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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 724531, 22 pages
Linear Robust Output Regulation in a Class of Switched Power Converters
1Department of Applied Mathematics IV, Technical University of Catalonia, 08800 Vilanova, i la Geltr####^~^~^~^~^~^####xfa;, Spain
2Institute of Industrial and Control Engineering, Technical University of Catalonia, 08028 Barcelona, Spain
Received 19 November 2009; Accepted 14 June 2010
Academic Editor: John Burns
Copyright © 2010 Josep M. Olm and Domingo Biel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The output regulation####^~^~^~^~^~^####x2014;or servomechanism####^~^~^~^~^~^####x2014;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-known internal model principle ; there is also an interesting algebraic approach contained in . The extension of the solution to nonlinear systems appeared almost two decades later in the celebrated paper . The reader is referred to  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 , 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. Section 2 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 .
The Output Regulation Theory addresses the problem of rendering the output of a linear control system, possibly with plant uncertainties , to asymptotically track any reference belonging to a given family and, at the same time, reject asymptotically any disturbance 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 error , there is no need to separate the roles of and , because both may be seen as components of an exogenous input that has to be rejected.
Therefore, consider the system with and . The exogenous input is assumed to satisfy the exosystem while stands for the plant uncertainty vector. , and are real matrices with appropriate dimensions, whose coefficients depend on the plant uncertainties, their nominal values matrices being , and . Notice, finally, that in (2.1) it is assumed that , and .
Let the dynamic state feedback law with , , , , and 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 Denote also by the matrix of the unforced closed-loop system, being its nominal value matrix.
Within this framework, the LROR problem may be posed as follows.
Definition 2.1. Let be an open subset of that contains the origin . The LROR problem of (2.1) with exosystem (2.2) in , 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 matrix defined from (2.5) is Hurwitz, that is, , where denotes the spectrum of .(ii)The matrix in (2.5) is such that , for all ; furthermore, for all and for all , the trajectories of (2.4) satisfy
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 ). Consider the plant (2.1) with exosystem (2.2). Assume that the pair is stabilizable and also that . Then, the LROR problem is solvable by means of a dynamic state feedback controller (2.3) if and only if
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 . Let be the minimal polynomial of . Then, consider the matrices with , , and define This selection of and , which ensures the controllability of , together with the assumed stabilizability of and the fulfillment of (2.7), yields the stabilizability of the following pair: where and are, respectively, and matrices. Thus, there exists such that
Remark 2.3. It is proved in  that, under the assumption , if a controller (2.3) solves the LROR problem for (2.1) and (2.2), then, for all , there exist unique matrices that satisfy the following matrix equations:
For every , (2.13) indicates that is an invariant manifold for the closed-loop system (2.4), while (2.14) means that the error is zero on the invariant manifold . Furthermore, let be any initial condition for the exosystem; then, the corresponding exogenous input is . If the initial state of the plant in (2.4) is set to , , it is immediate that , , and, subsequently, , for all .
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 current and the capacitor voltage as state variables, and a control variable . It is worth mentioning that the control action in the physical system is carried out by means of switches; hence, and 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: which make the system dimensionless:
The control gains take values in , with . The values of the parameters , 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 for the load resistance, due to the addition of a constant disturbance term to its nominal value ; that is, , with and . Consequently, the parameter may be written as , with being the only uncertain parameter of the system.
Assigning the dynamical system (3.2) may be written as Furthermore, notice that det####^~^~^~^~^~^####x2009; according to the admissible values for , indicated in Table 1. Hence, assuming , the state feedback control law transforms system (3.6) into
Let us now consider the problem of rendering the state of system (3.8) to asymptotically track a certain reference profile , which can be expressed as a linear combination of the solutions of a time-invariant, linear exosystem; that is, there exist real matrices and , of appropriate dimensions, such that Notice that an exosystem as (3.9) with 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
Let and denote, respectively, the identity matrix and the null matrix in the set of matrices. Then, identifying the elements of the original system (2.1) with those of the particular case (3.10) one gets while is defined in (3.5) and, subsequently, .
Proof. The proof is immediate.
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 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 of (3.6) remain entirely inside the state-space region defined as with Notice that the restriction det####^~^~^~^~^~^####x2009; is necessary and sufficient for guaranteeing the diffeomorphic character of the transformation (3.7), while the requirement in ensures nonsaturation of the controller performance. Eventually, this fact entails restrictions on the set of initial values and parametric uncertainties (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 profiles are especially interesting. They follow immediately from the assumption that lies in . Hence, assume that system (3.10) has achieved a steady-state under the action of the dynamic state feedback controller (2.3), and denote , , the corresponding stationary behavior for and ; it follows from Remark 2.3 that , , being the solution of (2.13) and (2.14), and by construction. Moreover, it is straightforward from the assignment (3.5)####^~^~^~^~^~^####x2013;(3.11) that , while is such that Hence, using (3.9), where satisfies (3.14). Furthermore, from (3.8) we obtain an alternative expression for :
Proposition 3.3. Let satisfying (3.15) be a reference profile for system (3.8), (3.9), and let be the set defined in (3.13). Then, , for all , if and only if the following relations are fulfilled: where denotes the first row of matrix , that is, .
The next result establishes sufficient conditions for a command profile 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 by the original system (3.6).
Theorem 3.4. Let satisfying (3.15) be a reference profile for system (3.6) in such a way that where denotes the first row of matrix . Let also (2.3) be a dynamic state feedback controller that solves the LROR problem for system (3.10). Then, there exist open subsets , and , with , such that, for all and for all , the controller (2.3) and (3.7), produces in system (3.6), with , for all , 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 of the origin where(i), for all , being the matrix defined in (2.5), which now is (see (3.5)####^~^~^~^~^~^####x2013;(3.11))
(ii)for all and for all , the trajectories of (3.10) satisfy .
Assume that satisfies (3.20). It then follows by continuity that there exists an open subset , containing the origin, such that also satisfies (3.19), for all ; moreover, is trivially nonempty and open. Continuity also guarantees the existence of three open subsets, , with , , with , satisfying (3.16) for , and , such that, for all and for all , it results that , for all .
The size of the open set (recall from (3.3) that ) 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 satisfying (3.9), is studied below.
On the one hand, depends on the features of the regulator (2.3). Indeed, coincides with the set where the matrix is exponentially stable , that is, with defined in (3.22). The next section contains a design procedure for (2.3) in such a way that a necessary condition for having is fulfilled; furthermore, this condition is also sufficient for regulation ( constant, i.e., .) tasks.
On the other hand, the size of can be tuned at will under certain restrictions. Indeed, assuming that the perturbed parameter belongs to a known, closed interval , with , and also that satisfies mild hypotheses that include periodicity, a technique based on semi-infinite programming methods developed in  allows the obtention of an also periodic reference for , with minimum Root Mean Square, in such a way that (3.19) are verified for all , that is, . Then, if the dynamic state feedback regulator (2.3) is also designed in order to have , then .
Finally, assume that is such that (3.19) are satisfied for all . The set is strongly dependent on the initial conditions , and the distance between the actual value of and . A good selection for is . However, depends on (see (3.16)); thus, the setting makes contain the values for which the distance is small enough. Otherwise, assuming that , alternative assignments such as , being a certain value of the interval , should be considered (see Remark 2.3).
4. Construction of a Dynamic State Feedback Regulator
Assume that . Let be the minimal polynomial of written as in (2.8), with , and consider the matrices and defined in (2.9). Therefore, using (2.10), let It was already commented in Section 2 that, with this selection of and , the pair defined in (2.11) is stabilizable. Hence, can be selected in such a way that . 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 matrices , , defined in (4.1). Then, the following pair is controllable:
Proof. Notice that Therefore, the controllability matrix is such that because of the fact that is controllable by construction.
Then, by Proposition 4.1, there exists that allows an arbitrary placement of the poles of the closed-loop system . The regulator is therefore ensured to be robust for all , being the set defined in (3.23).
As discussed in Section 3, it is of obvious interest to place the poles of the unperturbed system in such a way that . The design procedure suggested below, besides guaranteeing robustness in an open neighborhood of , gives a general necessary condition for having . However, arbitrary pole-placement must be replaced by stable pole-placement. This condition appears to be sufficient in the lowest dimensional case (i.e., ), that is, for regulation purposes. Other cases may demand further analysis of the resulting in order to establish the region where robustness is preserved.
Since the disturbance free pair 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 the characteristic polynomial of . It is well-known that achieves the controllable canonical form: on the base of defined by the column vectors of the matrix
Let us now assume that the feedback subsystem is in canonical form where , is expected to possess a spectrum such as , and let be the coefficients of the corresponding characteristic polynomial:
Proposition 4.2. Let , , for all , be a feedback matrix for system (4.13). If the gains are selected as then the characteristic polynomial of coincides with (4.14), which makes the system robust for all .
The final step should be the transformation of into the original -base:
Proof. Taking into account (4.9), the linear system associated to the pair may be written as Assigning Equation (4.19) takes the form: . Performing the base transformation and recalling from (4.9) that , one gets that with , and , being described in (4.11). Observe now that the state vector component may be expressed as , where stands for the first row of ; denoting the Kronecker product by , the reversion of the change (4.20) in (4.21) results in Therefore, is a matrix in controllable canonical form, and its characteristic polynomial coincides with that of due to the invariance property under base transformations: Hence, the result follows.
Consider the canonical feedback subsystem: On the one hand notice that since , an assignment of feedback gains equivalent to (4.15) guarantees robustness at least in a certain neighborhood of . 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 : , and let be the coefficients of the corresponding characteristic polynomial:
Proposition 4.4. Let = (), , be a feedback matrix for system (4.24).(i)If the gains are selected as then the characteristic polynomial of coincides with (4.25) for , which makes the subsystem robust in a neighbourhood of .(ii)Moreover, if with , for all , then all the coefficients of the characteristic polynomial of the feedback system are positive, for all ; furthermore, if then the system is stable for all .
Proof. It is straightforward from (4.18) that the characteristic polynomial of is (i) The replacement (4.26) in (4.28) yields from which issue (i) follows immediately.(ii) The assumption entails , for all and, since , for all by hypothesis, (4.27) results in the coefficients of (4.29) satisfying for all. The statement for the case is therefore trivial.
Remark 4.5. Propositions 4.2 and 4.4 allow to conclude the following.(i)The assignments (4.15) and (4.26) yield robustness in a neighborhood of . Indeed, the actual set is given by , with defined in (4.29).(ii)The assignments (4.15) and (4.26) and the restriction (4.27) constitute a necessary condition for having . In case that , that is, , the condition becomes necessary and sufficient.
Once the feedback matrix has been constructed following either (4.26) or (4.26)-(4.27), the transformation into the -base is to be carried out: Eventually, the dynamic state feedback control law (2.3) is now completely determined, with , selected as indicated in (4.1) and being the matrix associated to the change of variables defined in (4.7) for system (4.5):
5. Example: Output Voltage Regulation in a Noninverting Buck-Boost Converter
In this section we address the robust regulation of the output voltage of a NIBB, described by system (3.6) and (3.7) and Table 1, to a constant level 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 , a reference for the input current is selected in such a way that (3.19) are fulfilled for all and for all , where , are, respectively, lower and upper bounds for the uncertain parameter, with .
Hence, assume that the control goal is the robust regulation of the state variable to a constant level . Then, and, being , (4.1) indicates that , , which results in the dynamic state feedback regulator (2.3) appearing as The construction of matrices and 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 where is defined in (3.5). Therefore, let us assign complex conjugated, stable poles: for subsystems (4.13) and (4.24). Then, (4.15) and (4.26) yield and the base transformation matrices , being (see (4.12) and (4.33), resp.) equations (4.17), (4.31), and (4.32) provide
Proposition 5.2. Let the NIBB converter, described by (3.6), (3.7) and Table 1, be regulated by (5.1), with defined in (5.6). Let also be the set defined in (5.7). Assume that the output voltage is expected to attain a certain reference level , while the uncertain parameter belongs to the set with known bounds , . If the reference is such that then one has the following:(i)Restrictions (3.19) are fulfilled for all , for all .(ii)If , then ; furthermore, there exist open sets , and , with and , in such a way that, for all , and for all , the regulated system (3.6), (3.7), (5.1), (5.6) is able to accommodate any disturbance and, at the same time, maintain the system trajectories evolving inside the state-space region defined from (3.13), for all .
Proof. (i) The statement follows from the fact that, in this case, (3.19) answers to
####^~^~^~^~^~^####x2009;(ii) As by hypothesis, it follows from Remark 5.1 and item (i) that , this yielding . Then, the result follows from Theorem 3.4.
Remark 5.3. It is worth mentioning that the procedure described in  for the obtention of an input current reference when both , are assumed to be constant, yields the same result as that of Proposition 5.2.
6. Simulation Results
The parameters selected for the NIBB converter are V, H, F, , and it is expected to suffer an additive load disturbance at that may vary the nominal value in the range to , that is, admissible values for and are: and . These settings, translated to normalized variables, result in (see (3.3)) , and . Using in (5.3), the regulator obtained from (5.6) is Then, as , it follows from (5.7) and Remark 4.5(ii) that .
Let us now assign references for the state variables: , , corresponding to , ####^~^~^~^~^~^####x2009;V, respectively. It can be immediately checked that this selection guarantees the fulfillment of restriction (5.8) in Proposition 5.2: and, therefore, . Thus, considering that , Proposition 5.2(ii), ensures the existence of open sets , , , with , in such a way that, for all , the regulated system (3.6), (3.7), (5.1) is able to accommodate any disturbance and, at the same time, maintain the system trajectories evolving in the unsaturated region of the state-space region defined from (3.13), for all .
Finally, notice that the steady-state values for the state variables , are (see (5.10)): , .
Ideal simulations have been carried out with a SIMULINK model of the system for the cases and . The control signal is considered continuous. The duration of the simulations is time units, corresponding to ####^~^~^~^~^~^####x2009;s. In both cases the selected initial conditions have been: , .
Figures 1, 2, and 3 depict, respectively the state variables , the control actions , and the auxiliary variables for the case : 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 .
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 of for the inductor and the capacitor. Each switch is implemented by means of an IGBT with a saturation voltage of V and a power diode with a voltage drop of V. The implementation of the control law uses PWM with a frequency of .
The current and voltage references have been set to and V, respectively, and null initial conditions have been selected for both the state and the auxiliary variables. The total duration of the simulations is ####^~^~^~^~^~^####x2009;ms. Figure 7 depicts the inductor current and the output voltage reaching their reference levels under an actual output load of instead of the nominal , that is, with a disturbance . In turn, Figure 8 plots an equivalent situation for , that is, for . The results confirm the validity of the proposed approach.
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.
The authors thank the anonymous reviewers for their valuable suggestions. This work is partially supported by the spanish Ministerio de Educaci####^~^~^~^~^~^####xf3;n under project DPI2007-62582 (D. Biel and J. M. Olm) and the spanish Ministerio de Ciencia e Innovaci####^~^~^~^~^~^####xf3;n, through the Programa Nacional de Movilidad de Recursos Humanos of the Plan Nacional de I-D+i 2008-2011 (J. M. Olm).
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