LMI Robust Control of PWM Converters: An Output-Feedback Approach 17

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics.


Introduction
This chapter proposes a systematic approach for the synthesis of robust controllers for dc-dc converters. The approach is based on the Linear Matrix Inequalities (LMIs) framework and the associated optimization algorithms. The aim of this approach is to allow the designer to describe the uncertainty of the converter and to deal with the requirements of the application beforehand. The aforementioned dc-dc converters (see Figure 1) are devices that deliver a dc output voltage, with different properties from those in the input voltage (Erickson & Maksimovic, 1999). They are usually employed to adapt energy sources to the load requirements (or vice versa). These devices present several challenges regarding their robust control. First, the converter must maintain a tight regulation or tracking of the output. Moreover, the controller design is focused on maximizing the bandwidth of the closed-loop response in order to reject the usual disturbances that appear in these systems. Finally, the response of the converter must satisfy desirable transient characteristics, as for example, the shortest possible output settling time or the minimum overshoot. Besides of these common requirements, the converter can be affected by uncertainty in its components or by input or output disturbances that may appear.  Nevertheless, most of the modeling approaches in the literature disregard these uncertainties. Moreover, due to the switching nature of the system, pulse-width modulation (PWM) is commonly used in the industry applications, while the models that are usually employed disregard that part of the dynamics (i.e. the high frequency dynamics) and other inherent nonlinearities, such as saturations and bilinear terms. The chapter proposes a systematic approach to deal with these challenges, using the concepts of LMI control (Ben-Tal et al., 2009;Bernussou, 1996;Boyd et al., 1994;El Ghaoui & Niculescu, 2000;Pyatnitskii & Skorodinskii, 1982). Linear matrix inequalities have become an important topic in the field of Automatic Control due to the following facts. First of all, LMIs can be solved numerically by efficient computer algorithms (Gahinet et al., 1995;Löfberg, 2004;Sturm, 1999). Secondly, more and more methods have been developed to describe control problems in terms of LMI constraints. Finally, these methods are able to include descriptions of the uncertainty. Some of the previous literature on LMI control of dc-dc converters are (Montagner et al., 2005;Olalla et al., 2009a;2010a). In these papers, the uncertainty of the converter is taken into account and the control synthesis deals with different operating points. Nevertheless, they do not consider the stability of the system trajectories when the system changes from one operating point to another, nor they include other nonlinearities such as saturations. The versatility of LMI control has allowed to deal with some of these nonlinearities (Olalla et al., 2009b), . These approaches share the same feedback scheme, which is based upon state-feedback with error integration (Figure 2(a)). The main advantage of this approach is that the synthesis optimization problem can be posed as a convex semidefinite programming and that the implementation of the controller is simple. On the other hand, state-feedback requires sensing of the state variables, which may not be easily measurable or may require estimation in some cases. In practice, most of the designs that can be found in the power electronics literature employ output-feedback approaches since they usually rely on frequency-based concepts which are well-known by electrical engineers. This is the reason why this chapter focuses on LMI-based synthesis methods which may be applicable to the output-feedback scheme (Figure 2(b)), with the aim to derive robust controllers for dc-dc converters. In order to introduce such synthesis methods, the chapter is organized as follows. The first section deals with modeling of dc-dc converters, the averaging method, the sampling effect of the pulse-width modulator and the uncertainty. Section II reviews some of the results of previous works on LMI synthesis for state-feedback approaches. Section III puts forward the problem of output-feedback and some of the strategies that can be employed to pose such problem in terms of semidefinite programming. Concretely, Section III proposes the  following strategies. First, the classic dynamic output-feedback control problem is treated. This approach can be carried out with a change of variables as in (Scherer et al., 1997). However, with such an approach the uncertainty must be modeled with elaborated models, as for example, weighting functions Wallis & Tymerski (2000). Therefore, the chapter also proposes the synthesis of output-feedback controllers based on the static case. Both the static output-feedback and a parametrization (Peaucelle & Arzelier, 2001b) to deal with dynamic output-feedback are considered. The advantages and drawbacks of the three approaches shown in the chapter will be discussed and the results will be compared.

Notation
For symmetric matrices A and B, A > B means that A − B is positive definite. A denotes that the matrix A is an unknown variable. A ′ denotes the transpose of A.C o v j , j = 1,...,N denotes the convex hull defined by N vertices v j ∈ R n . The identity matrix of order n is noted as 1 n and the null n × m matrix is noted as 0 0 n,m .T h es y m b o l⋆ denotes symmetric blocks in partitioned matrices.

Modeling of uncertain dc-dc converters
This subsection shows the state-space averaged models of the buck and the buck-boost converters of Figures 3(a) and 3(b). The models are assumed to operate in Continuous Conduction Mode (CCM), i.e. the inductor current is always larger than zero. Besides of the averaged models, this section also introduces a model of the sampling effect caused by the PWM. Finally, at the end of the section, the uncertainty modeling of dc-dc converters is discussed and a simple example is shown.

Model of the buck converter
The first model that is introduced considers a buck converter, which is characterized by linear averaged control-to-output dynamics. As stated in (Olalla et al., 2010b), the transfer functions of dc-dc converters can strongly depend on the stray resistances of the converter. Since the chapter considers different output-feedback synthesis approaches, these stray resistances are considered in the models. Figure 3(a) shows the circuit diagram of a dc-dc buck converter where v o (t) is the output voltage, v g (t) is the line voltage and i load (t) is the load disturbance. The output voltage must be kept at a given reference V ref . The converter load is modeled as a linear resistor R.Th est ra y resistances of the switch during the on and the off position are combined with the resistance 319 LMI Robust Control of PWM Converters: An Output-Feedback Approach www.intechopen.com of the inductor and noted as: The measurable states of the converter are noted as x a (t). Note that the time dependence of the variables may be omitted to simplify the notation. The binary signal u b (t), which turns on and off the switches, is genereated by means of a Pulse Width Modulation (PWM) subcircuit, working at a constant frequency 1/T s .T h e switching period T s is equal to the sum of t on and t off . For a unit-amplitude sawtooth PWM, the duty-cycle d(t)=t on /(t on + t off ) is the control input of the converter. As shown in (Erickson & Maksimovic, 1999) and (Leyva et al., 2006), considering that the state-space matrices of the converter are [ A on , B on ]d u r i n gt on and [ A off , B off ]d u r i n gt off , the general state-space averaged model of a dc-dc converter can be written as: where the equilibrium (noted with capital letters) and the incremental vectors (noted with tildes) are as follows. X andx ∈ R n correspond to the state vectors, D andd ∈ R m are the control inputs, while W andw ∈ R l stand for the disturbance inputs.
In the buck converter, A on = A off , and the averaged model (2) can be rewritten as: where: being r eq = Dr on + D ′ r off and D ′ = 1 − D. The dimensions of the system matrices are defined Similarly, the averaged outputs of the buck converter can be written as: where in a general case C y ∈ R q×n , E yw ∈ R q×m . Considering the load voltage v o (t) as the only output, these matrices are written as:

Model of the buck-boost converter
In the buck-boost converter, matrices A on and A off are not equal, and therefore, the averaged model contains bilinear terms concerning the control input, the states and the disturbance inputs. According to those nonlinear terms, the linearized transfer function depends on the operating point, hence making the control subsystem design more difficult. In order to derive accurate transfer functions of the buck-boost converter for output-feedback approaches, the stray resistances are also taken into account. For the buck-boost converter, the averaged model in the form of (3) contains bilinear terms, and can be expressed as follows: where: being r eq = Dr on + D ′ r off . The dimensions of the system matrices are defined as The averaged output v o (t) of the buck-boost converter can be written as: Fig. 4. Waveforms of the PWM process. where: These models are employed in Section 3 to derive robust controllers for the buck and the buck-boost converters.

Delay model for the PWM actuator
The models presented above do not take into account the sampling effect of the modulation (Brown & Middlebrook, 1981;Erickson & Maksimovic, 1999) (see Figure 4). Usually, the sampling effect is not considered, and only the linear gain of the modulator is taken into account. In a voltage-mode modulator, the duty-cycle input is usually constrained between zero and the amplitude of the sawtooth signal V M , and therefore the linear gain of this modulator is 1/V M (Erickson & Maksimovic, 1999). For simplicity the amplitude V M can be considered equal to one, such that the linear model shown previously is valid for a duty-cycle input d ∈ [0, 1]. However, the sampling effect can be taken into account in order to limit the control-loop bandwidth in the automatic control synthesis algorithms. Such an effect can be incorporated to the power stage model as a sampling at the switching frequency 1/T s and a zero-order hold block, assuming that the switch is fired once every switching cycle T s (Maksimovic, 2000). The equivalent transfer function for this sampling model is then: The exponential factor e −sT s can be approximated by a Padé function: Taking the first order approximation n = 1 we obtain 322 Recent Advances in Robust Control -Theory and Applications in Robotics and Electromechanics

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The equivalent hold transfer function with the Padé approximation writes which is a strictly proper transfer function whose representation in state-space form could be: wherex p (t) is the state variable of the G ZOH (s),d(t) is its input andd 2 (t) is its output.

Modeling of uncertainty
As stated in (Gahinet et al., 1995), the notion of system uncertainty is of major importance in the field of robust control theory. First of all, one of the key features of feedback is that it reduces the effects of uncertainty. However, when designing a control system, the model used to represent the behavior of the plant is often approximated. The difference between the approximated model and the true model is called model uncertainty. Also the changes due to operating conditions, aging effects, etc... are sources of uncertainty.
The two main approaches shown in (Gahinet et al., 1995) when dealing with system uncertainties and LMI control are: • Uncertain state-space models, relevant for systems described by dynamical equations with uncertain and/or time-varying coefficients.
• Linear-fractional representation (LFR) of uncertainty, in which the uncertain system is described as an interconnection of known LTI systems.
While LFR models have had a main role in modern robust control synthesis methods such as in µ-synthesis (Zhou et al., 1996), state-space models have been used in convex optimization approaches (Boyd et al., 1994). Since this chapter presents approaches that do not employ the concept of structured singular value on which the µ-synthesis method is based, the following subsection is focused on uncertain state-space models. If some of the physical parameters are approximated or unknown, or if there exists nonlinear or non-modeled dynamic effects, then the system can be described by an uncertain state-space model: where the state-space matrices A, B, C, D depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. One of the state-space representations of relevance in LMI control problems is the class of polytopic models:

in which the matrix G(t)= A(t) B(t) C(t) D(t) varies within a fixed polytope of matrices
where G 1 ,...,G N are the vertices of the polytope.
In other words, G(t) is a convex combination of the matrices G 1 ,...,G N . Polytopics models are also called linear differential inclusions LDI in (Boyd et al., 1994).

Example: Buck converter polytopic model
Consider the buck converter model introduced in subsection 2.1, withw(t)=0. For simplicity, the stray resistances are disregarded. If we take R and V g as uncertain parameters of the converter, the uncertain system is described as follows The uncertain matrices A j and B uj are where R j = {R min R max R min R max },andV g j = {V g min V g min V g max V g max }, which represents a uncertain polytope of four vertices (2 power the number of uncertain parameters, that are R j and V g j in this example).

Robust control of dc-dc converters
Consider a general LTI model with states x(t), controlled outputs y(t) and performance outputs z(t): It is possible to assume that some elements involved in the system matrices are uncertain or time-varying. For the sake of simplicity, the performance and measurable outputs are discarded, hence these uncertain elements are concentrated in matrices A, B w and B u and they are grouped in a vector p. Thus, matrices A, B w and B u depend on such uncertainty vector, and we can express (21) as function of these parameters: Fig. 5. Block diagram of a state-feedback system with controller K and error integration.
This state-space representation has been previously used to derive robust control synthesis methods for dc-dc converters, which generally result in a state-feedback law that stabilizes the system for a certain range of uncertainty: parameter-dependent approaches for the linear dynamics of the converters are presented in (Montagner et al., 2005) and (Torres-Pinzon & Leyva, 2009) while (Hu, 2011) introduces a representation of the nonlinear dynamics. Consistent experimental results with tight performances are presented in (Olalla et al., 2009a;2010a;2011). The small-signal stabilization of nonlinear dc-dc converters is considered in (Olalla et al., 2009a;2010a), where the converter is ensured to be stable in a range of operating points, but its trajectory between those points is not ensured to be stable due to the disregard of the nonlinear dynamics. These nonlinearities are taken into account in  where also a less conservative polytopic uncertainty model is introduced. The state-feedback formulation of the control problem is of interest since (i) it may deliver better performance than some output-feedback approaches, (ii) it can be posed as a convex optimization problem with no conservatism or iterations and (iii) it is very simple to implement. However, the main disadvantage of state-feedback is that the full state vector must be available for measure, which is not always true. Therefore, it may require additional components and sensors to obtain the state or to implement estimators of the unaccessible states. Robust output-feedback approaches are then an alternative to derive robust controllers with known performances. Robust control via output-feedback has been the subject of extensive research in the field of automatic control (de Oliveira & Geromel, 1997;Garcia et al., 2004;Peaucelle & Arzelier, 2001a;b;Scherer et al., 1997;Skogestad & Postlethwaite, 1996), but it has been hardly employed in dc-dc converters (Rodriguez et al., 1999). Power electronics engineers tend to use current-mode approaches (Erickson & Maksimovic, 1999) that employ an inner current loop before applying the output-feedback loop and, in that way, ease the control of the dc-dc converter. However, current-mode approaches require current sensing, as state-feedback control, and they suffer from noise, since in some cases, as in peak-current control, the current waveform must be sensed accurately. Therefore, a plain output-feedback approach can be of interest in certain cases in which a simple control is required and the sensing of all the states of the converter is not possible.

State-feedback control
The most simple control problem in terms of an LMI formulation is the one in which all the system states are measurable. The state-feedback problem considers the stabilization of (22) with a simple controller u = Kx,whereK ∈ R m×n , as followṡ Since the state-feedback approach does not allow to eliminate steady-state error, an additional integral state can be introduced for the regulated output of the system, as shown in Figure 5. Once the augmented system has been rewritten in the form of (23), the following result, adapted from (Bernussou et al., 1989), points out a synthesis method to obtain a state-feedback controller that stabilizes quadratically the closed-loop system.
then, the state-feedback is given by Proof. The proof uses a quadratic Lyapunov function V(x)=x ′ Px, P = P ′ > 0,w h o s e time-derivative along the trajectories of the closed-loop systemẋ =( A + B u )Kx must be definite negative (Boyd et al., 1994). It follows that the following condition has to be satisfied. Finally, considering the left and right-hand multiplication of the previous condition by W = P −1 , and the substitution of KW = Y, LMI condition (24) follows.
A single Lyapunov function can be used to guarantee the stability of an uncertain system. The following theorem yields the state-feedback condition in the case of a polytopic representation.
then K = YW −1 is a state-feedback matrix.
The proof of this theorem is given in (Bernussou et al., 1989). It is worth to point out that there exist more recent works which have been concerned with the stability of polytopic uncertain systems considering in particular multiple Lyapunov functions instead of a single one (Apkarian et al., 2001;Bernussou & Oustaloup, 2002;Peaucelle & Arzelier, 2001c), in order to reduce the conservatism of the quadratic approach.
the linear dynamics of the buck-boost converter are then written as: Uncertainty: Polytopic uncertainty (19)  As in (Olalla et al., 2009a), additional variables are introduced, in order to remove the non affine dependence of the system matrices on the uncertain terms. The uncertainty parameter vector is defined Note that the uncertain model is inside a polytopic domain formed by N = 2 4 vertices. Also note that the multiplication between δ 2 and 1/R in the second row of B u does not imply a new variable because both functions are strictly decreasing.
Sampling effect: In this example the sampling effect has not been included in the converter, as the state variables of the modulator model can not be measured.  Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized is the H ∞ norm of the transfer function between the output disturbanceĩ load (t) and the output voltagẽ v o (t). In order to assure robust transient performances, the closed loop poles are constrained in an LMI region S (α, r, θ), where the desired minimum damping ratio is set to θ = 1 √ 2 , the required maximum damped frequency is r = 1 10 2π T s , and the minimum decay rate, for a settling time lower than 20 ms, is set to α = 200.

Results:
The robust control synthesis algorithm yields a controller K 1 : that ensures an H ∞ norm from output disturbance to output voltage of 3.80 (11.6 dB). Figure 6 shows  Fig. 7. Block diagram of an output-feedback system with controller G(s).
the simulation results of the controller over the sixteen vertices of the set of matrices. Since the nonlinear terms are disregarded, the robust stability and performance of the converter is guaranteed while the converter remains in the considered operating points, assuming that the change between these operation points is sufficiently slow. In order to account for the neglected dynamics, see (Hu, 2011;Olalla et al., 2009b;2011). Figure 7 shows the general diagram of an output-feedback control system. Depending on the structure of the controller G(s), two main approaches can be differentiated for the synthesis of output-feedback controllers: static and dynamic controllers. Given the system Σ as described in (21), for the buck and the buck-boost converter E yu = 0 can be considered. Then, in the case of a dynamic controller of order k with the following structure

Output-feedback control
the closed loop system has the form In the case of a static controller, K ∈ R m×q u = Ky = K(C y x + E yw w) and the closed loop system has the following structure For both problems the Lyapunov inequality is written, in a generic form: which depends non-linearly on P and the matrices of the controller (K in the static case or A c , B c , C c , D c in the dynamic case). There exist several methods to linearize the output-feedback synthesis problem. For the dynamic output-feedback case, the results of reference (Scherer et al., 1997) are summarized.

Dynamic output-feedback
The dynamic output-feedback synthesis method shown in (Scherer et al., 1997) employs the following transfer function parametrization defined from the exogenous input w = w j R j to the cost output z j = L j z as follows: To find a controller which stabilizes the closed-loop system, there must exist a quadratic Lyapunov function such that P > 0 0 A ′ P + PA < 0 0 The LMI constraints are formulated for a transfer function T j (s)=L j T(s)R j ,i nt e r m so ft h e state-space matrices A, B j , C j , D j . The goal is to synthesize an LTI controller Σ K that: • internally stabilizes the system • meets certain specifications (H 2 , H ∞ , pole placement,...) on a particular set of channels.
Generally, each transfer function T j will satisfy each specification S j , if there exists a Lyapunov matrix P j > 0 that satisfies some LMI constraints in P j . The control problem usually includes an u m b e ri of specifications. Therefore, the synthesis problem involves a set of matrix inequalities whose variables are: • the controller matrices A c , B c ,C c , D c .
•t h e i Lyapunov matrices P 1 ,...,P i , one per specification.
• additional auxiliary variables to minimize, for example, the norm cost H ∞ .
Since this problem is nonlinear and hardly tractable numerically, the method shown in (Scherer et al., 1997) requires that all the specifications are satisfied with a single Lyapunov function, that is: This restriction involves conservatism in the design, but it leads to a numerically tractable LMI problem, it produces controllers of reasonable order and it exploits all degrees of freedom in P (Scherer et al., 1997). Actually, if a single Lyapunov function P is considered, the following change of variable linearizes the control problem and makes it solvable with LMIs. Let n be the number of states of the plant, and let k be the order of the controller. Partition P and P −1 as where X and Y are ∈ S n ,a n d⋆ is a symmetric positive definite matrix such that PP −1 = 1 holds.
The change of variables is as follows whereÂ,B,Ĉ have dimensions n × n, n × m, q × n respectively. If M and N have full row rank, andÂ,B,Ĉ,D, X, Y are given, the matrices A c , B c C c , D c can be computed. If M and N are square n = k and invertible, then A c , B c , C c , D c are unique. The motivation for this change of variables lies in the following identities which can be used in a congruence transformation to derive the LMI constraints. A detailed proof is given in (Scherer et al., 1997). Once the variablesÂ,B,Ĉ,D, X, Y have been found, let us recover the original system by following this procedure. First we need to construct M, N and P that satisfy (45). M and N should be chosen such that NM ′ = 1 − YX. With the following LMI: we assure Y > 0 0 and X − Y −1 > 0 0 such that 1 − YX is nonsingular. Hence, M and N can always be found. After that, Π 1 and Π 2 are also nonsingular, and P = Π 2 Π −1 1 can be found.
Then D c , C c , B c and A c can be solved, in this order: For a list of LMI constraints which respond to several specifications with this change of variables, it is recommended to read (Scherer et al., 1997).

Example 2. Buck Converter
In this example, the synthesis of an output-feedback controller for a buck converter is carried out. The objective of the synthesis algorithm is, again, to minimize the H ∞ norm of the output disturbance to output voltage transfer function. In this case the stray resistances of the converter are taken into account, since only the output signals are used. This design considers a unique output signal v o (t) to set-up a voltage-regulation operation.

Sampling effect:
The sampling effect could be included in the converter model, in order to prevent the optimization algorithm to yield unrealistic results due to the switching action. However, in this case, a weighting function on the complementary sensitivity response can be used for this purpose.
Uncertainty: (Courties, 1997; where a cross-decomposition algorithm is described to obtain a local optimum controller giving an initial feasible solution. The solution proposed in this example exploits the weighting transfer functions to obtain the expected sensitivity and complementary sensitivity responses. The parameters of this example take the values shown in Table 2. The synthesis algorithm closely follows the linearizing change of variables of (Scherer et al., Parameter Value

Polytopic uncertainty (19) can be introduced in the model of the converter to cope with the variations of the uncertain parameters, as the load or the input voltage. However, in the case of output-feedback the polytopic representation of uncertainty introduces nonlinear relationship between the variables of the inequalities. This problem is treated in
150 mΩ r C 30 mΩ T s 5 µ s Table 2. Buck: converter parameters 1997) and the methodology explained in chapters 5 and 6 of (Gahinet et al., 1995): 1. First, the design specifications are expressed in terms of loop shapes and their corresponding shaping filters.

Then, the original plant is augmented with such filters to obtain a weighted plant.
3. Finally, the augmented plant is used in the optimization algorithm to derive a controller that meets certain LMIs.
The algorithm yields a controller of the same order as the augmented plant, that is, the order of the original plant plus the order of the shaping filters.

Performance Specifications:
The objective of the design procedure is to minimize the H ∞ norm of the disturbance to output transfer function. For such objective, a weighting function W 1 (s) for the error signal (i.e. for the sensitivity function) and a weighting function W 2 (s) for the output signal (i.e. for the complementary sensitivity function) are considered. Both weighting functions are depicted in Figure 8(b). In order to obtain small steady-state error, W 1 (s) is very large at low frequencies.
Other performance requirements could have been included (pole placement, H 2 , ...) in the optimization problem, but they have not been used to maintain all the degrees of freedom in the research of the minimum H ∞ norm.

Static output-feedback
An alternative to the use of weighting functions and frequency dependent uncertainty models is to consider the static output-feedback case. Static output-feedback considers a gain K to set up the feedback loop as u = Ky. The survey on output-feedback design methods (de Oliveira & Geromel, 1997) differentiates between several approaches to solve the synthesis of a static output gain as follows: 1. Nonlinear programming methods. They work on the parametric space defined by K and P to find an optimal value of a cost variable, if any. The search is done by means of classical optimization methods as, for example, a gradient algorithm, primal or dual Levine-Athans' method, etc. The solution of the algorithm, which converges to a local optimum, strongly depends on an initial stabilizing gain, which must be found beforehand.
2. Parametric optimization methods. These methods optimize the objective function for the parametric space defined by P, for some matrix K. The determination of the controller, if it exists, is decomposed in independent steps. These methods can be easily implemented using LMI solvers.
3. Convex programming methods. They solve a sufficient version of the Lyapunov inequality (38) obtained by the addition of constraints which lead to a convex feasibility set. The proposed parametrization is based on the elimination lemma (Boyd et al., 1994) and the introduction of additional variables to obtain an iterative algorithm. It has been extracted from (Peaucelle & Arzelier, 2001a).

Theorem 3.3. The Lyapunov inequality (38) can be rewritten as follows
where K s is a state-feedback gain that stabilizes the system. At the optimum point, which depends on the objective function, the output-feedback controller is given by K = F −1 R.
Proof. The equation (38) can be written as the following product of matrices: Applying the elimination lemma with

the previous inequality is written as
With the following change of variables it is verified that K = F −1 R = F −1 FK, and we obtain the result of (52).
This parametrization has been adapted to the minimization of the performance indexes H 2 and H ∞ (Peaucelle & Arzelier, 2001a;b).
With the new introduced variables we can split the optimization problem into two linear steps; in the first step, K s is kept constant, and the problem is solved for P, R and F. Then, in the next step, R and F are kept constant and the problem is solved for P and K s . With this iterative process, a stabilizing gain that satisfies a given cost function can be obtained.
A key point of this approach is however related to the initialization step, for which and admissible stabilizing state-feedback has to be selected. On the other hand, this algorithm presents the advantage that the Lyapunov matrix P is set as a free variable in both steps.

Initialization.
Step k = 1. Choose a stabilizing state-feedback gain. Since only the output signal is used, the sampling effect is included in the converter model in order to obtain an accurate model up to half the switching frequency. This addition also prevents the optimization algorithms to yield unrealistic results due to the switching action.
Performance Specifications: Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized is the H ∞ norm of the transfer function between the output disturbanceĩ load (t) and the output voltage. As in the state-feedback case, the closed loop poles are constrained in an LMI region S (α, r, θ), in order to assure robust transient performances. Again, the desired minimum damping ratio is set to θ = 1 √ 2 , the required maximum damped frequency is r = 1 10 2π T s , and the minimum decay rate, for a settling time lower than 40 ms, is set to α = 100.

Results:
The robust control synthesis algorithm yields a controller K 3 : that assures an H ∞ norm from output disturbance to output voltage of 0.656 (-3.66 dB). The waveforms of a numerical simulation and corresponding Bode plots of the converter with controller K 3 over the four vertices of the set of matrices are depicted in Figure 9. It can be observed that the static output-feedback controller does not achieve the performance of the dynamic output-feedback controller K 2 (s).

Dynamic output-feedback with static parametrization
The parametrization method shown in the previous subsection for the static case can be adapted to the synthesis of a dynamic output-feedback controller. The following theorem can be found in (Martenson, 1985;Nett et al., 1989).
Theorem 3.4. The synthesis of a dynamic controller Σ K for the system Σ (for simplicity, in absence of perturbations (w = 0) ) can be expressed as the synthesis of a static controller for the augmented system Σ aug :  where and Again, the sampling effect is included in the converter model in order to obtain an accurate model up to half the switching frequency.
Performance Specifications: As in the previous case, the objective is to minimize the H ∞ norm of the transfer function between the disturbance and the output. Also, the closed loop poles are constrained in the same LMI region S (α, r, θ), in order to assure robust transient performances. The initialization step is constrained with the corresponding pole placement inequalities and its objective function tries to enforce the states of the dynamic controller, using a weight β on R = FK.
where for this case of one input and one state of the dynamic controller can be considered β = β 1 0 0 β 2 , β 1 > β 2 . If an initial state-feedback is found then the iterative step minimizes the H ∞ norm γ.

Results:
For the present case the algorithm yields the following controller:   with the main drawback that the solution of the iterative algorithm may depend on the chosen initial state-feedback gain.

Conclusions
In the chapter, it has been shown how a control formulation based on linear matrix inequalities can cope not only with academic optimization problems, but also with large real-life complex problems, since the numerical solution can be found by efficient computer algorithms. The synthesis (or analysis) of the control system can be made by solving an optimization problem, using the concepts of Lyapunov stability and positive definiteness. Besides, frequency-based and time domain performance requirements can also be posed in form of LMIs, as H ∞ , H 2 or pole placement. LMI-based state-feedback synthesis methods have been already applied successfully in the field of power conversion, since they can be applied directly with no additional conservatism. However, output-feedback approaches require the linearization of the synthesis variables in order to be solvable with LMIs. Such linearization methods often impose changes in the matrix variables or require an initial feasible result.
The results presented in this chapter with LMI control of a buck and a buck-boost converter demonstrates the feasibility of this approach, but also shows some of the limitations. The buck-boost converter presents nonlinear dynamics; such control problem has been tackled with a state-feedback approach, so that the information about the converter states allows to consider the uncertainty coming from the disregarded dynamics, but also limits the achievable performance. Such limitation also allows to neglect the sampling effect of the modulation, since the effective feedback bandwidth of the control is well below the switching frequency. On the other hand, the control of the buck converter, whose averaged dynamics are basically linear, has been dealt with an output-feedback realization. In this case, the closed-loop performance and its associated bandwidth can reach high frequencies, and it is advisable to take into account the sampling effect of the modulator. From the comparison between the three output-feedback approaches, the best results have been achieved with the H ∞ -based dynamic controller, but this approach also presents some limitations, as the impossibility to deal with easy-to-derive uncertainty models. Another drawback of this technique is that the choice of appropriate weighting functions must be made by trial and error and therefore this task requires good knowledge of the plant limits. Nevertheless, such limitations also appear in the dynamic output-feedback approach by static parametrization, since the results strongly depend on the initial feasible solution for the iterative algorithm. Depending on the application of the dc-dc converter, it may be easier to implement a state-feedback controller or an output-feedback controller. For instance, the inductor current may not be accessible or the capacitor stray resistance cannot be assumed small enough. Future research on LMI control of power converters could focus on the improvement of the output-feedback synthesis algorithms, which still require the tedious task of selecting weighting functions, even for the initial controller of the static feedback parametrization. Besides of the inherent limitations of output-feedback, the synthesis algorithms are still very conservative and do not lead to tight performances, when compared with state-feedback approaches.