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MPC with Laguerre Functions-2

Example 3

Illustrate that as N increases, for a given Laguerre pole a, the control trajectory converges to the optimal solution generated by using a linear quadratic regulator. Use the same system and design parameters as in Example 2, where the augmented system model is described by the matrices:

We use the MATLAB program called ‘dlqr’ to find the state feedback control gain matrix K as:

K = [0.8350 0.6113].

The sum of squared errors between the DLQR and DMPC trajectories is:

Example 4. Continue the study from Example 3.3 with the identical system and design parameters, except that the weight on the control is reduced from R = 1 to R = 0.1. Examine the incremental control trajectory u(ki +m) for this performance specification, and compare the trajectory with the one from DLQR solution.

The optimal control gain vector:

• With a smaller weight R = 0.1, the feedback control gain K has increased from that in Example 3. The DLQR control trajectory has faster response speed.

Closed-loop Poles of a DLQR System

In the DLQR design, when we choose Q = CTC, the closed-loop poles are uniquely determined by the weighting parameter R = rω > 0 in the single inputand single-output case with their values given bythe inside-the-unit-circle zeros of the equation:

In the multi-input and multi-output case:

Verify this property by re-examining solutions from the previous examples 2, 3, 4.

The zeros are 1.5589 ± j0.9563, 0.4661 ± j0.2859 from which the closed-loop poles of the DLQR system are found to be 0.4661 ± j0.2859. We reduced R from 1 to 0.1,

The zeros are 1.8228 ± j2.3858, 0.2022 ± j0.2647 from which the closed-loop poles of the DLQR system are found to be 0.2022±j0.2647. Indeed, these poles confirmed with the computation of the actual closed-loop poles through the system matrix A−BKlqr.

When the weight parameter R is reduced, the closed-loop poles move towards the origin of the complex plane, hence the closed-loop response speed is faster, as we have observed from the responses in the examples.

Use of Laguerre Parameters as Tuning Parameters

The choice of a scaling factor a and a smaller N for the Laguerre functions affects the closed-loop performance of the DMPC system, although for a large N the performance converges to the underlying optimal DLQR system. However, when N is small, the potential here is to use a as the performance fine-tuning ‘knob’.

Example 5.

Examine the effect of parameter a and N on the closed-loop performance. Assume at time ki = 10, x(ki) = [0.1 0.2 0.3]T .

Case A. We fix the parameter N with a large value (N = 8), and vary the pole location α by choosing α = 0, α = 0.4, and α = 0.8. Table 3 shows the comparison results for the cases where N is large while a changes. It is seenfrom Table 3 that the closed-loop predictive control system is very similar to the underlying DLQR system and the effect of α is quite small when N is large.

Table 3- Closed-loop eigenvalues and feedback gain vector when N = 8 and α varies from 0 to 0.8

Case B. For the second case, we fix N with a smaller value (N = 2), and vary the parameter α. Table 4 summarizes the comparison results. It is seen that with a smaller N, the closed-loop poles and gain of the predictive control system are functions of the Laguerre pole α.

Table 4. Closed-loop eigenvalues and feedback gain vector when N = 2 and α varies from 0 to 0.9

We observe that the parameters α and N affect the closed-loop control performance, particularly when N is small. This is particularly useful in the situation when the optimal DLQR system does not provide us with satisfactory performance, and these additional tuning parameters help us with fine tuning of the closed-loop performance.

Constrained Control Using Laguerre Functions

With parameterization of the control signal trajectory using Laguerre functions, we have the flexibility to choose the locations of the future constraints.This could potentially reduce the number of constraints within the prediction horizon, and hence the on-line computational load for large-scale systems.In addition, because of the existing exponential decay factor in the Laguerre functions, the difference of the control signal is ensured to converge to zeroafter the transient period. Thus, it is sufficient in the majority of cases that the constraints are imposed in the transient period of the response. This in turn will reduce the number of constraints.

1 Constraints on the Difference of the Control Variable

Example 6. Suppose that a continuous-time system is described by the transfer function

Optimal solution without constraints. Key: line (1) DMPC; line (2) DLQR

It is seen from this figure that with this choice of α and N, without constraints, the responses are almostidentical. Incidentally, if the parameter α = 0, which corresponds to the traditional approach, then N is required to be approximately 40 in order to achieve similar performance. To ensure that the constraints are satisfied for the whole response, the first 15 samples of u are put into the constrained solution.

Optimal solution with constraints

The constrained results within one optimization window.Although there are 30 constraints imposed, there are only three active constraints, which can be found through visual inspection of Figure 3.10a. The first two samples and the 17th sample are the activated constraints in the solution. By comparing Figure 3.10 with Figure 3.9 it can be seen that activated constraints resulted in little performance deterioration.

2 Constraints on the Amplitudes of the Control Signal

Example 7

Optimal solution with constraints on u. Key: line (1) without constraints; line (2) with constraints

It is seen from Figure b that some of the constraints on the control signal were not met. By studying the multipliers, it is noticed that there were eight active constraints and the multipliers could not converge because the active constraints were in conflict. The values of the positive multipliers are listed after 88 iterations in the calculation as:

Numerically, the Mact matrix has row dependence, and as a result, the constraints are compromised (see Figure 12). One comment relates to the number of constraints used in this example. Although there are 192 constraints used in the optimization, there are only 8 active constraints. The rest of the constraints are inactive and have no effect on the optimal solution. Because of receding horizon control, where thefirst sample of the optimal control

is implemented, so from this point of view, only the constraint on the first sample had an effect in this example.

Stability Analysis