6.5 Simulation Results
6.5.3 Adaptive IDA-PBC with time-varying D
For the boost mode, Fig. 6.6 shows the profiles of the output voltage and the inductor current for the adaptive IDA-PBC—for different values of the control gain k1 and adaptation gain γ = 1—in the face of step changes in the extracted power D. It is seen that increasing the control gain k1 reduces the convergence time of the output voltage. It is also shown that the output voltage recovers very fast from the variations of the power D, always converging to the desired equilibrium. This is due to the fact that, as predicted by the theory, the power estimate converges—
exponentially fast—to the true value independently of the control signal. It should be remarked that the PD controller becomes unstable in this scenario.
Figure 6.6: Response curves for the adaptive IDA-PBC withγ = 1 to changes in the power D: (a) output voltage—with (b) and (c) zooms for it—and (d) the inductor current.
In Fig. 6.7 the step changes in the power D and the estimate ˆD for different values of the adaptation gain, with the initial condition ˆD(0) = D(0), are shown.
As predicted by the theory, for a largerγ, the speed of convergence of the estimator is faster. Notice, however, that in the selection of γ, there is a tradeoff between convergence speed and noise sensitivity.
Figure 6.7: Transient performance of the estimate ˆD under step changes of the parameterD for various adaptation gains γ and a zoom of the first step.
6.6 Summary
The challenging problem of regulation of the output voltage of a buck-boost con-verter supplying electric energy to a CPL with unknown power has been addressed in this chapter. First, assuming the power is known, an IDA-PBC that renders a desired equilibrium point asymptotically stable has been proposed. Subsequently, an on-line I&I estimator with global convergence property has been presented to renders the scheme adaptive, preserving the asymptotic stability property. It has also been illustrated the performance limitations of the classical PD controller stem-ming from the fact that, due to the presence of the CPL, the system is non-minimum phase. Some realistic simulations have been provided to confirm the effectiveness of the proposed method.
Technical Appendices of the Chapter 6.A Proof of Proposition 6.3
Proof of ClaimP1: It is shown first that the control (6.17) can be derived using the IDA-PBC method of Proposition 6.2.
First, assign the matrix Fd as Fd(x) :=
−xx21 −x2x2+12
2x2
x2+1 −(x22x+1)1 2
,
that, forxin the positive orthant ofR2, satisfies the condition (6.14).3 Now, observe that the system (6.4) can be rewritten in the form (6.10), with
f(x) :=
−x2 x1− xD2
, g(x) =
x2+ 1
−x1
.
3It is well-known [81] that a key step for the successful application of the method is a suitable selection of this matrix, which is usually guided by the study of the solvability of the PDE (6.12).
See [44] for some guidelines for its selection in this example.
The left annihilator of g(x) is then g⊥(x) := [x1 x2 + 1]. It follows that the PDE (6.12) can be written explicitly as
−x2∇x1Hd(x) + 2x1∇x2Hd(x) =D−x1+ D
x2. (6.28)
Using the symbolic language Mathematica, the solution of the PDE (6.28) is com-puted as where Φ(·) is any function with argument
x21+ x222
, which is selected as Φ(z) := k1
2(z+k2)2, with k1 and k2 arbitrary constants.
The existence of constants k1 and k2 guaranteeing that x? is a critical point of Hd and that, furthermore, is a local minimum of it, are verified next.
Evaluate the gradient of Hd(x), shown in Appendix 6.C, at the equilibrium x?4 and substitute k2 as presented in Appendix 6.F, which results in ∇Hd|x=x? = 0, implying thatx? is a critical point of Hd.
Now, a sufficient condition for x? to be a local minimum of Hd, is that M :=∇2Hd|x=x? >0.
Being∇2Hd a 2×2 real matrix,5 the above condition is equivalent to m11 >0, det(M)>0,
where m11 is the (1,1) entry of M. Regarding these expressions, the following equivalences hold.
where a0 and b0 are real constants defined in Appendix 6.E. Notice that both ex-pressions are linear with respect to k1 and that, furthermore, the coefficients of k1 are positive if
0< D < x22?
√2√
1 +x2?.
4Recall equation (6.6).
5For completeness, the entries of ∇2Hd are included in Appendix 6.D.
Consequently, M >0 if and only if the latter inequality on D, and k1 >max{− a0
4D2(x2?+1)2 x22?
, − b0 x42?−2D2(x2?+1)
x32?
}
⇒
m11>0 ∧ det(M)>0,
are satisfied simultaneously, which hold by assumption. The proof is completed showing that the IDA-PBC (6.17) results replacing the data in (6.15).
Proof of Claim P2: It has been shown that Hd(x) has a positive definite Hessian matrix atx?, therefore it is locallyconvex. Then, for sufficiently smallc, the sublevel set Ωx defined in (6.21) is bounded and strictly contained in the positive orthant of R2. The proof is completed recalling that sublevel sets of strict Lyapunov functions are inside the region of attraction of the equilibrium.
6.B Proof of Proposition 6.4
Computing the time-derivative of the estimation error, ˜D, along the trajectories of (6.4), and using the expression (6.23), it follows that
D˙˜ =−γx2x˙2+ ˙DI
=−γx1x2(1−u) +γD+ ˙DI. Substituting (6.24) in the last equation yields
D˙˜ =γD+ 1
2γ2x22−γDI
=−γD,˜ from which (6.25) follows immediately.
To prove the asymptotic stability of (x,D) = (xˆ ?, D), the adaptive controller (6.22) is written as
ˆ u:= 1
u0 (u1+u2)|D≡Dˆ. Then, the following holds
ˆ
u=u|D (exact)+δ(x,D),˜
where u|D (exact) denotes the controller (6.17). That is, the controller that assumes an exact knowledge of D is recovered, plus an additive disturbance δ(x,D); the˜ mapping δ can be proved to satisfyδ(x,0) = 0.
Invoking the proof of Proposition 6.3, the closed-loop system is now a cascaded system of the form
˙
x = Fd(x)∇Hd(x) +g(x)δ(x,D, k˜ 1) D˙˜ = −γD,˜
where
g(x) :=
x2+ 1
−x1
,
is the systems input matrix. The signal ˜D(t) tends to zero exponentially fast for all initial conditions, and for sufficiently large k1, i.e., such that (6.19) is satisfied, the system above with ˜D = 0 is asymptotically stable. Invoking well-known results of asymptotic stability of cascaded systems,e.g., Proposition 4.1 of [101], the proof of
(local) asymptotic stability is completed.
6.C Explicit form of ∇ H
d∇x1Hd=k1x1 2 k2+x21 +x22
−D(x2+ 1) 2x21+x22
+
√2Dx1arctanh
r x1
x21+x222
(2x21+x22)3/2
∇x2Hd= Dx1(x2+ 1) x32+ 2x21x2 +1
2k1x32+k1x21x2+ 2k1k2x2− 1 2
+
√2Dx2arctanh
r x1 x21+x222
(2x21+x22)3/2
6.D Components of the Hessian matrix ∇
2H
d6.E Values of the constants a
0and b
0.
6.F Value of the constant k
2k2 = 1
Damping Injection on a
Small-scale, DC Power System
Synopsis This chapter explores a nonlinear, adaptive controller aimed at in-creasing the stability margin of a direct-current (DC), small-scale, electrical net-work containing an unknown constant power load. Due to its negative incremental impedance, this load reduces the effective damping of the network, which may lead to voltage oscillations and even to voltage collapse. To overcome this drawback this chapter considers the incorporation of a controlled DC-DC power converter in par-allel with the load. The design of the control law for the converter is particularly challenging due to the existence of states that are difficult to measure in a practical context, and due to the presence of unknown parameters. To tackle these obstacles, a standard input-output linearization stage, in combination with a suitably tailored adaptive observer, is proposed. The good performance of the controller is evaluated through experiments on a small-scale network.
7.1 Introduction
Various techniques have been explored for the stabilization of DC networks with CPLs—a survey may be found in [109]. These techniques are categorized into pas-sive and active damping methods: the former are based on open-loop hardware al-terations, whereas the latter imply the modification of existing—or added—control loops. In an active damping strategy the control loops can be modified at three different network’s positions [109]: at the source’s side, at the load’s side, and at a midpoint between them. In the present chapter, the interest is in using the latter approach, which was firstly explored in [19], [129], and [55], for the stabilization of a small-scale network with a single CPL. In these references the network’s stabi-lization is achieved by adding acontrolled power converter in parallel with the load and then designing a suitable feedback control law for it: in [19] the converter is modeled as a simple controlled current source and a linear control law is designed to stabilize the overall network; a similar approach, but using a full model for the power converter, is used in [129]. Their stabilization result is based on the linearization of the network’s dynamics. Lastly, in [55] a large signal stability analysis, but using approximate techniques, such as the Takagi-Sugeno fuzzy model, is carried out to evaluate the performance of a linear controller.
The main contribution of this chapter is described next. Following [19] and [129],
the stabilization problem for a small-scale DC network supplying electrical energy to a CPL is studied here. First, the network is augmented by placing a controlled power converter between the load and the source. Then, for the converter’s con-troller design, instead of relying on linear-feedback techniques, this chapter proposes an adaptive observer-based nonlinear control law that provably achieves overall net-work’s stabilization. The control design is particularly challenging due to the ex-istence of unmeasured states—the current of the DC network—and the unknown power of the CPL. The construction of the proposed controller is based on the use of standardinput-output linearizationto which a suitably tailoredadaptive observer is added; its good performance is evaluated via experiments on a small-scale DC network.
The stabilization problem addressed here, as well as the proposed controller topology, have previously been studied in [66], where a full state-feedback adaptive passivity-based control has been proposed. As discussed in Subsection 7.4.1, besides the impractical requirement of full state measurement, the approach adopted in that paper suffers from significant energy efficiency drawbacks, which renders the proposed controller design practically unfeasible. Both limitations are overcome in this chapter.
The rest of the chapter is structured as follows. In Section 7.2 the model of the system under study is presented and its stability properties are summarized.
The proposed controller configuration, adopted from [19] and [129], is presented in Section 7.3. The main contributions of the chapter are developed in Section 7.4 and some preliminary realistic simulations are shown in Section 7.5. The results of two physical experimental are reported in Section 7.6 and the chapter is wrapped-up with a brief summary in Section 7.7.