliu Lazăr

Fig. 8 – Scenario 1: Clutch mode of operation. 5.4. Scenario 2: Tip-In Tip-Out Maneuvers

The results of a tip-in, tip-out maneuver simulation, in which the reference vehicle velocity goes from 30 km/h to 10 km/h and back to 30 km/h, are presented in Fig. 9. Note that the controller performance during deceleration is limited by the actuator authority. For this experiment the evolution of the CLF relaxation variable λk∗ and the corresponding upper bound defined by (18) are shown in Fig. 9, top left. Due to changing the reference vehicle velocity, the upper bound of the CLF relaxation variable defined by (18) may become unfeasible, so whenever a change in the reference vehicle velocity occurs, the value of the upper bound was re-initialized. However as k → ∞ , λk∗ is forced to converge to 0. 6. Conclusions This paper presents a method of bounding the disturbances produced by the network-induced time-varying delays for control purposes and a robust one step ahead MPC strategy, which can handle the performance/physical constraints and which can explicitly take into account the disturbances caused by the time-varying delay. Also, a flexible control Lyapunov function was employed to obtain a non-conservative ISS stability guarantee for the developed one step ahead MPC scheme. The proposed bounding method and control strategy were tested on a vehicle drivetrain controlled through CAN and the results obtained illustrate that the proposed controller has good performances and it meets the required timing constraints.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 1, 2012

81

Fig. 9 – Scenario 2: Tip-in, tip-out maneuver simulation.

REFERENCES Berriri M., Chevrel P., Lefebvre D., Active Damping of Automotive Powertrain Oscillation by a Partial Torque Compensator. Control Engineering Practice, 16, 874−883, 2008. Căruntu C.F., Bălău A.E., Lazăr M., Van den Bosch P.P.J., Di Cairano S., A Predictive Control Solution for Driveline Oscillations Damping. Hybrid Systems: Computation and Control, Chicago, USA, 181−190, 2011. Căruntu C.F., Lazăr C., Stabilizing MPC for Network-Controlled Systems with an Application to DC Motors. Proc. IEEE International Conference on Mechatronics, Istanbul, Turkey, 2011. Henriksson D., Cervin A., Arzn Pierrot K.E., TrueTime: Real-Time Control System Simulation with MATLAB/Simulink. Proc. Nordic MATLAB Conference, Copenhagen, Denmark, 2003. Ibeas A., Vilanova R., Balaguer P., Multiple-Delay Smith Predictor Based Control of LTI Systems with Bounded Uncertain Delay. Proc. 22nd IEEE International Symposium on Intelligent Control, Singapore, 2007. Jiang Z.-P., Wang Y., Input-to-State Stability for Discrete-Time Nonlinear Systems. Automatica, 37, 857−869, 2001. Kiencke U., Nielsen L., Automotive Control Systems: for Engine, Driveline, and Vehicle. Springer Verlag, 2005. Klehmet U., Herpel T., Hielscher K.-S., German R., Delay Bounds for CAN Communication in Automotive Applications. Proc. 14th GI/ITG Conference Measurement, Modelling and Evaluation of Computer and Communication Systems, Dortmund, Germany, 2008.

82

Constantin F. Căruntu and Corneliu Lazăr

Lazăr M., Flexible Control Lyapunov Functions. Proc. 28th American Control Conference, St. Louis, MO, USA, 102–107, 2009. Lazăr M., Heemels W.P.M.H., Optimized Input-to-State Stabilization of Discrete-Time Nonlinear Systems with Bounded Inputs. Proc. 27th American Control Conference, Seattle, USA, 2008. Lazăr M., Model Predictive Control of Hybrid Systems: Stability and Robustness. Ph. D. Diss., Eindhoven University of Technology, The Netherlands, 2006. Natori K., Tsuji T., Ohnishi K., Hace A., Jezernik K., Robust Bilateral Control with Internet Communication. Proc. 30th Annual Conference of the IEEE Industrial Electronics Society, Busan, Korea, 2004. Rostalski P., Besselmann T., Maric M., Van Belsen F., Morari M., A Hybrid Approach to Modelling, Control and State Estimation of Mechanical Systems with Backlash. International Journal of Control, 80, 1729−1740, 2007. Stewart J., Calculus, Early Transcendentals. Thomson Brooks/Cole, 2003. Stewart P., Zavala J., Flemming P., Automotive Drive by Wire Controller Design by MultiObjectives Techniques. Control Engineering Practice, 13, 257−264. 2005. Velagic J., Design of Smith-like Predictive Controller with Communication Delay Adaptation. Proc. World Congress on Science, Engineering and Technology, 5th International Conference on Control and Automation, Paris, France, 2008.

MODELAREA ÎNTÂRZIERILOR VARIABILE ÎN TIMP PENTRU COMPENSARE PRIN CONTROL PREDICTIV (Rezumat) Scopul acestei lucrări este de a oferi o metodologie de proiectare a unui regulator care să asigure performanŃele în buclă închisă ale unui proces fizic, în timp ce se compensează întârzierile variabile în timp, introduse de reŃeaua de comunicaŃii care conectează regulatorul cu procesul fizic. În primul rând, eroarea cauzată de întârzierile variabile în timp este modelată ca o perturbaŃie şi este propusă o metodă de mărginire a perturbaŃiilor. Apoi, este proiectat un regulator predictiv robust pe un pas bazat pe funcŃii de control flexibile Lyapunov, care ia în considerare, în mod explicit, limitele perturbaŃiilor cauzate de întârzierile variabile în timp şi garantează, de asemenea, stabilitatea intrare-stare a sistemului într-un mod neconservativ. În plus, se arată că, prin alegerea corespunzătoare a funcŃiei Lyapunov, algoritmul predictiv de control rezolvă un program liniar de complexitate scăzută la fiecare perioadă de eşantionare. Pentru a ilustra potenŃialul de aplicare a algoritmului de control propus, metoda de modelare şi strategia de control au fost testate pe un lanŃ de transmisie a puterii la autovehicule controlat prin intermediul unei reŃele CAN, cu scopul amortizării oscilaŃiilor, ceea ce este esenŃial pentru îmbunătăŃirea manevrabilităŃii şi a confortului pasagerilor. Mai multe simulări realizate cu ajutorul toolboxului TrueTime bazate pe scenarii realiste ilustrează faptul că strategia de control propusă poate satisface atât restricŃiile legate de performanŃă şi restricŃiile fizice, cât şi limitările stricte cu privire la complexitatea de calcul.