Computational issues

equals with1, other terms are zero. Other coefficient matrices are zero matrices with appropriate dimensions.

Terms depending onp4i−1(k)appear inA4i−1,B4i−1 andE4N−1(the effect ofρ˜N+1).

The related non-zero entries are summarized in Table 3.5.

Matrix entry Value a(3i−1,3i−2)

4i−1 νT

τ∆i

a(3i−1,3i−1)

4i−1 −_∆^δT

inr_i^∗ a(3i−1,3i+1)

4i−1 νT

τ∆i

b⁽³ⁱ⁻¹⁾_4i−1 −_∆^δT_inv^∗_i e^(4N_4N−1^−1),(2N+4) −_∆^δT

inv^∗_i

Table 3.5: The non-zero entries ofA_4i−1,B_4i−1 and E_4N−1

Finally, since only the control input direction B(x(k)) depends on the scheduling parameter family p_4i(k), only the coefficient matrices B_4i ∈ R^3N×N have non-zero entries withb⁽³ⁱ⁻¹⁾_4i being −_∆^δT

in. Polytopic description

Finally the transformation of the LPV dynamics of the underlying stretch into a poly-topic form can be performed by using the proposed TP-transformation. For this case the Ψdomain (3.47) and the corresponding gridM₁×M₂×. . .×M_2N should be extended by using detector measurements. According to our previous findings the higher dimensional polytopic description naturally inherits the assigned interconnected structure.

(q₀(k), v₀(k)andqN(k), vN(k)) and used for feeding the non-linear model, while the in-ternal detector measurements are treated as output sequences (qi(k), vi(k)). Collecting these variables and the systems states (ρ1(k), v1(k), . . . , ρN(k), vM(k)) in vectors the freeway dynamics can be rewritten in a compact non-linear model form (3.2a)-(3.2b) by emphasizing the dependency on the unknown parameters:

x(k+ 1) = f(x(k), u(k),Θ), (3.65)

y(k) = h(x(k),Θ), (3.66)

whereΘcontains the vector of unknown model parameters: vf ree, a, ρcr, τ, κ, ν, δ, β.

Through the detector measurements a sequence of data is available for identification:

u^m(k), y^m(k), k= 1,2, . . . , K. (3.67) The identification problem is then formulated as an optimization problem, by deter-mining the value ofΘ which minimizes the following weighted quadratic cost criterion by:

J(Θ) = XK

k=1

(y(k)−y^m(k))^T Q(y(k)−y^m(k)), (3.68) wherey(k)˜ denotes the discrepancy between the model response and real measurement data, M is the number of samples. Since the elements of the parameter vector have physical meanings, Θ has to take its value from a closed admissible region in the 8-dimensional parameter space. Qis a positive definite2×2weighting matrix determined in the basis of the stochastic components of the measured variables [CP81].

The optimization performed by an iterative non-linear programming routine, where in each step a new parameterΘvalue is computed, and a new simulation is performed with the latestΘby using the same measurement data. These steps are repeated until further improvement ofJ(Θ)is not possible. In order to determine the global optimum, the optimization algorithm needs to be initialized from several random starting point [CP81].

Once the parameter vectorΘhas been determined steady-states can be calculated in virtue of the corresponding non-linear algebraic equations. Obviously, these equations can be solved after the selection of physically meaningful free variables. At the same time, results for a general freeway stretch can still be unrealistic with very large variance in the density profile, caused by the overgeneralized freeway topology (see Fig. 2.5). If one investigates a real freeway network two important differences can be seen. Firstly, entering and exiting points are not located in every ∼ 500 meters, but only in ∼ 5 kilometers. Secondly, there is a certain distance (practically ∼ 500−1000 meters) between an off- and on-ramp, i.e.: segments with both ramps are rather unrealistic.

Accordingly, the following periodic structure can be observed in reality. An on-ramp is followed by a long straight freeway stretches without ramps, then finally an off-ramp (see Fig.3.1).

The steady-state values for such a topology can be constructed accordingly. We notice that the long stretch without ramps have a spatially smooth steady-state solu-tion, since each unit owns the following steady-state equations (as a simplified case of

∼5km

∼0.5−1km ∼0.5−1km

Figure 3.1: Illustration of a realistic freeway topology eqs. (3.14)-(3.16)):

0 = T

∆_in

ρ^∗_i−1·v_i−1^∗ ·n−ρ^∗_i ·v^∗_i ·n

, (3.69)

0 = T

τ [V(ρ^∗_i)−v_i^∗] + T

∆i

v^∗_i

v_i−1^∗ −v_i^∗

− ν τ

T

∆i

ρ^∗_i+1−ρ^∗_i

ρ^∗_i +κ . (3.70) The equations above have the special solution with ρ^∗_i−1 = ρ^∗_i = ρ^∗_i+1 and v_i−1^∗ = V(ρ^∗_i−1) =v_i^∗ =V(ρ^∗_i). Therefore it is sufficient to investigate the steady-state condi-tions of two interconnected segment, one with on-ramp (with subscriptr) and one with off-ramp (with subscripts), together with a cyclic boundary condition. The equations can be given as:

0 = T

∆rn[ρ^∗_s·v^∗_s·n−ρ^∗_r·v_r^∗·n+r^∗],

0 = T

τ [V(ρ^∗_r)−v_r^∗] + T

∆_rv_r^∗[v_s^∗−v_r^∗]−ν τ

T

∆_r

ρ^∗_s−ρ^∗_r ρ^∗_r+κ − δT

∆_rn r^∗v_r^∗ ρ^∗_r+κ,

0 = T

∆sn[ρ^∗_r·v_r^∗·n−ρ^∗_s·v_s^∗·n−s^∗],

0 = T

τ [V(ρ^∗_s)−v_s^∗] + T

∆s

v_s^∗[v_r^∗−v^∗_s]−ν τ

T

∆s

ρ^∗_r−ρ^∗_s ρ^∗_s+κ,

where the subscriptsr and srefer to segment with on-ramp and off-ramp respectively.

The equations above can be solved numerically, resulting in a reasonable solution in both physical and mathematical senses.

Computational issues of parameter-dependent representations

In general, most of the LPV based design techniques can be formulated as a con-vex problem with constraints represented by LMIs. The generic parameter-dependent structure implies infinite number of these constraints due to the non-linear dependence on scheduling parameters. In order to reduce the numerical complexity an approximate gridding technique is often used, where the problem is solved over an arbitrary dense grid covering the operation set [Wu95, Lee97, BP94, AG95]. To ensure global optimality a dense grid is necessary which implies large number of LMI constraints. On the other hand problems formulated by using affinely parameterized or polytopic models can be characterized by a finite number of LMI constraints, therefore gridding is not required.

The number of these constraints depends on the dimension of the scheduling parameter

vector respectively the number of LTI vertex systems. Accordingly a system with affine dependence onp(k)∈R^np result2^np number of LMI constraints, resulting from all the possible combination of the extreme values of the vector elements. Alternatively the number of LMI constraints equals with the nλ number of local LTI systems of the Ω polytope.

During the derivation of the freeway problem a large number of scheduling parame-ters were introduced to capture all types of non-linearities. By this way the developed models can be considered exact in the sense that they do not neglect any dynamics. We compare the computational needs of the different representations by considering a gen-eral freeway stretch withN interconnected segments, whereNrnumber of the segments have on-ramp connection. The generic LPV model is then scheduled with 2N number of variables according to the dimension of the state vector. Therefore a2N dimensional envelope should be gridded to solve LMI based design problems. This set can be reduced by considering the physical nature of the state variables, removing physically meaning-less points. The affine parametrization involves4N_rnumber of scheduling parameter to capture non-linearities for segments with on-ramps and3(N−Nr)number for segments without on-ramp. That is to say a 2^3N+Nr number of LMI constraints are necessary to ensure design objectives over the whole admissible operation domain. Finally, the polytopic description involves 6^N number of vertex systems, regardless of the number of on-rampsNr, and therefore the same number of LMI constraints.

Remark 3.2 In view of equation (2.52), the extension of the parameter varying struc-ture for destination dependent mode has no technical difficulties. Considering theγi,j(k) as time varying variables implies the extension of the scheduling parameter vectorp(k).

At the same time, the origin-destination information are difficult to measure, there-fore their application for scheduling is questionable. One could try to approximate the destination mode by considering only time independent parameters. Consequently, the derivation of the generic qLPV model for destination-dependent mode is straightforward.

As one can realize, larger dimensional freeway problems involve a powerful high number of LMI constraints. One possible way to reduce the computational demands of the representations is certainly the introduction of approximations throughout the derivation, i.e. to reduce the dimension of the scheduling vector. For this purpose one can conclude that the range of the non-linear scalar mapping _ρ˜ ¹

i(k)+ρ^∗_i+κ is much smaller compared to F(˜ρi(k)) or v˜i(k). Consequently the term _ρ˜ ¹

i(k)+ρ^∗_i+κ can be replaced with a constant value _ρ∗¹

i+κ⁺, where κ⁺ is introduced to fit the approximated model response to detector measurements. Since p3,i(k) and p4,i(k) depend on non-linearities in question, the neglection implies the use of only p_1,i(k) and p_2,i(k). This reduction lowers the number of LMI constraints to2^2N for the affine case and4^N for the polytopic model. Although the number of scheduling parameters is decreased drastically such approximation leads to decreased numerical accuracy.

Of course the numerical needs of the parameter-dependent realizations are still de-manding, therefore such representations can only be applied for lower dimensional prob-lems at this stage. At the same time we emphasize the interconnected structure which can be exploited through distributed or decentralized design methods [Wu03].

In document TamásLuspay LinearParameterVaryingConcepts AdvancedFreewayTrafficModelingandControl (Pldal 55-59)