7.2 Case study: DC-DC converter
7.2.6 Numerical results
and a block ofP. The submatrix∇zjCAj(zj∗, x∗j)> denoting the Jacobian of the active constraints are obtained appropriately from the active constraints of PN(p0). This shows that the KKT matrix of the linear system corresponding to the OCPPN−j(pj)can be constructed through the submatrices of the KKT matrix solved forPN(p0) which is already available. Finally, the right-hand side is a zero matrix except for the identityI2 appearing in∇pjCAj(zj∗, xj∗)>
corresponding toxj−pj.
7.2.6 Numerical results
We consider a low-power (2 Watt) step-down converter setup with the following design parameters: Vs= 6 V,rl = 15.5mΩ, Vo = 1V,ilmax= 4A,ro= 500 mΩ,C= 68µF,L= 1.5 µH andrc= 1.5 mΩ.
We formulate differentm-step and SBM MPC controllers by varying the sampling frequencyfs ∈[300kHz,400kHz] (wherefs := 1/Ts) and the number of steps m∈ {1,2, ...10,11}. Closed-loop simulations are performed in Matlab in order to measure the controller closed-loop performance and the required computing
Chapter 7. Numerical examples
power in terms of floating point operations (FLOPs)1. Closed-loop performance
For each m-step or SBM MPC scheme, we perform 103 simulations of the plant evolution of different initial values (using a set of random and uniformly distributed feasible initial state values) and evaluate the closed-loop performance function (7.5). These values are then averaged and assigned to the scheme.
Figure 7.2 shows the trend of the performance of the algorithm along increasing sampling frequencyfsfor varying multistepmboth form-step and SBM MPC.
Note first that from the discretization of (7.3) using sampling frequencyfs∈ [300kHz,400kHz]resulting in sampling time lengthTsof magnitude10−6seconds, the entries of the resulting submatricesM, P, Q, Rin (7.5) have magnitude10−6. With the prescribed state and control constraints of magnitudes100 and10−1, respectively, we expect Jcl to be not far from magnitude 10−6 as confirmed by the figure. In addition, the differences between values ofJcl ranging from 1.382×10−5 to1.402×10−5 can be, in this case, considered significant.
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
x 105 1.382
1.384 1.386 1.388 1.39 1.392 1.394 1.396 1.398 1.4 1.402x 10−5
Sampling frequency [Hz]
Closed−loop performance Jcl
m=1 m=3 m=5 m=7 m=9 m=11 sm=3 sm=5 sm=7 sm=9 sm=11
Figure 7.2: PerformanceJclfor varying sampling frequencyfs. The symbolm stands for the number of steps of them-step MPC whilesmfor the SBM MPC.
Observe that the scheme withm= 1gives the standard MPC where we solve an OCP at every sampling instant. As expected, this gives the best performance where the feedback is able to react to the disturbance at each time step. Also shown is that higher sampling frequency yields better closed-loop performance since faster reaction implies faster disturbance rejection.
Furthermore, the closed-loop performance worsens upon using higher value of m(in solid lines). This is as expected since the system runs in open loop for a longer time causing further propagation of the deviation between the measured and the predicted states. However, improvement is achieved through the use of the sensitivity updates. Unlike them-step feedback law, SBM MPC uses the perturbation magnitude and the sensitivity information to allow the controller to react to this measured and predicted state deviation. As seen in Figure 7.2 (in
1As opposed to FLOPS which means floating-point operations per second
7.2. Case study: DC-DC converter
dashed lines), the performance profiles get closer to that of the standard MPC indicating better closed-loop performance for the SBM MPC in comparison to the m-step MPC. The graph, however, gives little information to determine which number of stepssmgives for the best performance.
Computational complexity
We present here the details of the computational complexity of the active set method for solving a QP which is quantified by the number of FLOPs (i.e., addition, multiplication and division) to be executed per iteration as investigated in [44]. We quantify the number of FLOPs it takes for a fixed simulation time Tsim and aim to compare the number of FLOPs we save by increasing the multistepmand the additional operations we incur when updating the controls through computing sensitivities.
First, we consider the active set algorithm for solving a QP in each iteration of the SQP strategy as outlined in Algorithm 7.2.1 (given in [44]). The method begins with an initial guessW0 of the active set which is called theworking set.
The working set is then refined by adding or deleting a constraint until the exact active set is found.
Algorithm 7.2.1. (Active Set Method for Solving a QP) 1. Compute a feasible pointz0.
2. Set initial working setW0. 3. Fork= 0,1,2, . . .
4. Solve the system
H ∇CWk
∇CW>k 0
∆zk
η
=
−f −Hzk
0 5. If ∆zk= 0, then
6. If allηi ≥0,then 7. Terminate,z∗=zk.
8. Else
9. RemoveifromWk s.t. λi= min
i∈Wk
λi and thenzk+1←zk. 10. End if
11. Else
12. Dk ←
i /∈ Wk | ∇Ci∆zk >0, ri− ∇Cizk
∇Ci∆zk <1 13. If Dk=∅, then
14. zk+1←zk+ ∆zk andWk+1← Wk
15. Else
16. α← min
i∈Dk
ri− ∇Cizk
∇Ci∆zk
and zk+1←zk+α∆zk
17. ConstructWk+1 by adding one element ofDk toWk. 18. End if
19. End if 20. End for
We define the following variables
Chapter 7. Numerical examples
nx dimension of the state
nu dimension of the control
no= (nx+nu)N+nx number of optimization/decision variables ne=nx(N+ 1) number of equality constraints
ni= 2(nx+nu)N number of inequality constraints nc=ne+ni total number of constraints
We first consider the worst-case scenario which pertains to solving the system in line 4 of Algorithm 7.2.1 with the largest possible dimension, i.e., the maximum number of inequality constraints that can become active are active. This equals half of the box constraints in the formulation (7.4) which is ni/2. Let ξ :=
no+ne+ni/2. Since systems with banded matrices are best solved by Gaussian elimination with pivoting as pointed in [67], we use this technique to solve the system in line 4. It requires the following amount of operations
number of N(·)(ξ) = (ξ−1)ξ(ξ+ 1)/2 multiplication N(+)(ξ) =ξ2(ξ+ 1)/2 addition
N(÷)(ξ) =ξ division
LetNGE(N)be the total number of FLOPs needed to perform Gauss elimination as a function of the discrete time prediction horizonN. Asξ =ξ(N), this is given by
NGE(N) = N(·)+N(+)+N(÷) (ξ)
Let us now estimate the number of operations for Algorithm 7.2.1. The following lines require the corresponding amount of operations
line multiplication addition division 4 n20+ n0(n0−1) +n0+
N(·)(ξ) N(+)(ξ) N(÷)(ξ) 12 nc·2no nc((no−1) + 1) nc
+nc(no−1)
16 no no
Therefore, lettingNAS(N)be the total number of FLOPs performed in a single iteration of the active set method which is a function of the discrete time prediction horizonN, we obtain
NAS(N) = 2n2o+ 2no(2nc+ 1) +NGE(N) which is a polynomial inN of degree 3 (i.e. O(N3)).
This allows us to compute the number of operations for an MPC scheme over a simulation period. If we fix prediction horizonT (from which we determine N) and simulation time Tsim (from which we determineN˜) and assume¯kis the average number of iterations it takes the active-set method to converge, for the m-step MPC, the FLOPs amount to
N˜
m ·¯k· NAS(N),
7.2. Case study: DC-DC converter
while for SBM MPC, (m6= 1) , the FLOPs total to N˜
m ·(¯k· NAS(N) +
m−1
X
i=1
(NGE(N−i) + 2nu(nx+ 1))
| {z }
(∗)
)
where(∗)is additional the expense due to solving a sequence of linear systems for smaller dimension to compute the required sensitivities.
Figure 7.3 shows the trend in the amount of FLOPs of the algorithm along increasing sampling frequency for varying multistepmboth for MF and SBM MPC assuming¯k= 1. The standard MPC (m= 1) requires the most number of iterations. The number is divided bymasmincreases and additional amount is added if sensitivity updates are performed. Note that Figure 7.3 shows the worst-case scenario FLOPs requirement, i.e., with maximum number of active inequality constraints. In the reality, the number of active constraints is significantly much less than the maximum possible. The SBM MPC requires significantly less computing power compared to standard MPC, but requires more compared to anm-step approach whenm >1. In addition, by increasing the sampling frequencyfs, the measured FLOPs increase for any controller. This is related to the discretization step (see Section 7.2.2) in the sense that increasing fs means increasing the prediction horizonN and therefore the problem size and computational complexity.
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
x 105 0
0.5 1 1.5 2 2.5x 106
Sampling frequency [Hz]
Floating point operations (FLOPs)
m=1 m=3 m=5 m=7 m=9 m=11 sm=3 sm=5 sm=7 sm=9 sm=11
Figure 7.3: Worst case scenario FLOPs for varying sampling frequencyfsand variousm-step MPC andsmfor the SBM MPC.
In implementing SBM MPC, as mentioned in Section 6.2, one has to take care so as not to violate constraints or create changes in the active constraints when updating by sensitivities. To simplify the analysis on the reduction of cost by taking advantage of available information, we apply the following rule so as not to perform further computations (e.g. the post-optimal analysis in [17] for computing unavailable sensitivities, see Remark 5.5.3 (d)) when constraints are violated. At a given time step, if the control is already on the bound, we do not update in order to keep the corresponding constraint active. Otherwise, if
Chapter 7. Numerical examples
upon updating, the resulting updated control goes on or beyond the constraints, we use a control with a difference of 10−6 from the concerned bound in order to keep the corresponding constraint inactive. Similarly, we also prevent the predicted states and perturbed states to go beyond the state constraints. This, however, do not occur in this particular example where perturbation of5×10−3 is used. Figure 7.4 illustrates the state and control staying within the constraints indicated in (7.4) for SBM MPC implementations of varying sampling frequency fsand multistepm.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−3 1
1.5 2 2.5
State x1 (inductor current il)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−3 2
4
State x
2 (voltage V
0)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−3 0
0.2 0.4
Feedback control µ (duty cycle)
Figure 7.4: Perturbed state and updated control for SBM MPC implementations of varying sampling frequencyfsand multistepm.
Pareto optimality analysis
As shown in Figures 7.2 and 7.3, the closed-loop performance and computing power requirements are strongly correlated: (i) increasing the sampling frequency fsand decreasing the number of multistep mlead to controllers with lowerJcl (i.e., better closed-loop performance) and higher computing power requirement;
(ii) similarly, decreasingfsand using higher multistep myield controllers with worse closed-loop performance and limited computing power. This results in a design trade-off between closed-loop performance and computing power. We analyze these trade-offs and present them in terms of Pareto optimality and efficiency (for a single point solution) or compromise solutions (see the tutorial in [45]). Figure 7.5 shows the Pareto frontier, thus the design trade-off between closed-loop performanceJcl and computing power in terms of FLOPs. On one extreme, the points in red represent them-step schemes with higher value of mwhich we observe to be less computationally demanding algorithms, while
7.2. Case study: DC-DC converter
on the other extreme is the MPC scheme withm = 1 which is the one with the highest computing requirements but with the best closed-loop performance (indicated by the lowestJcl). Moreover, the points in blue represent the SBM MPC schemes which we observe to be the algorithms compromising a ’balance’
between the two opposing objectives of having a good algorithm performance and being computationally less demanding. This suggests a great potential for the suitability of the scheme for embedded systems with limited computing power.
0 0.5 1 1.5 2 2.5
x 106 1.382
1.384 1.386 1.388 1.39 1.392 1.394 1.396 1.398 1.4 1.402x 10−5
Floating point operations (FLOPs)
Closed−loop performance Jcl
Figure 7.5: A Pareto efficiency plot (solid circles and squares forming the Pareto frontier) on a set of feasible options for m-step (red circles) and SBM (blue squares) MPC.
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Ehrenwörtliche Erklärung
Hiermit versichere ich an Eides statt, dass ich die von mir vorgelegte Dissertation mit dem Thema
"Robust Updated MPC Schemes"
selbstständig verfasst und keine anderen als die angegebenen Quellen und
selbstständig verfasst und keine anderen als die angegebenen Quellen und