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\hyphenation{Mo-du-la-tion Con-ver-ter} \begin{document} \title{Dynamical Adaptive Pulse-Width-Modulation Control of DC-to-DC Power Converters: A Backstepping Approach. \thanks{This research was supported by the Consejo de Desarrollo Cient\'{\i}fico, Human\'{\i}stico and Tecnol\'{o}gico of the Universidad de Los Andes under Research Grant I-456-91}} \author{ Hebertt Sira-Ram\'{\i}rez \\ Departamento Sistemas de Control\\ Universidad de Los Andes\\ M\'{e}rida 5101, Venezuela\\ e-mail: isira@ing.ula.ve \and Mauricio Garc\'{\i}a-Esteban \thanks{On temporary leave from the Instituto de Cibern\'{e}tica, Universidad Polit\'{e}cnica de Catalunya, Barcelona, Spain}\\ Centro de Instrumentos\\ Universidad Nacional Aut\'{o}noma de M\'{e}xico\\ M\'{e}xico, D.F., M\'{e}xico\\ e-mail: imauricio@ing.ula.ve \and Alan S.I. Zinober\\ Applied Mathematics Section\\ School of Mathematics and Statistics\\ University of Sheffield\\ Sheffield S10 2TN, England\\ e-mail: A.Zinober@sheffield.ac.uk} \date{August 1994} \maketitle \thispagestyle{empty} \abstract{Adaptive regulation of Pulse-Width-Modulation (PWM) controlled dc-to-dc power supplies is proposed using a suitable combination of dynamical input-output linearization and the "backstepping" controller design method. A nominal parameter, input-dependent, state coordinate transformation of the average PWM converter models leads to a type of pure parameter feedback canonical form associated with the Fliess Generalized Observability Canonical Form of such average models. A backstepping design procedure can then be immediately devised which leads to a dynamical adaptive regulation scheme for the generation of the stabilizing duty ratio function. The validity of the proposed approach, regarding control objectives and robustness with respect to unmodelled, yet unmatched, and bounded stochastic perturbation inputs, is tested through digital computer simulations.}
\vspace{0.3in} {\bf Keywords :} DC-to-DC Power Converters, Adaptive Control, Genearalized Phase Canonical Forms, Pulse-Width-Modulation \clearpage
\section{Introduction \label{sect1}}
Feedback regulation of switchmode dc-to-dc power converters is usually accomplished by means of Pulse-Width-Modulation (PWM) feedback strategies. For fundamental background on this important subject the reader is referred to conference proceedings (such as the yearly Power Specialist Conference Records, the Multi-volume series edited by Middlebrook and C\'{u}k (1981), or the remarkable collection of articles recently edited by Bose (1992). Also, useful material may be found in specialized books such as, Kassakian {\em et al} (1991), Severns and Bloom (1985) and Csaki {\em et al} (1983).
PWM feedback regulation strategies for dc-to-dc power converters are usually based on perfect knowledge of the converter circuit parameters (see, among many other authors, the articles by Sira-Ram\'{\i}rez and co-workers (1989, 1991, 1992). This fundamental assumption is sometimes invalid due to imprecise knowledge of the values of the converter circuit components as well as of the external voltage source. The situation is often due to either measurement errors, or unavoidable ageing effects on the circuit components. Automatic control problems which efficiently handle uncertainty in the system parameter values, usually require adaptive solutions employing to different forms of the so called "Uncertainty Equivalence Principle" (see Sastry and Bodson 1989). In other words, the controller is designed as if the system parameters were perfectly known and then the values of the parameters appearing in the controller expression are regarded as "tuneable", in an on line fashion. Parameter tuning is accomplished by the specification of an updating, or parameter adaptation, law designed so as to simultaneously guarantee the demands of the regulation objectives and the stability of the adaptation process.
Adaptive feedback control techniques for PWM controlled dc-to-dc power supplies have been recently explored in Sira-Ram\'{\i}rez {\em et al} (1993a, 1993b). The approach in these contributions incorporated an extension of the results found in Sastry and Isidori (1989), for the adaptive stabilization of partially linearizable, minimum phase, discontinuously controlled nonlinear systems.
In this paper, a rather different adaptive feedback strategy is adopted by resorting to an approach inspired by the recently introduced {\em adaptive backstepping} controller design methodology. Backstepping adaptation was developed for the regulation of a large class of state linearizable nonlinear systems exhibiting constant, but otherwise unknown, parameter values. The basic ideas and rather useful variations, of the backstepping adaptive design procedure can be found in the excellent research articles authored by Kanellakopoulos {\em et al} (1991a, 1991b, 1991c) and by Krstic {\em et al} (1992) where the reader is referred to for enlightening details.
We specifically assume that the circuit converter components are only nominally known and their constant discrepancies from the given nominal values are totally unknown. The use of an input-dependent, nominal parameter based, input-output linearizing state coordinate diffeomorphism for the unknown system produces an imperfectly transformed system in {\em generalized phase variables} (see Fliess (1989)). The structure of the resulting systems, in generalized phase variables, strongly resemble the {\em Pure Parameter Feedback Canonical form}, presented in Kanellakopoulos (1991a), except for the presence of the control inputs in some of the "regressor vectors", as well as the control input time derivative in the transformed system equations.
Computation of the feedback controllers, and of the associated incremental parameter update laws, is then carried out by resorting to a backstepping calculation procedure applied to the obtained {\em Generalized Pure Incremental Parameter Feedback Canonical form}. The net result is that one yields adaptive {\em dynamical duty ratio synthesizers}, rather than traditional static feedback compensators. The advantage of dynamically generated duty ratio control signals lies in the enhanced smoothed character of this important feedback regulation signal during the actual (i.e. discontinuous) operation of the converters. Smoothing of the duty ratio function increases the precision, and qualitative performance features, of the closed loop behavior of the dc-to-dc power converter circuits.
Section \ref{sect2} is devoted to revisit, via a "Boost" converter example, the fundamentals of the input-output linearization of PWM controlled dc-to-dc power converters by means of a dynamical feedback duty ratio synthesizer. We assume that all parameters in the system are perfectly known. The input-output linearization scheme achieves indirect regulation of the average output capacitor voltage by means of average input inductor current regulation. This strategy, which essentially involves a "change of output" effectively avoids the non-minimum phase problem in the direct regulation of the output capacitor voltage variable. This method was already used in Sira-Ram\'{\i}rez et al (1991, 1993) and it has been later justified, from a general viewpoint, by Benvenuti {\em et al} (1992) for nonlinear systems and by Fliess and Sira-Ram\'{\i}rez (1993) for linear systems.
Section \ref{sect3} presents the developments leading to a dynamical adaptive PWM control strategy for dc-to-dc power supplies of the "Boost" and "Buck-Boost" types with unknown incremental parameter values. Computer simulations are presented which clearly indicate the effectiveness, and robustness, of the proposed adaptive feedback regulation scheme with respect to unmodelled, and unmatched, external stochastic perturbation inputs of bounded nature. Section \ref{sect4} contains the conclusions and suggestions for further work in this area.
\section{A Nominal Input-Output Linearization Strategy for Dc-to-Dc Power Converters \label{sect2}}
This section contains the developments leading to dynamical feedback duty ratio synthesizers for the PWM stabilization of nominal average models of dc-to-dc power converters. The scheme, already exploited by Sira-Ram\'{\i}rez (1991), is presented here only for the purpose of making the article self-contained. The fundamental idea is to obtain a generalized nonlinear phase variable representation of the input-output behaviour of the average circuit for which the control synthesis problem is straightforward. Due to non-minimum phase problems associated with the output capacitor voltage variable, the regulated output is chosen as the input inductor current. Thus, indirect output voltage regulation is achieved.
\subsection{ Boost Converter}
Consider the boost converter circuit shown in Figure \ref{fig1}. This circuit is described by the state equation model, \begin{eqnarray} {\dot I}(t) &=& -\frac{1}{L}(1-u) \; V(t) + \frac{E}{L}\nonumber\\ {\dot V}(t) & = & \frac{1}{C}(1-u) \; I(t) - \frac{1}{RC} V(t)\nonumber\\ y(t) & = & I(t) \label{2.1} \end{eqnarray} where $ I $ and $ V $ represent, respectively, the input inductor current and the output capacitor voltage variables. The positive quantity $ E $ is the external input voltage. The variable $ u $ denotes the switch position function, acting as a control input which takes values in the discrete set $ \{0,1\} $. The output, $ y $, of the system is represented by the input inductor current $ I $.
We define \begin{equation} \Theta_1 = \frac{1}{L} \ \ \ \ ; \ \ \ \ \Theta_2 = \frac{1}{C} \ \ \ \ ; \ \ \ \ \Theta_3 =\frac{1}{RC} \ \ \ \ ; \ \ \ \ \Theta_4 = \frac{E}{L}\label{2.2} \end{equation} as the system parameters assumed to be nominally known.
A PWM feedback control strategy for the regulation of the boost converter circuit is typically given by the following prescription of the switch position function (see the work of Sira-Ram\'{\i}rez (1989a) and Sira-Ram\'{\i}rez {\em et al} (1991)) \begin{eqnarray} u &=& \left\{\begin{array}{lc} 1 \ {\rm for} & t_k \le t < t_k+ \mu(t_k) T\\ 0 \ {\rm for} & t_k + \mu(t_k) T \le t < t_k + T \end{array} \right. \nonumber\\ & & t_{k+1} = t_k + T \ \ \ ; \ \ \ k = 0,1, \ldots \label{2.3} \end{eqnarray} where $ t_k $ represents a sampling instant; the parameter $T$ is the fixed sampling period, also called the {\em duty cycle}; the sampled values of the state vector $ x(t) $ of the converter are denoted by $x(t_k)$. The function, $\mu(\cdot) $, is the {\em duty ratio function} acting as a truly feedback policy. The value of the duty ratio function, $ \mu [t_k]$, determines, at every sampling instant, $t_k$, the width of the upcoming "pulse" (switch at the position $u = 1$) as $ \mu[t_k] T $. The actual duty ratio function, $ \mu(\cdot) $, is evidently a function limited to the closed interval $[0,1]$ on the real line.
The control problem, associated with the stabilization of the discontinuously controlled system (\ref{2.1}),(\ref{2.3}) towards some (feasible) prespecified constant desired equilibrium point, consists of specifying the duty ratio function, $\mu $, {\em as a static, or dynamical, feedback control policy\/}, i.e. as a function of the state vector $ x $, or as the solution of a time-varying differential equation based on the measured values of the state $x$. As formulated, the problem of synthesizing a suitable duty ratio function $ \mu $ is quite involved, owing to the difficulty in performing an exact discretization of the PWM system model (\ref{2.1}),(\ref{2.3}). A conceptually useful, and practical, alternative consists of resorting to the {\em Infinite Frequency Average PWM model}, also known as {\em State Space Average Model} of the PWM controlled converter (\ref{2.1})-(\ref{2.2}) (see Kassakian {\em et al} (1991) and Middlebrook and \'{C}uk (1981) ). The assumption of an infinite sampling frequency results in a smooth nonlinear average system model of (\ref{2.1}) in which the duty ratio function, $\mu$, is readily interpreted as a control input to the average system in formal replacement of the switch position function $u$. In fact, the duty ratio function becomes the {\em equivalent control} input in the corresponding {\em Sliding Mode\/} (see Utkin (1978)) interpretation of the obtained idealization (see also Sira-Ram\'{\i}rez (1989b))).
The above idealization has the fundamental advantage of reducing the duty ratio synthesis problem to a standard nonlinear feedback control design problem in which the duty ratio function acts as the required feedback control input. Any of the well known static, or dynamical, feedback controller design procedures established in the recent literature (see Isidori \ (1989), Rugh (1986) and Fliess (1989)) can be applied to the nonlinear average model of the circuit to obtain the required duty ratio function as a nonlinear feedback control law.
Consider then the following nominal average PWM model of the boost converter circuit, \begin{eqnarray} {\dot \zeta}_1 &=& -\Theta_1 (1-\mu) \; \zeta_2 + \Theta_4\nonumber\\ {\dot \zeta}_2 & = & \Theta_2 (1-\mu) \; \zeta_1 - \Theta_3 \zeta_2\nonumber\\ \eta & = & \zeta_1 \label{2.4} \end{eqnarray} where $\zeta_1$ and $\zeta_2$ represent the averaged values of the original state variables $I$ and $V$. The average output variable $\zeta_1$ is here denoted by $\eta $.
For a given constant value, $ \mu = U $, of the duty ratio function, the corresponding equilibrium values of the average state variables of the circuit are obtained as, \begin{equation} \zeta_1(U) = \frac{\Theta_3 \Theta_4}{\Theta_1 \Theta_2 (1-U)^2} \ \ \ \ ; \ \ \ \ \zeta_2(U) = \frac{\Theta_4}{\Theta_1 (1-U)} \label{2.5} \end{equation}
Note that by straightforward elimination of the constant parameter $U$, in the set of equations (\ref{2.5}), the equilibrium values for $\zeta_1$ and $\zeta_2$ are related by \[ \zeta_1(U) = \frac{\theta_3\theta_4}{\theta_2 \theta_1} \zeta_2^2(U) \] Hence, the prescription of a desired steady-state value for the average output capacitor voltage $\zeta_2(U)$, uniquely determines, both the required constant value of the duty ratio function, $U$, and the corresponding average value for the input inductor current $\zeta_1(U)$. This simple fact allows the {\em indirect} regulation of the average output capacitor voltage of the converter through regulation of the average input inductor current. For this reason our control problem will be formulated in terms of achieving a desired steady state equilibrium value, denoted by $Y$, for the average input inductor current $\zeta_1 $.
Consider then the following, locally invertible, {\em nominal} input-dependent state coordinate transformation of the nonlinear average model (\ref{2.4}), \begin{equation} x_1 = \zeta_1 \ \ \ \ ; \ \ \ \ x_2 = -\Theta_1(1-\mu)\zeta_2 + \Theta_4 \label{2.6} \end{equation} and its associated inverse transformation, \begin{equation} \zeta_1 = x_1 \ \ \ \ ; \ \ \ \ \zeta_2 =\frac{\Theta_4-x_2}{\Theta_1(1-\mu)} \label{2.7} \end{equation}
Using the above state coordinate transformation (\ref{2.6}), (\ref{2.7}), on the average circuit equations (\ref{2.4}), one obtains the following Fliess Generalized Observability Canonical Form of the average boost converter model \begin{eqnarray} {\dot x}_1 &=&x_2\nonumber\\ {\dot x}_2 & = & -\theta_1\theta_2(1-\mu)^2x_1-\theta_3(x_2-\theta_4) - \frac{{\dot \mu}}{1-\mu} \left(x_2 - \theta_4\right) \nonumber\\ \eta &= & x_1 \label{2.8} \end{eqnarray} The {\em zero dynamics} associated with the equilibrium point $x_1 = \zeta_1(U)$, $ x_2 = 0 $, is given by the first order dynamics, \begin{equation} \dot \mu = - \frac{\theta_3}{(1-U)^2}\left(1-\mu\right)\left(\mu-U\right) \left(2-\mu-U\right)\label{2.9} \end{equation} The zero dynamics exhibits two unstable equilibrium points $ \mu = U-2 < 0$ and $\mu = 1$. The only stable equilibrium point $ \mu = U $ makes the average system {\em minimum phase} in the region of interest.
Let the desired behaviour of the transformed system (\ref{2.8}) be prescribed as the following asymptotically stable second order linear time-invariant dynamics: \begin{eqnarray} {\dot x}_1 &=&x_2\nonumber\\ {\dot x}_2 & = & -2\xi \omega_n x_2 - \omega_n^2( x_1 - Y) \nonumber\\ \eta &= & x_1 \label{2.10} \end{eqnarray} where $\xi$ and $\omega_n$ are design parameters reflecting our need for particular transient features of the average regulated output $\eta = x_1$. The constant $Y$ represents a desired steady state closed-loop equilibrium value of the average output variable $\eta$. In our case of perfectly known parameter values, the desired output is obtained as $ Y = \zeta_1(U) = {\Theta_3\Theta_4}/{\Theta_1\Theta_2 (1-U)^2}$.
>From (\ref{2.8}) and (\ref{2.10}) one readily obtains an expression for the dynamical feedback controller yielding the required linearizing and stabilizing duty ratio function. In terms of the transformed variables $x_1$ and $x_2$ the duty ratio function, $\mu$, is obtained as the solution of the time-varying differential equation \begin{equation} {\dot \mu} = \frac{1-\mu}{x_2-\theta_4} \left[ 2\xi\omega_nx_2+\omega_n^2(x_1-Y)-\theta_1\theta_2(1-\mu)^2x_1 -\theta_3(x_2-\theta_4) \right]\label{2.11} \end{equation} In terms of the average state variables $\zeta_1$ and $\zeta_2$, the dynamical duty ratio synthesizer is equivalently obtained as \begin{equation} {\dot \mu} = \frac{1}{\theta_1\zeta_2} \left[ \left(\theta_1\theta_2(1-\mu)^2-\omega_n^2\right)\zeta_1 + \left(2\xi\omega_n\theta_1-\theta_1\theta_3\right)(1-\mu)\zeta_2 -2\xi\omega_n\theta_4+ \omega_n^2 Y \right] \label{2.12} \end{equation} The values of $\mu$, obtained from the on-line solution of equation (\ref{2.11}), actually represent the {\em computed duty ratio function}, which we still denote by $\mu$. However, in order to implement this dynamical feedback control strategy on the actual (i.e. discontinuously regulated) converter system the values of $\mu$ must be necessarily limited to the closed interval $[0,1]$. We define then the {\em actual duty ratio function}, denoted by $\mu_a $ as, \begin{eqnarray} \mu_a(t) = \left\{ \begin{array}{ll} 1 & \ \ {\rm for} \ \ \mu(t) \ge 1\\ \mu (t) & \ \ {\rm for} \ \ 0 \ < \ \mu(t) \ < \ 1\\ 0 & \ \ {\rm for} \ \ \mu(t) \ \le \ 0 \end{array}\right.\label{2.13} \end{eqnarray} Finally, it should be noted that when the sampling period $T$ is sufficiently small, the actual values of the state variables $V$ and $I$, rather than their average values, $\zeta_1$, $\zeta_2$, may be used for the on-line solution of the computed duty ratio function $\mu$. This procedure is precisely at the heart of the state average method for PWM designs. A theoretical justification of this procedure has been given by Sira-Ram\'{\i}rez {\em et al} (1993).
Summarizing, a dynamical PWM controller achieving the asymptotic stabilization of the average input inductor current of the boost converter circuit (\ref{2.1}) to a desired constant equilibrium value $ y = Y$, is given by \begin{eqnarray} u &=& \left\{\begin{array}{lc} 1 \ {\rm for} & t_k \le t < t_k+ \mu_a(t_k) T\\ 0 \ {\rm for} & t_k + \mu_a(t_k) T \le t < t_k + T \end{array} \right. \nonumber\\ & & t_{k+1} = t_k + T \ \ \ ; \ \ \ k = 0,1, \ldots \label{2.14} \end{eqnarray} where $ \mu_a (t_k)$ represents the sampled values of the actual duty ratio function at time $t_k$, given by (\ref{2.13}). The computed duty ratio function $ \mu(t) $ is obtained from the on-line solution of \begin{eqnarray} {\dot \mu} &=& \frac{1}{\theta_1V(t)} \left[ \left(\theta_1\theta_2(1-\mu)^2-\omega_n^2\right)\;I(t) + \left(2\xi\omega_n\theta_1-\theta_1\theta_3\right)(1-\mu)\;V(t)\right. \nonumber\\ & &\quad \left. -2\xi\omega_n\theta_4+ \omega_n^2 Y \right]\label{2.15} \end{eqnarray}
\section{An Adaptive Feedback Control Strategy for Indirect Output Voltage Regulation in Dc-to-Dc Power Converters \label{sect3}}
\subsection{Boost Converter} With reference to the boost converter circuit, consider the following version of the average boost converter model \clearpage \begin{eqnarray} \dot{\zeta_{1}}&=&-\theta_{1}(1-\mu)\zeta_{2}+\theta_{4}\nonumber \\ \dot{\zeta_{2}}&=&\:\;\theta_{2}(1-\mu)\zeta_{1}-\theta_{3}\zeta_{2}\nonumber\\ y & = & \zeta_1\label{3.1} \end{eqnarray} where the $ \theta_i \; , \; i = 1,2,3,4 $, represent the {\em actual parameter values}, modelled by \begin{equation} \theta_i = \Theta_i + \Delta \theta_i \ \ , \ \ i = 1,2,3,4\label{3.2} \end{equation} with $\Theta_i \ , \ i = 1,2,3,4$, being the {\em nominal parameters}, assumed to be perfectly known. The quantities, $\Delta \theta_i \ ; \ i = 1,2,3,4 $ denote the corresponding constant, but unknown, incremental variations of the parameters from their nominal values.
Consider then the {\em nominal} input-dependent state coordinate transformation, used for exact input-output linearization of the average boost converter model in the previous section, \begin{eqnarray} x_{1} & = & \zeta_{1} \nonumber \\ x_{2} & = & -{\Theta_{1}(1-\mu)}\zeta_{2} + \Theta_4\label{3.3} \end{eqnarray} Clearly this control-parametrized transformation is invertible everywhere except when the duty ratio function $ \mu $ is identically equal to one. The associated inverse transformation is readily found to be \begin{eqnarray} \zeta_{1}&=& x_{1} \nonumber \\ \zeta_{2}&=&\frac{\Theta_{4}-x_{2}}{\Theta_{1} (1-\mu)}\label{3.4} \end{eqnarray} When the state coordinate transformation (\ref{3.3})-(\ref{3.4}) is applied to the actual boost converter model (\ref{3.1})-(\ref{3.2}), the transformed system is not quite in the Fliess Generalized Observability Canonical form (\ref{2.8}) but rather in what we call the {\em Generalized Pure Incremental Parameter Feedback Canonical Form}. The transformed system is easily shown to be given by \begin{eqnarray} \dot{x_{1}}&=&x_{2}+\Delta\theta^{T} \gamma_{1}(x_1,x_2)\nonumber \\ \dot{x_{2}}&=&\dot{\mu} \left(\frac{-x_2+\Theta_4}{(1-\mu)}\right) -\Theta_1 \Theta_2(1-\mu)^2 x_1-\Theta_3(x_2-\Theta_4) +\Delta\theta^{T} \gamma_{2}(x_1,x_2,\mu) \nonumber\\ y & = & x_1 \label{3.5} \end{eqnarray} where \begin{equation} \begin{array}{*8{c}} \gamma_{1}^T(x_1,x_2) &=&[ &{\frac{\displaystyle x_{2}-\Theta_{4}} {\displaystyle\Theta_{1}}} & 0 & 0& 1 &] \\ \\ \gamma_{2}^T(x_1,x_2,\mu)&=&[&0 &{-\Theta_{1}(1-\mu)^{2}x_{1}}& {-x_2+\Theta_4} & 0 &]\\ \\ \multicolumn{8}{c} \:\\ \Delta\theta^T&=&[&\Delta\theta_{1}&\Delta\theta_{2}&\Delta\theta_{3}& \Delta\theta_{4}&] \end{array}\label{3.6} \end{equation} For ease of reference we let \begin{equation} f(x,\mu,\dot{\mu},\Theta) = \dot{\mu} \left(\frac{-x_2+\Theta_4}{1-\mu}\right) -\Theta_1 \Theta_2(1-\mu)^2 x_1-\Theta_3(x_2-\Theta_4)\label{3.7} \end{equation} The transformed system (\ref{3.5}) strongly resembles the more traditional {\em Pure Parameter Feedback Canonical form} developed by Kanellakopoulos {\em et al} (1991), except for the presence of the control input (duty ratio) $\mu$ in the "regressor vector" $\gamma_2$ and the presence of the first order time derivative, ${\dot \mu}$, of the control input in the second differential equation. This control input derivative will, in fact, act as the actual control input, while the control input $\mu$ may be regarded as playing the role of an additional state variable.
We now proceed to apply the adaptive backstepping algorithm, as developed by Kanellakopoulos {\em et al} (1991),, to the transformed model (\ref{3.5})-(\ref{3.6}).
\vspace{0.2in} {\bf Step 0}
\vspace{0.2in} Let $ Y $ be the desired steady state equilibrium value of the output variable $x_1$ and define the stabilization error $z_1$ as, \begin{eqnarray} z_1&=&x_1-Y \label{3.8} \end{eqnarray}
\vspace{0.2in} {\bf Step 1}
\vspace{0.2in} Consider the stabilization error equation \begin{equation} {\dot z}_1 = x_2 + \Delta \theta^T \gamma_1(x_1,x_2)\label{3.9} \end{equation} Suppose that the transformed variable $x_2$ can be used as a "pseudo-control" in (\ref{3.9}) and proceed to compute the required value of $x_2$ which stabilizes the error variable $z_1$ to zero. Computation of $x_2$ requires the unknown vector, $\Delta \theta$. Using the "certainty equivalence principle" (see Kanellakopoulos {\em et al} (1991),) we replace the vector $\Delta \theta$ by an estimate in the fictitious stabilizing "control law". We proceed to devise also a parameter update law for the hypothesized estimate of the incremental parameter vector $\Delta \theta$, denoted here by ${\widehat{\Delta \theta}}^1$. This specification must result in simultaneous stable adaptation and convergence to zero of the error variable $z_1$. The superscript "1" will denote a {\em first estimate} of $\Delta \theta$. Let $c_1$ be a strictly positive design parameter. We then have, as a plausible " pseudo-control" action, the following expression for $x_2$ \begin{equation} x_2 = - c_1\:z_1 - \frac{\widehat{\Delta\theta_{1}}^1}{\Theta_1}(x_2-\Theta_4) -\widehat{\Delta\theta_{4}}^1 \ \ \ ; \ \ \ c_1 > 0 \label{3.10} \end{equation} where $ \widehat{\Delta \theta_{i}}^1 \ , \ i=1,4 $, denotes a first estimate of ${\Delta \theta_{i}}\ , \ i=1,4$. A simple Lyapunov stability argument shows that the "pseudo controller" (\ref{3.10}) and the update law \begin{equation} {\dot {\widehat{\Delta \theta}} }^1 = z_1 \gamma_1(x_1,x_2) \label{3.11} \end{equation} yield a closed-loop stable system for which $z_1$ is guaranteed to converge to zero.
Since $x_2$ is not really a control input, one defines the "pseudo-control" error variable, $z_2$, as the difference between $x_2$ and its required value, computed in (\ref{3.10}). Let
\begin{equation} z_2=x_2-\left[-c_1\:z_1-\frac{\widehat{\Delta\theta_{1}}^1}{\Theta_1} \left(x_2 - \Theta_4\right)-\widehat{\Delta\theta_{4}}^1 \right]\label{3.12} \end{equation}
Solving for $x_2$ from (\ref{3.12}) and using (\ref{3.8}) one obtains a new state coordinate transformation defining, respectively, the stabilization error and the "pseudo-control" error variables, $z_1$ and $z_2$ \begin{eqnarray} z_1 & = & x_1 - Y \nonumber\\ z_2 & = & c_1 z_1 + x_2 \left( 1 + \frac{{\widehat{\Delta \Theta_1}}^1}{\Theta_1}\right) - \frac{{\widehat{\Delta\theta_1}}^1}{\Theta_1}\Theta_4 + {\widehat{\Delta\theta_4}}^1 \label{3.13} \end{eqnarray} The corresponding inverse transformation is simply obtained as \begin{eqnarray} x_1 & = & z_1 + Y\nonumber\\ x_2&=&\frac{\Theta_1}{\Theta_1+\widehat{\Delta\theta_{1}}^1} \left[ z_2-c_1z_1+\frac{\widehat{\Delta\theta_1}^1}{\Theta_1}\Theta_4 -\widehat{\Delta\theta_{4}}^1 \right]\label{3.14} \end{eqnarray} The first equation of the transformed system may then be written as \begin{eqnarray} \dot{z_1} &=& z_2-c_1 z_1+ \nonumber\\ & & + \left(\Delta\theta_{1}-\widehat{\Delta\theta_{1}}^1\right) \left[(\frac{1}{\Theta_1+\widehat{\Delta\theta_1}^1} \left[ z_2-c_1z_1+\frac{\widehat{\Delta\theta_1}^1}{\Theta_1}\Theta_4 -{\widehat{\Delta\theta_4}}^1 \right]-\frac{\Theta_4}{\Theta_1}\right] + \nonumber\\ & & \hspace{0.3in} +\left(\Delta\theta_{4} -\widehat{\Delta\theta_{4}}^1\right)\label{3.15} \end{eqnarray} which can be briefly expressed as \begin{equation} \dot{z_1}=z_2-c_1z_1+(\Delta\theta-\widehat{\Delta\theta}^1)^T w_1(z_1,z_2, {\widehat {\Delta \theta}}^1)\label{3.16} \end{equation} where \begin{equation} \begin{array}{*5{c}} w_1^T(z_1,z_2,{\widehat{\Delta \theta}}^1) =[&(\frac{\displaystyle 1}{\displaystyle \Theta_1+\widehat{\Delta\theta_1}^1} \left[ z_2-c_1z_1+ \frac{{\widehat{\Delta\theta_1}^1}}{\Theta_1}\Theta_4 -\widehat{\Delta\theta_{4}}^1 \right]-\frac{\displaystyle \Theta_4}{\displaystyle\Theta_1}) &0 &0 &1] \end{array} \label{3.17} \end{equation}
Note that the update laws corresponding to ${\widehat{\Delta\theta}_2}^1$ and ${\widehat{\Delta\theta}_3}^1$ will yield constant values for such estimates. Note, moreover, that these two estimated parameters are {\em not needed} in this first step of the backstepping calculation.
We let $W_1^1$ denote the first component of the regressor vector $w_1(z_1,z_2,{\widehat{\Delta \theta}}^1)$.
The first adaptation law (\ref{3.11}) may then be rewritten, in terms of the new error variables $z_1$, $z_2$, as \begin{equation} {\dot{\widehat{\Delta\theta}}^1} \;=\; z_1\;w_1(z_1,z_2,{\widehat{\Delta \theta}}^1) \label{3.18} \end{equation}
\vspace{0.2in} {\bf Step 2}
\vspace{0.2in} We proceed to complete the state coordinate transformation (\ref{3.13})-(\ref{3.14}) of the original phase variables by considering now the differential equation for the "pseudo controller" error $z_2$. Using the definition of $z_2$, $z_1$ and the first incremental parameter update laws for the involved components of the vector $\Delta \theta$, one obtains, after long but straightforward manipulations, the following expression,
\begin{eqnarray} \dot{z_2}&=& \left(1+\frac{{\widehat{\Delta\theta_{1}}}^1}{\Theta_1}\right) \Bigg[ -\;\frac{{\dot \mu}}{1-\mu}\left( x_2(z_1,z_2,{\widehat{\Delta\theta}}^1) -\Theta_4\right) -\Theta_1 \Theta_2(1-\mu)^2 (z_1+Y) \nonumber\\ & & \quad -\Theta_3\left(x_2(z_1,z_2,{\widehat{\Delta\theta}}^1) -\Theta_4\right) \nonumber \\ & & \quad - \Delta\theta_2\Theta_1(1-\mu)^{2}(z_1 + Y) -\Delta\theta_3 \left(x_2(z_1,z_2,{\widehat{\Delta\theta}}^1) -\Theta_4 \right)\Bigg] + \nonumber\\ & & \quad+ c_1\left[ z_2-c_1z_1-{\widehat{\Delta\theta_1}}^1 W_1^1 -{\widehat{\Delta\theta_4}}^1 \right] + z_1\left( 1 + (W_1^1)^2\right) \nonumber\\ & &\quad + c_1 \left( \Delta\theta_1W_1^1 + \Delta\theta_4 \right) \label{3.19} \end{eqnarray} where $ x_2(z_1,z_2,{\widehat{\Delta\theta}}^1) $ is given by the second equation of (\ref{3.14}), which we do not substitute just to avoid lengthy intermediate equations. In the rest of this section $x_2$ stands for $x_2(z_1,z_2,{\widehat{\Delta\theta}}^1)$.
If we now equate the dynamics obtained in (\ref{3.19}) for $z_2$ to the dynamics of an asymptotically stable behavior for $z_2$, given by \begin{equation} {\dot z}_2 = -c_2 z_2 \ \ \ ; \ \ \ c_2 > 0 \label{3.20} \end{equation} one can immediately solve for the required control input derivative ${\dot \mu}$ upon invoking, once more, the certainty equivalence principle. In this instance the unknown value of the vector $\Delta\theta$ will be replaced by a {\em new} vector of parameter estimates, denoted by $ \widehat{\Delta\theta}^2 $. One then obtains \begin{eqnarray} \dot{\mu}&=&\frac{(1-\mu) \Theta_1} {\left(\Theta_1+{\widehat{\Delta\theta_1}}^1\right)(x_2-\Theta_4)} \Bigg\{ c_2z_2 + c_1\left[z_2 -c_1z_1- ({\widehat{\Delta\theta}}^1)^T w_1(z_1,z_2,{\widehat{\Delta\theta}}^1)\right]\nonumber\\ & & \hspace{-0.2in} +z_1[1+ (W_{1}^1)^2 ] +c_1({\widehat{\Delta\theta}}^2)^Tw_1(z_1,z_2, {\widehat{\Delta\theta}}^1) -\left(1 + \frac{{\widehat{\Delta\theta}_1}^1}{ \Theta_1}\right) \left[ \Theta_1 \Theta_2(1-\mu)^2 (z_1+Y)\right. \nonumber\\ & & \hspace{-0.2in} \left. + \Theta_3(x_2-\Theta_4) + {\widehat{\Delta\theta_2}}^2\Theta_1 (1-\mu)^2(z_1 + Y) +{\widehat{\Delta\theta_3}}^2(x_2-\Theta_4) \right] \Bigg\} \label{3.21} \end{eqnarray} where $ {\widehat {\Delta\theta_j}}^2 \ , \ j = 2,3 $, represent the new estimates of the incremental parameter vector components $ \Delta\theta_{j} \ ; \ j = 2,3 $, and $x_2$ is given by (\ref{3.14}). The expression for the dynamically controlled error variable $z_2$ (i.e. the closed loop behaviour of $z_2$ ) is found to be \begin{eqnarray} \dot{z_2}&=&-c_2z_2+ \left(1 + \frac{{\widehat{\Delta\theta_1}}^1}{\Theta_1}\right) \left[-(\Delta\theta_{2}-{\widehat{\Delta\theta_2}}^2) \Theta_1(1-\mu)^2(z_1+Y) \right.\nonumber \\ & & \quad \left. - (\Delta\theta_3-{\widehat{\Delta\theta_3}}^2)(x_2-\Theta_4) \right] + c_1 \left[\left(\Delta\theta_1 -{\widehat{\Delta\theta_1}}^2\right) W_1^1 +\left( \Delta\theta_4-{\widehat{\Delta\theta_4}}^2\right) \right] \nonumber\\ & & \label{3.22} \end{eqnarray} which can also be briefly expressed as \begin{equation} \dot{z_2}=-c_2z_2+(\Delta\theta-{\widehat{\Delta\theta}}^2)^T w_2(z_1,z_2,\mu, {\widehat{ \Delta \theta}}^1, {\widehat{\Delta\theta}}^2) \label{3.23} \end{equation}
The regressor vector for the new estimation process is thus given by \begin{equation} w_2(z_1,z_2,\mu,{\widehat{\Delta\theta}}^1,{\widehat{\Delta\theta}}^2) = \left(1 + \frac{{\widehat{\Delta\theta}_1}^1}{\Theta_1}\right) \gamma_2(z_1,z_2,\mu,{\widehat{\Delta\theta}}^1,{\widehat{\Delta\theta}}^2) + c_1 w_1(z_1,z_2,{\widehat{\Delta\theta}}^1) \end{equation} Note that the dependance of $w_2 $ on ${\widehat{\Delta\theta}}^2$ is implicit through its dependence on $\mu $, as given by the solution of (\ref{3.21}).
As in the previous step, an incremental parameter adaptation law for the vector of new estimates, ${\widehat{\Delta\theta}}^2$, can be devised so as to simultaneously achieve a stable adaptation process and an asymptotic convergence to zero of the "pseudo-control" error variable $z_2$. Such a new incremental parameter update law is given by \begin{equation} {\dot{\widehat{\Delta\theta}}}^2 = z_2\;w_2(z_1,z_2,\mu, {\widehat{\Delta\theta}}^1,{\widehat\Delta\theta}^2) \label{3.24} \end{equation} \subsubsection{Summary of Adaptive Controller Expressions for the Boost Converter} The adaptive PWM controller is next summarized in terms of the original state variables of the system. The constant $Y$ stands for the desired value of the input inductor current $I(t)$. The constants $c_1$, $c_2$ are positive design constants satisfying $c_1$, $c_2 > 2 $.
The adaptive feedback regulated switch position function is synthesized as
\begin{eqnarray} u &=& \left\{ \begin{array}{ll} 1 & {\rm for} \ t_k \le t < t_k + \mu_a(t_k) T\\ 0 & {\rm for} \ t_k + \mu_a(t_k) T \le t < t_k + T \end{array} \right. \nonumber\\ & & t_k + T = t_{k+1} \ \ ; \ \ k = 0,1,2,\ldots \label{3.25} \end{eqnarray} where $\mu_a (t) $ is obtained from a bounding operation carried out on the computed duty ratio $\mu$ in the following manner, \begin{equation} \mu_a(t) = \left\{ \begin{array}{ll} 1 & {\rm if } \ \mu(t) \ \le \ 1\\ \mu(t) & {\rm if} \ 0 \ < \ \mu(t) \ < \ 1 \\ 0 & {\rm if } \ \mu(t) \ \le \ 0 \end{array} \right. \label{3.26} \end{equation}
The duty ratio function $\mu$ is obtained as the solution of the following time-varying differential equation from an initial condition which does not cause permanent saturation of the actual duty ratio $\mu_a(t)$,
\begin{eqnarray} {\dot \mu} & = & \frac{1}{V(t) \left(\Theta_1 + {\widehat{\Delta\theta_1}}^1\right)} \Bigg\{-(I(t) - Y)(1-\mu)^2 V^2(t) + (\Theta_1+{\widehat{\Delta\theta_1}}^1) (1-\mu) \times \nonumber\\ & &\quad \left[(\Theta_2+{\widehat{\Delta\theta}_2}^2)(1-\mu)I(t) - (\Theta_3 + {\widehat{\Delta\theta}_3}^2) V(t) \right] -(I(t)-Y) \nonumber\\ & & \quad -c_1\left[ -(\Theta_1+ {\widehat{\Delta\theta}_1^2})(1-\mu)V(t) + (\Theta_4 + {\widehat{\Delta\theta_4}}^2) \right] \nonumber\\ & & \quad+ c_2\left[ (\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu) V(t) -(\Theta_4 + {\widehat{\Delta\theta}_4}^1) -c_1(I(t) - Y) \right] \Bigg\} \label{3.27} \end{eqnarray}
The estimated values of the controller parameters are obtained as the on-line solution of the following system of differential equations
\begin{eqnarray} {\dot {\widehat{\Delta\theta}_1}^1} &=& - (I(t) - Y) (1-\mu) V(t)\nonumber\\ {\dot {\widehat{\Delta\theta}_2}^1} &=& 0\nonumber\\ {\dot {\widehat{\Delta\theta}_3}^1} &=& 0 \nonumber\\ {\dot {\widehat{\Delta\theta}_4}^1} &=& - (I(t) - Y) \nonumber\\ {\dot {\widehat{\Delta\theta}_1}^2} & = & \left[ - (\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)V(t)+ (\Theta_4 + {\widehat{\Delta\theta}_4}^1) + c_1(I(t) - Y) \right] \left[-c_1(1-\mu) V(t)\right]\nonumber\\ {\dot{\widehat{\Delta\theta}_2}^2} & = & \left[ -(\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)V(t)+ (\Theta_4 + {\widehat{\Delta\theta}_4}^1) + c_1(I(t) - Y) \right] \left[-(\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)^2 I(t) \right] \nonumber\\ {\dot{\widehat{\Delta\theta}_3}^2} & = & \left[ -(\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)V(t)+ (\Theta_4 + {\widehat{\Delta\theta}_4}^1) + c_1(I(t) - Y) \right] \left[(\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)V(t) \right] \nonumber\\ {\dot{\widehat{\Delta\theta}_4}^2} & = & \left[ -(\Theta_1 + {\widehat{\Delta\theta}_1}^1)(1-\mu)V(t)+ (\Theta_4 + {\widehat{\Delta\theta}_4}^1) + c_1(I(t) - Y) \right] c_1 \label{3.28} \end{eqnarray}
\subsubsection{Simulation Results}
Simulations were carried out for a perturbed version of the boost converter model in conjunction with the adaptive controller described by (\ref{3.25})-(\ref{3.28}). An unmodelled stochastic but bounded, yet unmatched, uncertain signal (denoted by $\nu(t) $ ) was hypothesized to be acting on the circuit through the external source voltage $E$. The designed dynamical adaptive PWM controller (\ref{3.25})-(\ref{3.28}) was then directly used for the regulation of the input inductor current variable $I(t)$ of the converter using the actual discontinuously regulated state variables $I(t) $ and $V(t)$ rather than the averaged values $\zeta_1$, $\zeta_2$.
The perturbed circuit model, used in the computer simulations, was taken to be
\begin{eqnarray} {\dot I}(t) &=& -\frac{1}{L}(1-u) \; V(t) + \left( \frac{E + \nu (t) }{L} \right) \nonumber\\ {\dot V}(t) & = & \frac{1}{C}(1-u) \; I(t) - \frac{1}{RC} V(t)\nonumber\\ y & = & I(t) \end{eqnarray}
The simulation results, depicting the behaviour of the controlled converter, are shown in Figure \ref{fig2}. The nominal values of the converter parameters were chosen as $L = 20$ mH, $C = 20$ mF, $R= 30 \ \Omega$ and $E= 15 $ Volts. These values rendered: $\Theta_1 = 50 $; $\Theta_2 = 5.0 \ \times \ 10^4 $; $\Theta_3 = 1.667 \times 10^3 $ and $\Theta_4 = 750$. The actual parameter values used in the simulations were set to $\theta_1 = 55 $, $ \theta_2 = 4.5 \times 10^3 $, $\theta_3 = 1.5 \times 10^3 $ and $\theta_4 = 825 $. These values, however, were assumed to be completely unknown in the controller implementation. In Figure \ref{fig2} the response of the input inductor current $I(t) $ evolves towards the preassigned (nominal) equilibrium value $I(t) = Y = 3.125$ Amp., which corresponds to a nominal output capacitor voltage, $V(t) = 37.5 $ and a duty ratio $U = 0.6$ Volts. The PWM sampling frequency was set to $10 {\rm KHz}$ The asymptotically stable evolution of the duty ratio function $\mu(t)$ towards its equilibrium value, $\mu = U = 0.6 $, along with a small portion (0.1 ms) of the PWM switching actions is also presented in this figure. The trajectories of the estimated incremental parameter values, ${\widehat{\Delta\theta}_1}^1 (t) $, ${\widehat{\Delta\theta}_4}^2 (t)$, ${\widehat{\Delta\theta}_1}^2$,$ {\widehat{\Delta\theta}_2}^2$ ${\widehat{\Delta\theta}_3}^2$ and ${\widehat{\Delta\theta}_4}^2$, are also depicted in this figure. Finally, a sample of the computer generated stochastic perturbation input, $\nu(t)$, is shown at the end of Figure \ref{fig2}.
\subsection{The buck-boost converter}
In this section we briefly summarize the controller expressions obtained from the backstepping calculation procedure applied to a nominally transformed parameter uncertain average PWM buck-boost converter model.
Consider the buck-boost converter circuit shown in Figure \ref{fig3}.
\vspace{0.2in} {\bf State-space model of the buck-boost converter} \begin{eqnarray} {\dot I}(t) &=& \frac{1}{L}(1-u) \; V(t) + \frac{E}{L} u\nonumber\\ {\dot V}(t) & = & -\frac{1}{C}(1-u) \; I(t) - \frac{1}{RC} V(t)\nonumber\\ y(t) & = & I(t) \end{eqnarray} where $ I(t) $ and $ V(t) $ represent respectively the input inductor current and the output capacitor voltage variables. The positive quantity $ E $ is the constant external input voltage. The variable $ u $ denotes the switch position function taking values in the discrete set $\{0,1\}$. The output, $ y(t) $, of the system is represented by the input inductor current $ I(t) $.
\vspace{0.2in} {\bf Nominal parameters definitions}
\begin{equation} \Theta_1 = \frac{1}{L} \ \ \ \ ; \ \ \ \ \Theta_2 = \frac{1}{C} \ \ \ \ ; \ \ \ \ \Theta_3 =\frac{1}{RC} \ \ \ \ ; \ \ \ \ \Theta_4 = \frac{E}{L} \end{equation}
\vspace{0.2in} {\bf Uncertainty model for the parameters}
\begin{equation} \theta_i = \Theta_i + \Delta\theta_i \ \ \ \ ; \ \ \ \ i = 1,2,3,4 \end{equation}
{\bf Average PWM model of the buck-boost converter} \begin{eqnarray} {\dot \zeta}_1 &=& \theta_1(1-\mu) \; \zeta_2 + \theta_4 \mu\nonumber\\ {\dot \zeta}_2 &=& -\theta_2(1-\mu) \; \zeta_1 - \theta_3 \zeta_2\nonumber\\ \eta & = & \zeta_1 \end{eqnarray}
\vspace{0.2in}
{\bf Nominal transformation of average PWM buck-boost converter model to \hspace{0.2in} Fliess' Generalized Observability Canonical Form}
\begin{eqnarray} x_1 & = & \zeta_1 \ \ \ \ ; \ \ \ \ x_2 = \Theta_1(1-\mu)\zeta_2 + \Theta_4 \mu \nonumber\\ \zeta_1 & = & x_1 \ \ \ \ ; \ \ \ \ \zeta_2 = \frac{x_2- \Theta_4 \mu}{\Theta_1 (1 -\mu)} \end{eqnarray}
\vspace{0.2in} {\bf Uncertain buck-boost converter model transformed to generalized phase \hspace{0.2in} variables}
\begin{eqnarray} {\dot x}_1 &=& x_2 + \Delta\theta^T \gamma_1(x_1,x_2)\nonumber\\ {\dot x}_2 &=& f(x_1,x_2,\mu,{\dot \mu}) + \Delta\theta^T \gamma_2(x_1,x_2,\mu) \end{eqnarray} where,
\begin{eqnarray} \Delta\theta^T & = & \left[ \Delta\theta_1 \ \ \Delta\theta_2 \ \ \Delta\theta_3 \ \ Delta\theta_4 \right] \nonumber\\ \gamma_1^T(x_1,x_2) &=& \left[\frac{x_2-\Theta_4 \mu}{\Theta_1} \ \ 0 \ \ 0 \ \ \mu \right] \nonumber\\ \gamma_2^T(x_1,x_2,\mu) &=& \left[0 \ \ -\Theta_1(1-\mu)^2x_1 \ \ -x_2 + \Theta_4 \mu \ \ 0 \right] \nonumber\\ f(x_1,x_2,\mu,{\dot \mu}) & = & {\dot \mu}\left( \Theta_4 - \frac{x_2 - \Theta_4 \mu}{1-\mu}\right) -\Theta_1\Theta_2 (1-\mu)^2x_1 - \Theta_3(x_2 - \Theta_4 \mu) \end{eqnarray}
\vspace{0.2in} \subsubsection{Summary of adaptive controller expressions for the buck-boost converter}
\begin{eqnarray} u &=& \left\{ \begin{array}{ll} 1 & {\rm for} \ t_k \le t < t_k + \mu_a(t_k) T\\ 0 & {\rm for} \ t_k + \mu_a(t_k) T \le t < t_k + T \end{array} \right. \nonumber\\ & & t_k + T = t_{k+1} \ \ ; \ \ k = 0,1,2,\ldots \label{3.38} \end{eqnarray}
where $\mu_a (t) $ is obtained from the following bounding operation
\begin{equation} \mu_a(t) = \left\{ \begin{array}{ll} 1 & {\rm if } \ \mu(t) \ \le \ 1\\ \mu(t) & {\rm if} \ 0 \ < \ \mu(t) \ < \ 1 \\ 0 & {\rm if } \ \mu(t) \ \le \ 0 \end{array} \right. \end{equation}
We let ${\widehat{\theta}_i}^j \ , \ i = 1,\ldots, 4 \ ; \ j = 1,2 $ stand for $\Theta_i + {\widehat{\Delta\theta}_i}^j $ in the following expressions.
\begin{eqnarray} {\dot \mu} & = & \frac{1}{\left({\widehat{\theta_4}}^1 -{\widehat{\theta_1}}^1 V(t)\right)} \Bigg\{-(I(t) - Y)(1-\mu)^2 V^2(t) + {\widehat{\theta_1}}^1 {\widehat{\theta}_2}^2(1-\mu)^2 I(t) \nonumber\\ & &+ {\widehat{\theta}_1}^1{\widehat{\theta}_3}^2(1-\mu)V(t) - (I(t)-Y)\mu -c_1\left( {\widehat{\theta}_1^2}(1-\mu)V(t) + {\widehat{\theta_4}}^2 \mu \right) \nonumber\\ & & + c_2\left[ {\widehat{\theta}_1}^1(1-\mu) V(t) +{\widehat{\theta}_4}^1 +c_1(I(t) - Y) \right] \Bigg\} \end{eqnarray}
\begin{eqnarray} {\dot {\widehat{\theta}_1}^1} &=& (I(t) - Y) (1-\mu) V(t)\nonumber\\ {\dot {\widehat{\theta}_2}^1} &=& 0\nonumber\\ {\dot {\widehat{\theta}_3}^1} &=& 0 \nonumber\\ {\dot {\widehat{\theta}_4}^1} &=& (I(t) - Y) \mu\nonumber\\ {\dot {\widehat{\theta}_1}^2} & = & \left[ {\widehat{\theta}_1}^1(1-\mu)V(t)+ {\widehat{\theta}_4}^1\mu + c_1(I(t) - Y) \right] \left[c_1(1-\mu) V(t)\right]\nonumber\\ {\dot{\widehat{\theta}_2}^2} & = & \left[ {\widehat{\theta}_1}^1(1-\mu)V(t)+ {\widehat{\theta}_4}^1\mu + c_1(I(t) - Y) \right] \left[-{\widehat{\theta}_1}^1(1-\mu)^2 I(t) \right] \nonumber\\ {\dot{\widehat{\theta}_3}^2} & = & \left[ {\widehat{\theta}_1}^1(1-\mu)V(t)+ {\widehat{\theta}_4}^1\mu + c_1(I(t) - Y) \right] \left[-{\widehat{\theta}_1}^1(1-\mu)V(t) \right] \nonumber\\ {\dot{\widehat{\theta}_4}^2} & = & \left[ {\widehat{\theta}_1}^1(1-\mu)V(t)+ {\widehat{\theta}_4}^1\mu + c_1(I(t) - Y) \right] c_1\mu \label{3.41} \end{eqnarray}
\subsubsection{Simulation results}
Simulations were carried out for the following perturbed version of the buck-boost converter model
\begin{eqnarray} {\dot I}(t) &=& \frac{1}{L}(1-u) \; V(t) + \left( \frac{E + \nu (t) }{L} \right) u \nonumber\\ {\dot V}(t) & = & -\frac{1}{C}(1-u) \; I(t) - \frac{1}{RC} V(t)\nonumber\\ y & = & I(t) \end{eqnarray} in conjunction with the adaptive controller described by (\ref{3.38})-(\ref{3.41}).
The simulation results, depicting the behaviour of the controlled converter, are shown in Figure \ref{fig4}. The nominal values of the converter parameters were chosen to be the same as for the boost converter case: $L = 20$ mH, $C = 20$ mF, $R= 30 \ \Omega$, and $E= 15 $ Volts. The actual parameter values used in the simulations were the same as before. The response of the input inductor current $I(t) $ is seen to evolve towards the preassigned (nominal) equilibrium value $I(t) = Y = 1.5 $ amp., which corresponds to a nominal output capacitor voltage, $V(t) = -21.38 $ volts. The PWM sampling frequency was also set at $ 10 {\rm KHz} $ The asymptotically stable evolution of the duty ratio function $\mu(t)$ towards its equilibrium value, $\mu = U = 0.55 $, along with a small portion of the switching action is also presented in this figure. The trajectories of the estimated incremental parameter values, ${\widehat{\Delta\theta}_1}^1 (t) $, ${\widehat{\Delta\theta}_4}^2 (t)$, ${\widehat{\Delta\theta}_1}^2$,$ {\widehat{\Delta\theta}_2}^2$ ${\widehat{\Delta\theta}_3}^2$ and ${\widehat{\Delta\theta}_4}^2$, are also depicted in this figure. Finally, a sample of the computer generated stochastic perturbation input, $\nu(t)$, is shown at the end of Figure \ref{fig4}.
\section{Conclusions \label{sect4}} An adaptive feedback control approach has been proposed which is based on nominal dynamical input-output linearization of the average model of PWM regulated dc-to-dc power converters and the backstepping algorithm. The approach achieves indirect average output capacitor voltage regulation by considering the input inductor current as the regulated output. This procedure sidesteps the non-minimum phase problems associated with direct output capacitor voltage regulation. The simulated behavior of the closed loop system exhibits remarkable robustness with respect to unmatched and unmodelled external perturbation signals of bounded and stochastic nature.
Overparametrization is implicit in the backstepping procedure when applied to systems in Pure Parameter Feedback Canonical Form Kanellakopoulos {\em et al} (1991). This feature substantially contributes to increase the complexity of the controller. An alternative approach to the one presented here is constituted by the possibility of avoiding the overparametrization associated with the incremental parameter update estimation process. The fundamental developments regarding this technique may be found in Krsti\'{c} (1992), and an alternative approach to that in this paper has been presented by Sira-Ram\'{\i}rez {\em et al} (1995b).
Simulations show that the scheme is quite robust with respect to unmodelled stochastic, but bounded, external perturbation inputs of the unmatched type. This type of robust behaviour is inherited from (i) the underlying input-output viewpoint present in the Generalized Observability Canonical Form, used for the derivation of the dynamical feedback controller, and (ii) the robustness features traditionally associated with discontinuous feedback control policies of the Pulse-Width-Modulation type.
A topic for further study is the direct output capacitor voltage regulation problem, which exhibits a non-minimum phase property and, hence, an input-output linearization approach fails. In a recent work Sira-Ram\'{\i}rez {\em et al} (1995a) proposed the possibility of handling the non-minimum phase case by means of a piece-wise unstable dynamical compensator in which a controller output "resetting" strategy is enforced. The adaptive version of this resetting controller did not use the backstepping algorithm.
Another very interesting development has been recently given by Karsenti and Lamnabhi-Lagarrigue (1995) in which the backstepping method is generalized to include sliding mode control strategies in systems with nonlinear parameter dependencies. Application of this latter technique to dc-to-dc power converters represents a welcome contribution since a more realistic class of (nonlinear) incremental circuit parameter variations may be efficiently handled with such a method.
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