Iterative Predictions

As shown in Lemma 3.1, the quasi-LPV MPC problem can be solved given that the parameter trajectory depends on the state and can thus be predicted as a function of previous states and inputs. However, the nonlinear nature of the problem would make its solution on a real-time environment difficult to implement. To overcome the complexity arising from the nonlinear optimization, an iterative algorithm is proposed (henceforth referred to as qLMPC, quasi-Linear Model Predictive Control) which requires only the solution of a sequence of Quadratic Programs (QP) to find the solution to the underlying nonlinear optimization problem.

Note that for fixed scheduling trajectories ^P𝑘, the predicted states 𝑋_𝑘 in (3.5) are linear in the control inputs𝑈_𝑘, and one can - just as in the case of LTI systems - find a solution to problem (P.1) by solving a QP. This motivates the following iterative approach:

• Initially problem (P.1) is solved with the quasi-LPV model (3.1) replaced by the LTI model that is obtained when the state-dependent scheduling sequence ^P𝑘 is replaced by P⁰

𝑘 =1⊗ 𝜚(𝑥(0), 𝑢(0)).

• A scheduling sequence ^P𝑙𝑘 is then iteratively driven towards a possibly sub-optimal se-quence^P∗𝑘 = 𝜚(𝑋^∗

𝑘, 𝑈^∗

𝑘), where 𝑋^∗

𝑘 and𝑈^∗

𝑘 denote the state and input trajectories corre-sponding to the (sub-) optimal solution to (P.1).

• This is achieved by solving at iteration𝑙 the optimization problem (P.1) with^P𝑘 replaced by ^P𝑙𝑘, and by generating a new scheduling sequence from the resulting optimal state sequence𝑋^𝑙

𝑘 as^P𝑙𝑘⁺¹= 𝜚(𝑋^𝑙

𝑘

, 𝑈^𝑙

𝑘).

• After the last iteration, 𝑋^∗

𝑘 and𝑈^∗

𝑘 are used in the next time step to warm-start^P0_𝑘+1, i.e.

P⁰_𝑘+

1= 𝜚(𝑋^∗

𝑘, 𝑈^∗

𝑘)by appropriately shifting𝑋^∗

𝑘 and𝑈^∗

𝑘.

Thus the idea is to solve a sequence of optimization problems where the quasi-LPV model (3.1) is replaced by an LTV model that is generated from (3.1) by imposing a fixed scheduling sequence, which is then updated at each iteration step using the optimized state sequence: the initial scheduling sequence ^P0𝑘 yields an LTV system; this system is referred to as Σ⁰. The optimization problem yields 𝑈0

𝑘 as an estimate of the control input, where the superscript is

used to indicate that the sequence corresponds to the systemΣ⁰. The computation of the state sequence 𝑋⁰

𝑘 is done using (3.5), which is then used to calculate a parameter trajectory for a subsequent iteration, i.e., ^P1𝑘 = 𝜚(𝑋0

𝑘

, 𝑈0

𝑘), this LTV system is now called Σ¹. A new input trajectory𝑈¹

𝑘 can then be found by solving (P.1) again. Note that input and state trajectories are calculated iteratively.

When 𝑋^𝑙

𝑘 ≈ 𝑋^𝑙−1

𝑘 , the input sequence𝑈^𝑙

𝑘 gives an approximation of the optimal solution𝑈^∗

𝑘 to (P.1). The first element of the sequence is then applied to the plant and the procedure is repeated for all subsequent time steps. Implementation of the proposed approach is summarized in the algorithm in Figure 3.1a and a graphical depiction is shown in Figure 3.1b.

Stop Criterion

A crucial factor of any iterative procedure is the stop criterion. Whereas Sequential Quadratic Programming (SQP) and SQP-like methods (which qLMPC could be considered to be) for nonlinear optimization often use a stop criterion of the form ||𝑋^𝑙

𝑘 − 𝑋^𝑙−1

𝑘 || ≤ 𝜖 where 𝜖 is a predefined tolerance, in a real-time context such a stop criterion could cause the computation time to exceed the sampling time with potentially catastrophic effects. In the opposite end of the spectrum one could carry out only 1 iterative step per sampling time and leave convergence to the warm-start kind ofiteration; this is a common practice in MPC as seen in e.g. [44]. In the case of qLMPC, experience has shown that 1-2 iterations give satisfactory results (see Figure 3.3 below). However, one should consider that by not driving the optimization to convergence at each time step, any stability guarantee would be lost in the first few time steps before convergence is achieved.

10: untilstop criterion 11: Apply𝑢_𝑘 to the system Figure 3.1: qLMPC Algorithm

Example 3.2(qLMPC Iterative Algorithm). Consider the nonlinear system 𝑥

1(𝑘+1) 𝑥2(𝑘+1)

=

4/3+0.2𝑥

1(𝑘) −2/3+0.1𝑥

2(𝑘)

1 0

𝑥1(𝑘) 𝑥2(𝑘)

+ 0.1

0

𝑢_𝑘 𝑦(𝑘) =

−2/3 1

with the constraint𝑢 ∈ [−1 1]. The system is similar to the one presented in Example 2.1, albeit perturbed with nonlinear terms and with a scaled input. A linearization around the origin reveals that the origin is a stable equilibrium, however contrary to Example 2.1, this system is not globally stable nor can it be globally stabilized, given the input constraint.

A quasi-LPV model can be readily obtained by defining 𝜌

1(𝑘) = 𝑥

1(𝑘), 𝜌

2(𝑘) = 𝑥

2(𝑘). Applying the algorithm in Figure 3.1a with 𝑄 = 𝐶^>𝐶, 𝑅 = 0.1, 𝑃 = 10𝑄^a and 𝑁 = 10 yields the closed-loop response shown below.

-1 -0.5 0 0.5

-1 -0.5 0 0.5

yu

k

10 20 30 40 50 60 70 80 90 100

0

Figure 3.2: Closed-loop response of Example 3.2

In order to investigate the convergence properties of the iterative algorithm, 10 iterations were carried out. Prediction of the state 𝑥

2 at 𝑘 = 0 after each iteration is shown in Figure 3.3a which highlights that predicted trajectories are indeed convergent. The norm of the prediction error at each iteration w.r.t the nonlinear prediction given 𝑈10

𝑘 (i.e. the prediction resulting from nonlinear simulation given the predicted input sequence after 10 iterations) at several time steps is shown in Figure 3.3b, where it is clear that not only are the iterations converging, indeed so are the initial guesses (i.e. the real-timeiterations) thanks to the warm-start. This leads to the conclusion that in this example even not using iterations would lead to a satisfactory response. Indeed, relying only on warm-start and not using intra-time-step iterations leads to a qualitatively identical closed-loop response.

1

Figure 3.3: (a) Predicted𝑥

2trajectories at 𝑘 =0 after 1 ( ), 2 ( ), 3 ( ), 4 ( ), 5 ( ) and 6 ( ) iterations, iterations 7-10 are not shown for clarity.

(b) Prediction error given𝑈¹⁰

𝑘 w.r.t nonlinear simulation at𝑘 =0 ( ),𝑘 =5 ( ),𝑘 =10 ( ), and𝑘 =15 ( ).

aThe terminal cost function is not positive definite in this case, this does not represent a problem as no stability analysis is to be made.

Convergence of the iterative scheme is addressed in Appendix A. The discussion is based on drawing an analogy to Newton-type Sequential Quadratic Programming. Indeed under some conditions, qLMPC can be understood as a version of Newton-SQP, for this reason an overview of SQP is given and conditions for local convergence are stated. The velocity-based qLMPC (Chapter 4) is seen to have a similar structure to SQP, enabling the same convergence analysis to be carried out.

Im Dokument 1.1.1 LPV Model Predictive Control (Seite 53-56)