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Explore the performance of DMPC control systems with Laguerre functions compared to DLQR, analyzing control trajectory convergence. Use MATLAB dlqr program for state feedback control. Verify closed-loop poles property and examine Laguerre parameters impact.
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دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي استاد درس دكتر فرزاد توحيدخواه بهمن 1387 MPC with Laguerre Functions-2 کنترل پيشبين-دکتر توحيدخواه
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.
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 input and single-output case with their values given by the inside-the-unit-circle zeros of the equation:
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.
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’.
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 a by choosing α = 0, α = 0.4, and α = 0.8. Table 3.3 shows the comparison results for the cases where N is large while a changes. It is seen from Table 3 that the closed-loop predictive control system is very similar to the underlying DLQR system and the effect of a is quite small when N is large.
Table 3- Closed-loop eigenvalues and feedback gain vector when N = 8 and a 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 a 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.
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 zero after 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.
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 a and N, without constraints, the responses are almost identical. 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.
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.
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 the first 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.
When the constraints in model predictive control are activated, the control law effectively becomes a nonlinear control problem. Thus, the stability properties of linear time-invariant systems do not apply.
Stability with Terminal-State Constraints The core to establishing closed-loop stability is based on an equality constraint on the terminal state, which is:
Stability with Large Prediction Horizon In order to have constrained optimal solutions, firstly, the number of decision variables should be larger than the number of active constraints. Thus, having the terminal-state constraints, the number of decision variables, i.e., the dimension of η, is likely to be increased. Secondly, there is a possibility that the active constraints from the terminal-state could cause a linear dependence with other inequality constraints such as constraints on input and output signals. When this happens, the active constraints (including the terminal-state constraints) may not be satisfied. For these reasons, the terminal-state constraints are seldom used in practice. Instead, we explore under what conditions the term x(k + Np | k)TQx(k + Np | k) is sufficiently small so that the difference of the Lyapunov function is guaranteed to be negative.
Without constraints, for a large prediction horizon and a large N in the Laguerre functions, the optimal control trajectory u(k + m) converges to the underlying optimal control dictated by a discrete-time linear quadratic regulator with the same weight matrices Q and R. The cost functions of predictive control and DLQR become nearly identical. Because of the asymptotic stability of the DLQR system, with a large N, for some large prediction horizon Np, Consequently, this leads to the closed-loop stability of the predictive control system from the Lyapunov function analysis.
