# Linear Robust Output Regulation in a Class of Switched Power Converters

Authors: Josep M. Olm; Domingo Biel
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

This article addresses the robust output regulation problem for a class of nonlinear switched power converters after its linearization by means of a change of the control vector variable. The methodology employs a dynamic state feedback control law and considers parametric uncertainty due to unknown values of resistive loads. Restrictions arising from the fact that the control gains exhibit fixed values are taken into account. The proposed technique is exemplified with the output voltage regulation of a Noninverting Buck-Boost converter and tested through realistic numerical simulations.


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## Body

## 1. Introduction

The output regulation—or servomechanism—problem deals with the design of feedback control laws that provide output tracking of any reference belonging to a family of command profiles and, at the same time, are able to reject any perturbation from a certain set, with both references and disturbances being generated by a known, autonomous system of ordinary differential equations, the so-called exosystem. The control design is robust when the objective is achieved despite the presence of parametric uncertainties.This capital subject, with long trajectory in control theory, was solved for linear systems in the early 1970's with the introduction of the well-knowninternal model principle [1]; there is also an interesting algebraic approach contained in [2]. The extension of the solution to nonlinear systems appeared almost two decades later in the celebrated paper [3]. The reader is referred to [4] for a summary of main topics in Output Regulation Theory. On the other hand, any of the excellent monographic text-books [5, 6] offer a complete overview of the subject.In this article we address the robust output regulation problem for a family of nonlinear power converters with two control switches that includes the NonInverting Buck-Boost (NIBB), the Watkins-Johnson (WJ), the Inverse of Watkins-Johnson (IWJ) and the Full-Bridge NonInverting Buck-Boost (FBNIBB). The NIBB, the WJ and the IWJ are among the class of eight elementary single-input (i.e., possessing a single voltage source) single-output converters containing a single inductor [7], while the FBNIBB is derived through the substitution of the original switches by a full-bridge in a NIBB [7, 8].The converters being nonlinear, the first thought may be to face the problem by means of nonlinear output regulation techniques. However, performing a change of control variable the resulting system appears to be linear. Thus, linear output regulation tools are used from this stage on.The approach, which considers resistive loads with uncertain output resistance, proves the existence of a dynamic state feedback law that solves the linear robust output regulation problem and provides an algorithmic-like construction of the regulator for the general case, that is, either output voltage regulation or tracking. Nevertheless, the eventual achievement of control objectives in the physical system is limited by a possible control action saturation due to the fixed values of the control gains. Hence, guaranteeing a dynamical evolution of the converter in an unsaturated region of the phase plane involves restrictions on the system parameters, state variables and reference profiles that are also studied.The article is structured as follows. Section2 contains the main results of linear robust output regulation by means of dynamic state feedback. Section 3 introduces a family of switched power converters and establishes the solution of its output regulation problem. Section 4 is specifically devoted to the construction of a dynamic state feedback regulator for the target system. A numerical example based on the methods outlined in this section is presented in Section 5, while the corresponding simulation results are in Section 6. Finally, conclusions are outlined in Section 7.


## 2. Linear Robust Output Regulation

This section is focused on the major highlights of Linear Robust Output Regulation (LROR) and follows the exposition in [6].The Output Regulation Theory addresses the problem of rendering the outputy(·) of a linear control system, possibly with plant uncertainties w, to asymptotically track any reference yR(·) belonging to a given family and, at the same time, reject asymptotically any disturbance d(·) that may be found in a certain set, while maintaining the internal stability of the closed loop system.The formulation of the problem to be solved takes advantage of the following fact: if one thinks in nullifying the output errore=y-yR, there is no need to separate the roles of yR and d, because both may be seen as components of an exogenous input that has to be rejected.Therefore, consider the system(2.1)ẋ=Awx+Bwu+Pwv,e=Cwx+Qwv,
with x∈ℝn and u,e∈ℝm. The exogenous input v∈ℝp is assumed to satisfy the exosystem(2.2)v̇=Sv,
while w∈ℝq stands for the plant uncertainty vector. Aw,Bw,Cw,Pw, and Qw are real matrices with appropriate dimensions, whose coefficients depend on the plant uncertainties, their nominal values matrices being A0,B0,C0,P0, and Q0. Notice, finally, that in (2.1) it is assumed that d=Pwv,y=Cwx, and yR=-Qwv.Let the dynamic state feedback law(2.3)ż=Φz+Ne,u=H1x+H2z,
with z, Φ, N, H1, and H2 being real vector and matrices of appropriate dimensions. Then, the forced closed-loop system consisting of the plant (2.1), the exosystem (2.2) and the control law (2.3) is(2.4)(ẋż)=(Aw+BwH1BwH2NCwΦ)(xz)+(PwNQw)v,v̇=Sv,e=Cwx+Qwv.
Denote also by(2.5)Acw=(Aw+BwH1BwH2NCwΦ)
the matrix of the unforced closed-loop system, Ac0 being its nominal value matrix.Within this framework, the LROR problem may be posed as follows.Definition 2.1.
LetW be an open subset of ℝq that contains the origin w=0. The LROR problem of (2.1) with exosystem (2.2) in W, by means of dynamic state feedback, consists of designing a control law of the form (2.3) such that one has the following:(i)
The matrixAc0 defined from (2.5) is Hurwitz, that is, σ(Ac0)⊂ℂ-, where σ(Ac0) denotes the spectrum of Ac0.(ii)
The matrixAcw in (2.5) is such that σ(Acw)⊂ℂ-, for all w∈W; furthermore, for all x(0),z(0),v(0) and for all w∈W, the trajectories of (2.4) satisfy
(2.6)limt→∞e(t)=limt→∞[Cwx(t)+Qwv(t)]=0.A necessary and sufficient condition for the solvability of the LROR problem using dynamic state feedback is established in the next result.Theorem 2.2 (see [6]).
Consider the plant (2.1) with exosystem (2.2). Assume that the pair (A0,B0) is stabilizable and also that σ(S)⊂ℂ+¯={λ∈ℂ;Re(λ)≥0}. Then, the LROR problem is solvable by means of a dynamic state feedback controller (2.3) if and only if
(2.7)rank(A0-λ𝕀B0C00)=n+m,∀λ∈σ(S).Assume that system (2.1)-(2.2) satisfies condition (2.7). Then, the construction of a linear robust regulator with a dynamic state feedback control law of the form (2.3) may be carried out as follows [6]. Let(2.8)mpS(λ)=λr+ar-1λr-1+⋯+a1λ+a0
be the minimal polynomial of S. Then, consider the matrices(2.9)Φi=(010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1-a0-a1-a2⋯-ar-1),Ni=(00⋮01),
with Φi∈Mr×r(ℝ), Ni∈Mr×1(ℝ), and define(2.10)Φ=diag(Φ1,…,Φm)∈Mrm×rm(ℝ),N=diag(N1,…,Nm)∈Mrm×m(ℝ).
This selection of Φ and N, which ensures the controllability of (Φ,N), together with the assumed stabilizability of (A0,B0) and the fulfillment of (2.7), yields the stabilizability of the following pair:(2.11)A¯0=(A00NC0Φ),B¯0=(B00),
where A¯0 and B¯0 are, respectively, (n+rm)×(n+rm) and (n+rm)×m matrices. Thus, there exists H∈Mm×(n+rm)(ℝ) such that(2.12)σ(A¯0+B¯0H)=σ[(A0+B0H1B0H2NC0Φ)]⊂ℂ-.SelectingΦ, N, H1, and H2 as indicated above, the dynamic state feedback regulator (2.3) solves the LROR problem for system (2.1)-(2.2).Remark 2.3.
It is proved in [6] that, under the assumption σ(S)∈ℂ+¯, if a controller (2.3) solves the LROR problem for (2.1) and (2.2), then, for all w∈W, there exist unique matrices (Πw,Σw) that satisfy the following matrix equations:
(2.13)(ΠwΣw)S=(Aw+BwH1BwH2NCwΦ)(ΠwΣw)+(PwNQw),(2.14)0=CwΠw+Qw.
For everyw∈W, (2.13) indicates that Mcw={(x,z,v);x=Πwv,z=Σwv} is an invariant manifold for the closed-loop system (2.4), while (2.14) means that the error is zero on the invariant manifold Mcw. Furthermore, let v(0)=v* be any initial condition for the exosystem; then, the corresponding exogenous input is v*(t)=exp(St)v*. If the initial state of the plant (x,z) in (2.4) is set to x(0)=Πwv*, z(0)=Σwv*, it is immediate that x(t)=Πwv*(t), z(t)=Σwv*(t), and, subsequently, e=0, for all t≥0.


## 3. Output Regulation in a Class of Nonlinear Switched Power Converters

The basic nonlinear switched power converters NonInverting Buck-Boost, Full-Bridge NonInverting Buck-Boost, Watkins-Johnson and Inverse of Watkins-Johnson have a general state-space representation in terms of an averaged model consisting of a two-dimensional system with the inductor currentiL and the capacitor voltage vC as state variables, and a control variable û=(û1,û2)⊤. It is worth mentioning that the control action in the physical system is carried out by means of switches; hence, û1 and û2 are to be actually implemented through an appropriate PWM signal.For a systematic analysis it is advisable to minimize the number of parameters of the system. This goal may be achieved with the change of variables and parameters:(3.1)x1=1VgLCiL,x2=1VgvC,t=1LCτ,μ=1RLC,
which make the system dimensionless:(3.2)ẋ1=û1-x2û2+k1(û2-1)+k2x2(1-û1),ẋ2=-μx2+x1û2-k2x1(1-û1).The control gainsû1,û2 take values in [û-,û+]×[û-,û+], with û-<û+. The values of the parameters k1, k2 and of the lower and upper bounds of the control gains for the different converters are summarized in Table 1. Moreover, assume an unknown value R for the load resistance, due to the addition of a constant disturbance term Rw to its nominal value RN; that is, R=RN+Rw, with RN>0 and Rw∈(-RN,+∞). Consequently, the parameter μ may be written as μ=μN-w, with(3.3)μN=1RNLC>0,w=μNRwRN+Rw∈(-∞,μN),w being the only uncertain parameter of the system.Table 1
State space descriptors of a class of switched power converters.
Convertersk1k2û-û+NonInverting Buck-Boost0001Full-Bridge NonInverting Buck-Boost00-11Watkins-Johnson1001Inverse of Watkins-Johnson0101Assigning(3.4)x=(x1x2),û=(û1û2),δ=(-k10),AN=(0k2-k2-μN),(3.5)Aw=(000w),B(x)=(-k2x2+1-x2+k1k2x1x1),
the dynamical system (3.2) may be written as(3.6)ẋ=(AN+Aw)x+δ+B(x)û.
Furthermore, notice that det B(x)=x1(1-k1k2)=x1 according to the admissible values for k1, k2 indicated in Table 1. Hence, assuming x1≠0, the state feedback control law(3.7)û=B-1(x)[u-ANx-δ]
transforms system (3.6) into(3.8)ẋ=Awx+u.Let us now consider the problem of rendering the statex of system (3.8) to asymptotically track a certain reference profile x=xR(t), which can be expressed as a linear combination of the solutions of a time-invariant, linear exosystem; that is, there exist real matrices S and Q, of appropriate dimensions, such that(3.9)v̇=Sv,v(0)=v0,xR=Qv.
Notice that an exosystem as (3.9) with σ(S)⊂ℂ+¯ can generate a large class of functions, including combinations of step functions with arbitrary amplitude, ramps with arbitrary slope or sinusoidal signals with arbitraries amplitude and initial phase. These are the type of references/disturbances usually faced by power converters.Therefore, the problem may be posed as that of finding a linear robust output regulator for(3.10)ẋ=Awx+u,v̇=Sv,e=x-Qv.Let𝕀2 and 02 denote, respectively, the identity matrix and the null matrix in the set of 2×2 matrices. Then, identifying the elements of the original system (2.1) with those of the particular case (3.10) one gets(3.11)Bw=B0=𝕀2,Pw=P0=02,Cw=C0=𝕀2,Qw=Q0=-Q,
while Aw is defined in (3.5) and, subsequently, A0=02.Proposition 3.1.
Let us consider system (3.10) and the equivalences (3.5)–(3.11). Then,(i)
The pair(A0,B0)=(02,𝕀2) is controllable.(ii)
The following matrix is nonsingular for allλ∈ℂ:
(3.12)(A0-λ𝕀2B0C002)=(-λ𝕀2𝕀2𝕀202).Proof.
The proof is immediate.Theorem 3.2.
Consider the plant (3.10), and assume that σ(S)⊂ℂ+¯. Then, the LROR problem is solvable by means of a dynamic state feedback controller (2.3).Proof.
The result follows using Proposition3.1 and Theorem 2.2.Let (2.3) be a dynamic state feedback controller that solves the LROR problem for the system (3.10). Turning back now to the original system (3.6), the corresponding feedback control law is to be obtained using (2.3) in (3.7). However, recalling that the original control vector û has fixed gain values and taking into account (3.7), it is easily realizable that (2.3) will be actually useful for output regulation situations in which the trajectories x(·) of (3.6) remain entirely inside the state-space region X defined as(3.13)X={x∈ℝ2;detB(x)≠0∧B-1(x)[u-ANx-δ]∈Û},
with Û=[û-,û+]×[û-,û+]. Notice that the restriction det B(x)≠0 is necessary and sufficient for guaranteeing the diffeomorphic character of the transformation (3.7), while the requirement û in Û ensures nonsaturation of the controller performance. Eventually, this fact entails restrictions on the set of initial values x(0),z(0),v(0) and parametric uncertainties W (see issue (ii) in Definition 2.1) from which robust output regulation is attainable.At this point, specific conditions to be accomplished by candidate reference profilesxR are especially interesting. They follow immediately from the assumption that xR lies in X. Hence, assume that system (3.10) has achieved a steady-state x=xR under the action of the dynamic state feedback controller (2.3), and denote zR, uR, the corresponding stationary behavior for u and z; it follows from Remark 2.3 that xR=Πwv, zR=Σwv, (Πw,Σw) being the solution of (2.13) and (2.14), and uR=H1xR+H2zR by construction. Moreover, it is straightforward from the assignment (3.5)–(3.11) that Πw=Q, while Σw is such that(3.14)QS=(Aw+H1)Q+H2Σw,ΣwS=ΦS.
Hence, using (3.9),(3.15)xR=Qv=Qexp(St)v0,(3.16)zR=Σwv=Σwexp(St)v0,(3.17)uR=(H1Q+H2Σw)exp(St)v0,
where Σw satisfies (3.14). Furthermore, from (3.8) we obtain an alternative expression for uR:(3.18)uR=ẋR-AwxR=QSv-AwQv=(QS-AwQ)exp(St)v0.Proposition 3.3.
LetxR=(x1R,x2R)⊤ satisfying (3.15) be a reference profile for system (3.8), (3.9), and let X be the set defined in (3.13). Then, xR∈X, for all  t≥0, if and only if the following relations are fulfilled:
(3.19)x1R=q1v=q1exp(St)v0≠0,∀t≥0,B-1(xR)[(QS-ANQ-AwQ)exp(St)v0-δ]∈Û,∀t≥0,
where q1 denotes the first row of matrix Q, that is, Q=col(q1,q2).Proof.
The proof is immediate using (3.4)-(3.5), Table 1, (3.15), and (3.18) in (3.13).The next result establishes sufficient conditions for a command profilexR in such a way that the dynamic state feedback regulator (2.3) that solves the LROR for system (3.10) also yields robust tracking of xR by the original system (3.6).Theorem 3.4.
LetxR=(x1R,x2R)⊤ satisfying (3.15) be a reference profile for system (3.6) in such a way that
(3.20)x1R=q1v=q1exp(St)v0≠0,∀t≥0,B-1(xR)[(QS-ANQ)exp(St)v0-δ]∈Û,∀t≥0,
where q1 denotes the first row of matrix Q. Let also (2.3) be a dynamic state feedback controller that solves the LROR problem for system (3.10). Then, there exist open subsets X0⊂ℝ2, Z0⊂ℝ2r and W0∈ℝ, with 0∈W0, such that, for all (x(0),z(0))∈X0×Z0 and for all  w∈W0, the controller (2.3) and (3.7), produces
(3.21)limt→∞x(t)=xR(t)
in system (3.6), with x(t)∈X, for all t≥0, X being the region defined in (3.13).Proof.
As, by hypothesis, (2.3) solves the LROR problem for (3.10), Definition 2.1 ensures the existence of a neighborhood W⊂ℝ of the origin w=0 where(i)
σ(Acw)⊂ℂ-,  for all w∈W, Acw being the matrix defined in (2.5), which now is (see (3.5)–(3.11))
(3.22)Acw=(Aw+H1H2NΦ);(ii)
for allx(0),z(0) and for all w∈W, the trajectories of (3.10) satisfy limt→∞x(t)=xR(t).
Assume thatxR satisfies (3.20). It then follows by continuity that there exists an open subset Ŵ⊂ℝ, containing the origin, such that xR also satisfies (3.19), for all w∈Ŵ; moreover, W∩Ŵ is trivially nonempty and open. Continuity also guarantees the existence of three open subsets, X0∈ℝ2, with Qv0∈X0, Z0∈ℝ2r, with Σ0v0∈Z0, Σ0 satisfying (3.16) for w=0, and W0⊆W∩Ŵ, such that, for all (x(0),z(0))∈X0×Z0 and  for all w∈W0, it results that x(t)∈X, for all t≥0.The size of the open setW0⊆W∩Ŵ⊆(-∞,μN) (recall from (3.3) that w∈(-∞,μN)) of possible parametric uncertainties that can be accommodated by the control system (3.6), (3.7), (2.3) during the tracking task of a certain reference xR satisfying (3.9), is studied below.On the one hand,W depends on the features of the regulator (2.3). Indeed, W coincides with the set where the matrix Acw is exponentially stable [6], that is,(3.23)W={w∈(-∞,μN);σ(Acw)⊂ℂ-},
with Acw defined in (3.22). The next section contains a design procedure for (2.3) in such a way that a necessary condition for having W=(-∞,μN) is fulfilled; furthermore, this condition is also sufficient for regulation (xR constant, i.e., S=0.) tasks.On the other hand, the size ofŴ can be tuned at will under certain restrictions. Indeed, assuming that the perturbed parameter w belongs to a known, closed interval [wm,wM], with wm≤0≤wM, and also that x2R satisfies mild hypotheses that include periodicity, a technique based on semi-infinite programming methods developed in [9] allows the obtention of an also periodic reference x1R for x1, with minimum Root Mean Square, in such a way that (3.19) are verified for all w∈[wm,wM], that is, Ŵ=[wm,wM]. Then, if the dynamic state feedback regulator (2.3) is also designed in order to have W=(-∞,μN), then W∩Ŵ=Ŵ.Finally, assume thatxR is such that (3.19) are satisfied for all w∈W∩Ŵ. The set W0⊆W∩Ŵ is strongly dependent on the initial conditions x(0), z(0) and the distance between the actual value of w and w=0. A good selection for x(0) is x(0)~xR(0)=Qv0. However, zR(t) depends on w (see (3.16)); thus, the setting z(0)~Σ0v0 makes W0 contain the values w for which the distance ∥(Σw-Σ0)v0∥ is small enough. Otherwise, assuming that w∈[wm,wM], alternative assignments such as z(0)~Σw¯v0, w¯ being a certain value of the interval [wm,wM], should be considered (see Remark 2.3).


## 4. Construction of a Dynamic State Feedback Regulator

The construction of a dynamic state feedback regulator (2.3) for system (3.10) is carried out using the results of Section 3.Assume thatσ(S)⊂ℂ+¯. Let mpS(λ) be the minimal polynomial of S written as in (2.8), with degmpS(λ)=r, and consider the matrices Φi and Ni defined in (2.9). Therefore, using (2.10), let(4.1)Φ=diag(Φ1,Φ2)∈M2r×2r(ℝ),N=diag(N1,N2)∈M2r×2(ℝ).
It was already commented in Section 2 that, with this selection of Φ and N, the pair (A¯0,B¯0) defined in (2.11) is stabilizable. Hence, H=(H1H2)∈M2×(2+2r)(ℝ) can be selected in such a way that σ(A¯0+B¯0H)⊂ℂ-. However, the situation for system (3.10) is even better, because the corresponding pair is controllable, as stated in the next result.Proposition 4.1.
Consider the matricesΦ, N, defined in (4.1). Then, the following pair is controllable:
(4.2)A¯0=(0202×2rNΦ),B¯0=(𝕀202r×2).Proof.
Notice that(4.3)A¯0kB¯0=(02Φk-1N),∀k≥1.
Therefore, the controllability matrix 𝒞(A¯0,B¯0)=(B¯0,A¯0B¯0,…,A¯02r+1B¯0) is such that
(4.4)rank𝒞(A¯0,B¯0)=rank(𝕀20202⋯0202r×2NΦN⋯Φ2rN)=rank𝕀2+rank[𝒞(Φ,N),Φ2rN]=2+2r,
because of the fact that (Φ,N) is controllable by construction.Then, by Proposition4.1, there exists H=(H1H2) that allows an arbitrary placement of the poles of the closed-loop system A¯+B¯H. The regulator is therefore ensured to be robust for all w∈W, W being the set defined in (3.23).As discussed in Section3, it is of obvious interest to place the poles of the unperturbed system in such a way that W=(-∞,μN). The design procedure suggested below, besides guaranteeing robustness in an open neighborhood of w=0, gives a general necessary condition for having W=(-∞,μN). However, arbitrary pole-placement must be replaced by stable pole-placement. This condition appears to be sufficient in the lowest dimensional case S=0 (i.e., r=1), that is, for regulation purposes. Other cases may demand further analysis of the resulting Acw in order to establish the region W where robustness is preserved.Hence, consider the perturbed system associated to (4.2): (4.5)(ẋż)=A¯w(xz)+B¯wu,
with (4.6)A¯w=(Aw02×2rNΦ),B¯w=B¯0=(𝕀202r×2),
and A¯w(w=0)=A¯0. The change of variables (4.7)x¯1=(x¯11x¯12⋮x¯1,r+1)=(x1z1⋮zR),x¯2=(x¯21x¯22⋮x¯2,r+1)=(x2zr+1⋮z2R)
transforms system (4.5) into the block-diagonal form: (4.8)(x¯̇1x¯̇2)=(A¯10r+10r+1A¯2)(x¯1x¯2)+(B¯10(r+1)×10(r+1)×1B¯2)(u1u2),
where(4.9)A¯1=(001×rN1Φ1),A¯2=(w01×rN2Φ2),B¯1=B¯2=(10r×1).Since the disturbance free pair(A¯1,B¯1) is trivially controllable, its poles can be arbitrarily placed by means of appropriate feedback and are not affected by the perturbation. Thence, let us denoted by(4.10)pA¯1(λ)=-λpΦ1(λ)=-λmpS(λ)=-(λr+1+ar-1λr+⋯+a0λ)
the characteristic polynomial of A¯1. It is well-known that (A¯1,B¯1) achieves the controllable canonical form:(4.11)Ã1=(010⋯0001⋯0⋮⋮⋮⋮⋮000⋯10-a0-a1⋯-ar-1),B̃1=(00⋮01)
on the base of ℝn+1 defined by the column vectors of the matrix(4.12)T=col(A¯1rB¯1+ar-1A¯1r-1B¯1+⋯+a0B¯1,…,A¯1B¯1+ar-1B¯1,B¯1).Let us now assume that the feedback subsystem is in canonical form(4.13)x̃̇1=Ã1x̃1+B̃1u1,u1=H̃1x̃1,
where x̃1=T-1x¯1, is expected to possess a spectrum such as σ(Ã1+B̃1H̃1)={λ11,…,λ1,r+1}⊂ℂ-, and let {α10,…,α1r}⊂ℝ+ be the coefficients of the corresponding characteristic polynomial:(4.14)(λ-λ11)(λ-λ12)⋯(λ-λ1,r+1)=λr+1+α1rλr+⋯+α10.Proposition 4.2.
LetH̃1=(h̃11⋯h̃1,r+1), h1i∈ℝ, for all i∈{1,…,r+1}, be a feedback matrix for system (4.13). If the gains are selected as
(4.15)h̃11=-α10,h̃1k=-α1,k-1+ak-2,k=2,…,r+1,
then the characteristic polynomial of Ã1+B̃1H̃1 coincides with (4.14), which makes the system robust for all w∈(-∞,μN).Proof.
It is immediate from (4.11) that the characteristic polynomial of Ã1+B̃1H̃1 is
(4.16)p1(λ)=λr+1+(ar-1-h̃1,r+1)λr+⋯+(a0-h̃12)λ-h̃11.
The substitution of (4.15) in (4.16) yields the result.The final step should be the transformation ofH̃1 into the original x¯1-base:(4.17)Hx¯1=H̃1T-1.Proposition 4.3.
Let(A¯2,B¯2) be defined from (4.9). The base transformation with associated matrix T introduced in (4.12) reduces the pair to the canonical form (Ã2,B̃2), with
(4.18)Ã2=(01⋯000⋯0⋮⋮⋮⋮00⋯1wa0(wa1-a0)⋯(w-ar-1)),B̃2=(00⋮01).Proof.
Taking into account (4.9), the linear system associated to the pair (A¯2,B¯2) may be written as
(4.19)x¯̇2=A¯2x¯2+B¯2u¯2=A¯1x¯2+(wx¯210r×1)+(10r×1)u2.
Assigning
(4.20)u¯2=wx¯21+u2,
Equation (4.19) takes the form: x¯̇2=A¯1x¯2+B¯2u¯2. Performing the base transformation x̃2=T-1x¯2 and recalling from (4.9) that B¯2=B¯1, one gets that
(4.21)x̃̇2=Ã1x̃2+B̃2u¯2,
with x̃2=T-1x¯2, B̃2=B̃1 and Ã1, B̃1 being described in (4.11). Observe now that the state vector component x¯21 may be expressed as x¯21=T1x̃2=T11x̃21+⋯+T1,r+1x̃2,r+1, where T1=(T11,…,T1,r+1) stands for the first row of T; denoting the Kronecker product by ⊗, the reversion of the change (4.20) in (4.21) results in
(4.22)x̃̇2=Ã2x̃2+B̃2u2,withÃ2=T-1A¯2T=Ã1+wB̃2⊗T1.
Therefore, Ã2 is a matrix in controllable canonical form, and its characteristic polynomial coincides with that of A¯2 due to the invariance property under base transformations:
(4.23)pÃ2(λ)=pA¯2(λ)=(w-λ)pΦ2(λ)=-[λr+1+(ar-1-w)λr++(ar-2-war-1)λr-1+⋯+(a0-wa1)λ-wa0].
Hence, the result follows.Consider the canonical feedback subsystem:(4.24)x̃̇2=Ã2x̃2+B̃2u2,u2=H̃2x̃2.
On the one hand notice that since Ã2|w=0=Ã1, an assignment of feedback gains equivalent to (4.15) guarantees robustness at least in a certain neighborhood of w=0. On the other hand, it is well-known that a necessary condition for the stability of a polynomial is that all its coefficients have the same sign, which is also sufficient for polynomials of degree 2.Therefore, assume that (4.24) is expected to possess the following spectrum for w=0: σ(Ã2|w=0+B̃2H̃2)={λ21,…,λ2,r+1}⊂ℂ-, and let {α20,…,α2r}⊂ℝ+ be the coefficients of the corresponding characteristic polynomial:(4.25)(λ-λ21)(λ-λ22)⋯(λ-λ2,r+1)=λr+1+α2rλr+⋯+α20.Proposition 4.4.
LetH̃2 = (h̃21⋯h̃2,r+1), h̃2i∈ℝ,foralli∈{1,…,r+1}, be a feedback matrix for system (4.24).(i)
If the gains are selected as(4.26)h̃21=-α20,h̃2k=-α2,k-1+ak-2,k=2,…,r+1,
then the characteristic polynomial of Ã2+B̃2H̃2 coincides with (4.25) for w=0, which makes the subsystem robust in a neighbourhood of w=0.(ii)
Moreover, if(4.27)α2k>ϵk+μN|ak|,k=0,…,r-1,α2r>μN,
with ϵk>0, for all k=0,…,r-1, then all the coefficients of the characteristic polynomial of the feedback system Ã2+B̃2H̃2 are positive, for all w∈(-∞,μN); furthermore, if r=1 then the system is stable for all w∈(-∞,μN).Proof.
It is straightforward from (4.18) that the characteristic polynomial of Ã2+B̃2H̃2 is
(4.28)p2(λ)=λr+1+(ar-1-w-h̃2,r+1)λr+⋯+(a0-wa1-h̃22)λ+(wa0+h̃21).(i)
The replacement (4.26) in (4.28) yields
(4.29)p2(λ)=λr+1+(α2r-w)λr+⋯+(α21-wa1)λ+(α20-wa0),
from which issue (i) follows immediately.(ii)
The assumptionσ(S)⊂ℂ+¯ entails ak∈ℝ, for all k=0,…,r-1 and, since ϵk>0,  for all k=0,…,r-1 by hypothesis, (4.27) results in the coefficients of (4.29) satisfying
(4.30)α2k-wak>ϵk+(μN-w)|ai|>0,k=0,…,r-1,α2r-w>μN-w>0,
for allw∈(-∞,μN). The statement for the case r=1 is therefore trivial.Remark 4.5.
Propositions4.2 and 4.4 allow to conclude the following.(i)
The assignments (4.15) and (4.26) yield robustness in a neighborhood of w=0. Indeed, the actual set W is given by W={w<μN;σ[p2(λ)]⊂ℂ-}, with p2(λ) defined in (4.29).(ii)
The assignments (4.15) and (4.26) and the restriction (4.27) constitute a necessary condition for having W=(-∞,μN). In case that S=0, that is, r=1, the condition becomes necessary and sufficient.Once the feedback matrixH̃2 has been constructed following either (4.26) or (4.26)-(4.27), the transformation into the x¯2-base is to be carried out:(4.31)Hx¯2=H̃2T-1.
Eventually, the dynamic state feedback control law (2.3) is now completely determined, with Φ, N selected as indicated in (4.1) and(4.32)(H1H2)=(Hx¯101×(r+1)01×(r+1)Hx¯2)M-1,M-1 being the matrix associated to the change of variables defined in (4.7) for system (4.5):(4.33)(x¯1x¯2)=M-1(xz).


## 5. Example: Output Voltage Regulation in a Noninverting Buck-Boost Converter

In this section we address the robust regulation of the output voltagex2 of a NIBB, described by system (3.6) and (3.7) and Table 1, to a constant level x2R∈ℝ is addressed. Firstly, dynamic state feedback regulators are constructed following Section 4. Later, restrictions arising from control saturation as discussed in Section 3 are considered and, for a certain command profile x2R, a reference x1R for the input current is selected in such a way that (3.19) are fulfilled for all t≥0 and for all w∈Ŵ=[wm,wM]⊂(-∞,μN), where wm, wM are, respectively, lower and upper bounds for the uncertain parameter, with wm≤0≤wM.Hence, assume that the control goal is the robust regulation of the state variablex to a constant level x=xR. Then, S=0 and, being r=degmpS(λ)=1, (4.1) indicates that Φ=02, N=𝕀2, which results in the dynamic state feedback regulator (2.3) appearing as(5.1)ż=e,u=H1x+H2z.
The construction of matrices H1 and H2 is made from the pole-placement design method provided by Propositions 4.2 and 4.4. Notice that now the closed-loop system (3.22) is(5.2)Acw=(Aw+H1H2𝕀202),
where Aw is defined in (3.5). Therefore, let us assign complex conjugated, stable poles:(5.3)λ11,12=λ21,22=λ1,2;λ¯2=λ1∧Re(λ1)<0,
for subsystems (4.13) and (4.24). Then, (4.15) and (4.26) yield(5.4)H̃1=H̃2=(-|λ1|2-2|Re(λ1)|),
and the base transformation matrices T, M-1 being (see (4.12) and (4.33), resp.)(5.5)T=(0110),M-1=(1000001001000001).
equations (4.17), (4.31), and (4.32) provide(5.6)(H1⋮H2)=(-2|Re(λ1)|0⋮-|λ1|200-2|Re(λ1)|⋮0-|λ1|2).Remark 5.1.
According to Remark4.5(i)
(5.7)W={w<μN;σ[λ2+(2|Re(λ1)|-w)λ+|λ1|2]⊂ℂ-},
with W=(-∞,2|Re(λ1)|)⊂(-∞,μN) in case that 2|Re(λ1)|<μN. Otherwise, the assumption 2|Re(λ1)|≥μN results in the fulfillment of (4.27), this yielding W=(-∞,μN) in accordance to Remark 4.5(ii).On the other hand, letv0=1, Q=(q1,q2)⊤∈ℝ2 in (3.9), which yield x1R=q1, x2R=q2. Next result provides a selection criteria for x1R which is shown to be sufficient for the fulfillment of (3.19).Proposition 5.2.
Let the NIBB converter, described by (3.6), (3.7) and Table 1, be regulated by (5.1), with (H1∣H2) defined in (5.6). Let also W be the set defined in (5.7). Assume that the output voltage x2 is expected to attain a certain reference level x2R=q2∈ℝ∖{0}, while the uncertain parameter w belongs to the set with known bounds [wm,wM], wm≤0≤wM<μN. If the reference x1R=q1∈ℝ+ is such that
(5.8)q1>(μN+|wm|)max{|q2|,q22},
then one has the following:(i)
Restrictions (3.19) are fulfilled for all t≥0, for all w∈Ŵ=[wm,wM].(ii)
If2|Re(λ1)|>wM, then W∩Ŵ=Ŵ; furthermore, there exist open sets X0, Z0 and W0, with W0⊆Ŵ and 0∈W0, in such a way that, for all (x(0),z(0))∈X0×Z0, and for all w∈W0, the regulated system (3.6), (3.7), (5.1), (5.6) is able to accommodate any disturbance w∈[wm,wM] and, at the same time, maintain the system trajectories evolving inside the state-space region X defined from (3.13), for all t≥0.Proof.
(i) The statement follows from the fact that, in this case, (3.19) answers to(5.9)q1≠0,0<(μN-w)q22q1<1,0<(μN-w)q2q1<1.
(ii) As2|Re(λ1)|>wM by hypothesis, it follows from Remark 5.1 and item (i) that Ŵ⊂W, this yielding W∩Ŵ=Ŵ. Then, the result follows from Theorem 3.4.Remark 5.3.
It is worth mentioning that the procedure described in [9] for the obtention of an input current reference x1R when both x1R, x2R are assumed to be constant, yields the same result as that of Proposition 5.2.Finally, recall that the stationary valuesxR, zR, obtained following (3.15), (3.16), are:(5.10)xR=(q1q2),zR=Σw=-H2-1(Aw+H1)(q1q2),
where S=0, v0=1 and Φ=02 has been taken into account.


## 6. Simulation Results

The parameters selected for the NIBB converter areVg=40V, L=0.001H, C=0.00006F, RN=10Ω, and it is expected to suffer an additive load disturbance at t=0 that may vary the nominal value RN in the range -50% to +100%, that is, admissible values for Rw and R are: Rw∈[Rwm,RwM]=[-5Ω,10Ω] and R∈[Rm,RM]=[5Ω,20Ω]. These settings, translated to normalized variables, result in (see (3.3)) μN=0.4082, w∈[wm,wM]=[-0.4082,0.2041] and μ∈[0.1361,0.8165]. Using λ1,2=-1/4±(1/4)i in (5.3), the regulator obtained from (5.6) is(6.1)(H1⋮H2)=(-120⋮-1800-12⋮0-18).
Then, as 1/2=2|Re(λ1)|>μN=0.4082, it follows from (5.7) and Remark 4.5(ii) that W=(-∞,μN)=(-∞,0.4082)⊃[-0.4082,0.2041]=[wm,wM].Let us now assign references for the state variables:x1R=5, x2R=2, corresponding to iL=48.9898A, vC=80 V, respectively. It can be immediately checked that this selection guarantees the fulfillment of restriction (5.8) in Proposition 5.2: 5=q1>(μN+|wm|)max{|q2|,q22}=3.2660 and, therefore, Ŵ=[-0.4082,0.2041]⊂W. Thus, considering that Ŵ=W∩Ŵ=[-0.4082,0.2041]=[wm,wM], Proposition 5.2(ii), ensures the existence of open sets W0⊂W∩Ŵ, X0, Z0, with 0∈W0, in such a way that, for all (x(0),z(0))∈X0×Z0, the regulated system (3.6), (3.7), (5.1) is able to accommodate any disturbance w∈W0 and, at the same time, maintain the system trajectories evolving in the unsaturated region of the state-space region X defined from (3.13),  for all t≥0.Finally, notice that the steady-state values for the state variablesx, z are (see (5.10)): xR=(5,2)⊤, zR=Σw=-(20,8-16w)⊤.Ideal simulations have been carried out with a SIMULINK model of the system for the casesRw=-5Ω and Rw=10Ω. The control signal û is considered continuous. The duration of the simulations is 204.1241 time units, corresponding to 0.05 s. In both cases the selected initial conditions have been: x(0)=(4,2)⊤, z(0)=Σ0=-(20,8)⊤.Figures1, 2, and 3 depict, respectively the state variables x1andx2, the control actions û1,û2, and the auxiliary variables z1andz2 for the case Rw=-5Ω: the results show an excellent agreement with the theory developed in Section 5, which is confirmed by plots in Figures 4, 5, and 6, corresponding to Rw=10Ω.Figure 1
Rw=-5Ω: inductor current (x1) and output voltage (x2).Figure 2
Rw=-5Ω: equivalent controls û1 and û2.Figure 3
Rw=-5Ω: auxiliary variables z1 and z2.Figure 4
Rw=10Ω: inductor current (x1) and output voltage (x2).Figure 5
Rw=10Ω: equivalent controls û1 and û2.Figure 6
Rw=10Ω: auxiliary variables z1 and z2.The section closes with realistic simulations that use the software package PSIM. Besides the converter's main parameters indicated above, the model incorporates internal resistances of10mΩ for the inductor and the capacitor. Each switch is implemented by means of an IGBT with a saturation voltage of 2V and a power diode with a voltage drop of 0.5V. The implementation of the control law uses PWM with a frequency of 50kHz.The current and voltage references have been set to49A and 80V, respectively, and null initial conditions have been selected for both the state and the auxiliary variables. The total duration of the simulations is 9 ms. Figure 7 depicts the inductor current and the output voltage reaching their reference levels under an actual output load of R=5Ω instead of the nominal RN=10Ω, that is, with a disturbance Rw=-5Ω. In turn, Figure 8 plots an equivalent situation for R=20Ω, that is, for Rw=10Ω. The results confirm the validity of the proposed approach.Figure 7
Rw=-5Ω: output voltage vC (in Volts) and inductor current iL (in Ampères).Figure 8
Rw=10Ω: output voltage vC (in Volts) and inductor current iL (in Ampères).


## 7. Conclusions

The robust output regulation problem for a family of nonlinear switched power converters that includes the NonInverting Buck-Boost, the Full-Bridge NonInverting Buck-Boost, the Watkins-Johnson and the Inverse of Watkins-Johnson has been addressed. Linear techniques, available after a transformation of the control variable, render an efficient solution of the problem. The methodology employs a dynamic state feedback control law and considers resistive loads with load resistance uncertainty. Restrictions due to fixed values of the control gains are considered. The proposed technique is successfully tested via realistic numerical simulations of the robust output voltage regulation in a NonInverting Buck-Boost converter.Further research should explore the possibility of using state feedback linearization plus linear robust output regulation techniques in converters with a single control switch, such as the boost or the buck-boost.



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*Source: 724531-2010-07-27.xml*