- Email: [email protected]
LQ Optimal Control Synthesis for a Class of Pulse Frequency Modulated Systems*
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
LQ Optimal Control Synthesis for a Class of Pulse Frequency Modulated Systems C.-Y. Kao ∗
H. Fujioka ∗∗
∗
Dept. of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]). ∗∗ Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]) Abstract: We consider linear quadratic optimal control for a class of pulse frequency modulated systems. The problem is motivated from a practical application — digital control of a new type of twin-buck switching power converters. The pulse frequency modulated technique is deemed necessary in order to achieve the so-called “zero current switching”. The control synthesis problem is posed based on a sampled data model of the original switching dynamics and a linear quadratic criterion that takes the at-sampling behavior into account. Keywords: Control of switched systems; Switching stability and control; Control design for hybrid systems 1. INTRODUCTION In this manuscript, we consider a class of systems consisting of four affine vector fields that are switched in a given order. The system is operated in cycles. In each cycle, the duration that two of the vector fields are activated is determined by the state of the system at the beginning of the cycle, which is not directly placed under the control of the control mechanism (controller). The duration that the other two vector fields are activated, on the other hand, is determined by the controller. Thus, in each operating cycle, there are two state-dependent switches and two controlled switches. More specifically, in each operating cycle, the duty ratio defined as the faction of the cycle in which two of the vector fields are activated is determined by a certain algorithm and the duration of the cycle is adjusted accordingly so that the duty ratio is realized. As such, the length of the operating cycles is varied and as the matter of fact, the mean of our control. Such switching systems may seem peculiar and special, but is nevertheless of practical interest. One application where such systems arise is control of DC-to-DC power converters with zero current switching via the pulse frequency modulation (PFM) technique, which is precisely what motivates this study (Sun et al. [2000], Hasanien and Sayed [2008], Chen [2009]). Feedback controllers for power converters are often designed based on the averaged dynamics. The classical averaging approach is well suited for fast switching frequency but may be unsuitable if the switching frequency is relatively low, as is the case in many high-power converter applications. Even if the averaging approach were to be applied, the resulting model is still nonlinear with input/state constraints and the control design process remains complicated unless further simplification of the
model is introduced. Such simplification often leads to a conservative design which reduces the controller dynamical performance. In the recent past the advances of computing technology and the corresponding availability of computational power have made digital control as an increasing viable option for power electronics and created interest in developing alternative control methods that could overcome the limitations of the classical approach. There have been several recent studies that apply hybrid control techniques for fixed-frequency, pulse-width-modulated (PWM), DC-to-DC converters. See for example Lincoln and Rantzer [2006], Fujioka et al. [2007], Bˆaja et al. [2008], Wernrud [2008], Geyer et al. [2008], Mari´ethoz and Morari [2009], Beccuti et al. [2009], Patino et al. [2009], Fujioka et al. [2009], Mari´ethoz et al. [2010] and the technical report Morari et al. [2006] for a comprehensive survey of related works in the power electronics area. In this manuscript we consider feedback control design of PFM systems based on a sampled data model involving the lifted dynamics of the original switching system with a quadratic cost function. This is similar to our previous work on the sampled-data LQ and H∞ control for the PWM systems (Fujioka et al. [2007, 2009]). While the underlying principles of the control design we proposed for the two type of systems are similar, the PFM systems is different from the PWM systems in that the frequency modulation results in aperiodic sampling. As such, the sampled-data model for the PFM systems is distinctively more complicated compared to that of the PWM systems, and the corresponding control design problem is likely to be intractable. To simplify the problem, we consider an approximation of the cost function which only takes into account the at-sampling behavior of the system. In this way, the control design problem becomes tractable and we still take into account with high frequency behavior due to switching.
⋆ C.-Y. Kao is supported by Nation Science Council, Taiwan, under the grant 99-2628-E-110-006.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
6313
10.3182/20110828-6-IT-1002.02538
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Notations For a given continuous-time signal f , a discrete-time signal obtained by ideal sampling of f will be denoted by f¯: f¯[k] = f (tk ), where tk , k = 0, 1, 2, · · · , are the sampling instances. fˆ denotes the lifted signal of f : fˆ[k](θ) = f (tk + θ), θ ∈ [0, tk+1 − tk ].
In this paper, the standard Hilbert spaces Rn , L2 [0, T ], n and L2 [0, ∞) are considered. Symbol S+ ⊂ Rn×n denotes the set of symmetric positive definite matrices. Given M ∈ Rn×n , M ′ denotes the transpose of M . Finally, for a given interval T ⊂ R, |T | denotes the length of T . 2. DESCRIPTION OF THE SYSTEM AND MOTIVATION In this section, we describe the class of linear switched systems considered in this manuscript and the motivation of considering such systems. vref
-
K
d-
- v P
y
x(tk ) and can not be adjusted freely. On the other hand, the length of Tk,2 is determined by control algorithms. Let τk := |Tk,1 | = tˆk − tk and dk := |Tk,2 |/|Tk,1 | = (tk+1 − tˆk )/τk . By the above discussion, τk is a state (x(tk )) dependent parameter while parameter dk , which dictates the length of Tk,2 and essentially the length of sampling cycle Tk , can be assigned independently. We will referred to dk (and the sequence d := {dk }∞ k=0 ) as “frequency modulation ratio”. The frequency modulation ratio signal d is to be regarded as the control input of system P by which one can manipulate the output of the system. The discrete time controller to be designed generates the frequency modulation ratio signal d based on a constant reference input vref and the discrete-time measurement output y which is determined by usual ideal sampling at tk . The primal objective of the control design is to regulate the signal v so that the error between v and the reference vref is minimized. Note that, contrast to the systems where the sampling period is fixed and the width of certain pulses is modulated, the controller modulates the sampling period, and hence the technique is called Pulse Frequency Modulation (PFM, as opposed to Pulse Width Modulation, or PWM). L
S1
Fig. 1. Feedback Control System Ls
iL1
The switching system under consideration in this manuscript S2 L is illustrated Fig. 1. Block P represents a linear switching C system governed by + iL2 S3 Vin S4 − A1 x(t) + B1 when t ∈ Tk,1 x(t) ˙ = A2 x(t) + B2 when t ∈ Tk,2 (1) Fig. 2. A Type of Twin-Buck converter. v(t) = Cx(t) y[k] = M x(tk ) where x, v are continuous time signals which denote the Tk+1 Tk states, and the output signal to be regulated. The precise meaning of “regulation” will be clarified later. The signal S1 y is a discrete time signal (or, sampled-data signal) to be fed into a discrete time controller, which is identified in Fig. 1 as the block K. We make a standing assumption S4 ˜ such that that there exists a real matrix M ˜ M. S2 C=M This assumption implies that v(tk ) can be obtained diS3 rectly from the measurement y[k]. The discrete time components of the system are synchronized to operate in cycles, in particular, the sampling operation and the switching operation. The sampling operation occurs only once per cycle and is assumed always to take place at the beginning of each cycle. The switching operation, on the other hand, happens two times in a cycle. Let Tk := [tk , tk+1 ) denotes the k th sampling cycle. At the k th sampling cycle, the dynamics of the system is governed by the first affine vector field during time interval Tk,1 := [tk , tˆk ) and by the second affine vector field during time interval Tk,2 := [tˆk , tk + 1). The switching instance tk and tˆk must obey F1′ x(tk ) + F2′ x(tˆk ) + F3 (tˆk − tk ) = 0, (2)
where F1 , F2 ∈ Rn×1 , and F3 is a real number. Thus the length of Tk,1 , i.e., tˆk −tk , is determined by the state vector
Tk,1 tk
tˆk
Tk,2 tk+1
tˆk+1
R
Vout
tk+2
Fig. 3. Timing diagram of four switching signals S1 to S4 . The motivation of considering such control systems comes from controlling a certain type of twin-buck switching power converters (Sun et al. [2000], Hasanien and Sayed [2008], Chen [2009]). An example of the circuitry of such converters is shown in Fig. 2, while the four square-wave signals in Fig. 3 illustrates the gate signals in one period. The controlled switch is turned on (connected) when the corresponding gate signal has value 1 (at high voltage level), and is turned off (disconnected) when the gate
6314
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
signal has value 0. Let IL := iL1 + iL2 . It can be shown that the circuitry is governed by the following equations 2 1 1 1 I˙L = − Vout + Vin , V˙ out = IL − Vin L L C CR when (S1 , S2 , S3 , S4 ) is equal to (1, 0, 0, 1) and (0, 1, 1, 0) (i.e., at time intervals Tk,1 , Tk+1,1 , Tk+2,1 , · · · ), and by 2L + Ls 2L + Ls I˙L = − Vout + Vin , L(L + Ls ) L(L + Ls ) 1 1 V˙ out = IL − Vin C CR when (S1 , S2 , S3 , S4 ) is equal to (1, 0, 0, 0) and (0, 1, 0, 0) (i.e., at time intervals Tk,2 , Tk+1,2 , Tk+2,2 , · · · ). To minimize the power loss, it is desirable to disconnect gates 3 and 4 when there is no current flowing through these two gates. This technique is called “zero current switching” (ZCS). In order to achieve ZCS, it can be verified that τk must satisfy Ls Ls τk = 0. (IL (tk ) + IL (tk + τk )) − 2 + Vin L Moreover, suppose the system reaches steady state, one can also verify that 2d0 + 1 the average of Vout = 0 Vin , (3) 2d + 2 where d0 is the stationary frequency modulation ratio. For the detailed working principle of such twin-buck converter, readers are referred to Chen [2009]. Note that due to the switching nature of the system, the signal v is periodic at stationarity; therefore, the error between v and vref can never converge to 0. Thus, the primal objective of our control design task is to make the error v(t) − vref small. 3. THE LIFTING REPRESENTATION AND THE PROBLEM FORMULATION In this section, we introduce the so-called lifting representation of system (1) and formulate the control synthesis problem to be considered in this manuscript. 3.1 The Lifting Representation The lifting representation of system (1) is given by ( x ¯[k + 1] = Φ(dk , τk )¯ x[k] + Γ(dk , τk ) vˆ[k] = Ψ(dk , τk )¯ x[k] + Θ(dk , τk ) (4) y[k] = M x ¯[k] where operators Φ(d, t): Rn → Rn , Γ(d, t): R → Rn , Ψ(d, t): Rn → L2 [0, (d + 1)t), and Θ(d, t): R → L2 [0, (d + 1)t) are defined with parameters d ∈ (0, ∞) and t ∈ (0, ∞) as follows: ˇ ˇ [Φ(d, t) Γ(d, t)] := [In 0] eA2 dt eA1 t [Ψ(d, t) Θ(d, t)] := Ω(d, t) and Ω(d, t): Rn+1 → C[0, ∞) is an operator defined as (Ω1 (d, t)x)(θ), when θ ∈ T1 (Ω(d, t)x)(θ) := (Ω2 (d, t)x)(θ), when θ ∈ T2 ˇ Aˇ1 θ x, (Ω1 (d, t)x)(θ) := Ce ˇ Aˇ2 (θ−t) eAˇ1 t x. (Ω2 (d, t)x)(θ) := Ce T1 := [0, t), T2 := [t, (d + 1)t).
In the above expression, Aˇ1 , Aˇ2 , and Cˇ are matrices of the following forms A1 B1 ˇ A2 B2 Aˇ1 := , A2 := , Cˇ := [C 0] . 0 0 0 0
Moreover, the constraint (2) that τk has to obey can be expressed as F1 (τk )′ x ¯[k] + F2 (τk ) = 0 (5) where F1 (t)′ := F1′ + F2′ eA1 t Z t F2 (t) := F2′ eA1 (t−s) B1 ds + F3 t 0
The derivation of (4) is straightforward by observing that one has the following expressions for x: Z θ A1 θ eA1 (θ−s) B1 ds, θ ∈ [0, τk ) e x ˆ [k](0) + 0 Z θ x ˆ[k](θ) = eA2 (θ−τk ) x ˆ[k](τk ) + eA2 (θ−s) B2 ds, τk θ ∈ [τk , (dk + 1)τk ) and the standard formula for matrix exponentials: Z θ Aθ A(θ−τ ) A B e B dτ . θ = e exp 0 0 0 0 I We assume that the plant (1) attains at least one periodic solution x0 of period T0 := (d0 + 1)τ 0 if all dk are set to d0 and the system is initialized by x0 := x0 (0). The periodicity of x0 implies that x0 = Φ0 x0 + Γ0 (6) or equivalently 0 ˇ2 d0 τ 0 A ˇ1 τ 0 0 x A 0 , (7) (I − e e )ˇx = 0, ˇx := 1
where Φ0 := Φ(d0 , τ 0 ) and Γ0 := Γ(d0 , τ 0 ). The continuous time output v corresponding to x0 , denoted by v 0 , is also periodic and satisfies vˆ0 [k] = Ψ0 x0 + Θ0 = Ω0 ˇx0 (8) for any k, where Ψ0 := Ψ(d0 , τ 0 ), Θ0 := Θ(d0 , τ 0 ), Ω0 := Ω(d0 , τ 0 ). (9) Finally, τ 0 and x0 must obey constraint (5); i.e., F1 (τ 0 )′ x0 + F2 (τ 0 ) = 0. (10) Now, let us consider the difference of the transient response of the plant and the nominal periodic solution at the sampling instances. More specifically, let ¯ := x ξ[k] ¯[k] − x0 , ¯ e[k] := v(tk ) − v 0 (T0 ) = C ξ[k], η[k] := y[k] − M x0 , where the discrete-time signal e measures the deviation of output v at the sampling instances to the desired value v 0 (T0 ). Using (4), (6), and (8), we can derive the error dynamics at sampling instants: ¯ + 1] = Φ(dk , τk )ξ[k] ¯ + Γ(d ˜ k , τk ) ξ[k ¯ (11) e[k] = C ξ[k] ¯ η[k] = M ξ[k] ˜ t) ∈ Rn is defined as follows where Γ(d, ˜ t) := [Φ(d, t) Γ(d, t)] − Φ0 Γ0 ˇx0 . Γ(d,
6315
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
0
0
Mx
-c η −6
-
Kd
d u ? c d-
P
by O (A, C, T ). Proposition 1. For given x0 , τ 0 and d0 satisfying (7) and (10), one has 0 ′ 0 1 ˇx ˇx 0 0 Js (d0 , τ 0 , x0 ) = 0 Q (d , τ ) s vref (d + 1)τ 0 vref
- v y
where Qs (d0 , τ 0 ) has the form ˆ d0 τ 0 eAˆ1 τ 0 ˆ τ 0 + eAˆ′1 τ 0 O Aˆ2 , C, O Aˆ1 , C,
Fig. 4. Feedback Control Configuration 3.2 Problem Formulation The primal objective of our control design task is to make the error v(t) − vref small. To achieve this, we consider a two step design procedure. (1) We first design d0 and x0 such that the power of the stationary error Z 1 τ 0 |v (t) − vref |2 dt Js := lim τ →∞ τ 0 (12) Z T0 1 0 2 = 0 |ˆ v [k](θ) − vref | dθ T 0 is minimized. (2) Then we design a dynamic output feedback controller which ensure asymptotic convergence to x0 in such a way that the rate of convergence measured by the cost function J of the following form ∞ X q1 |e[k]|2 + q2 |ed [k]|2 + q3 ||Tk | − T0 |2 + r|u[k]|2 k=0
(13) is minimized, where qi ≥ 0, i = 1, 2, 3, and r > 0 are design parameters, and ed and u are respectively defined as k−1 X η1 [i], u[k] := dk − d0 . ed [k] := i=0
Note the term ||Tk | − T0 |2 is introduced to penalize the aperiodicity so that, by minimizing this term, appropriate control action is produced to drive the system to the periodic state. Also note that introducing the term ed in (13) leads to an integral action in the resultant controller.
The resultant feedback configuration is illustrated in Fig. 4. Note that in the design step (2), we only take into account the at-sampling error e[k]. Ideally one would also like to take into account the inter-sampling error. This action, however, would result in a very difficult optimization problem due to the aperiodic nature of PFM. Hence, in this manuscript we only consider the simple discrete-time cost function (13) as the first step towards establishing a more sophisticate sampled-data control paradigm for PFM systems. 4. DESIGN OF THE STATIONARY FREQUENCY MODULATION RATIO The following proposition provides a computational formula for Js in (12). To facilitate the development, let us denote the finite observability grammian by Z T ′ eA t C ′ CeAt dt 0
and Aˆi , i = 1, 2, and Cˆ Aˇi Aˆi := 0
are defined as follows 0 , Cˆ := Cˇ −1 . 0
Proof. The proof follows straightforwardly that
vˆk0 (θ) − vref
0 ˆ1 θ ˇx A ˆ , θ ∈ T10 Ce vref 0 = ˆ2 (θ−τ 0 ) A ˆ1 τ 0 ˇx A ˆ Ce e , θ ∈ T20 vref
where T10 := [0, τ 0 ), and T20 := [τ 0 , (d0 + 1)τ 0 ).
By this proposition, the minimization of the stationary error can be formulated as min Js (d0 , τ 0 , x0 ) d0 ∈R,x0 ∈Rn ,τ 0 ∈R (14) subject to (7) and (10) Optimization problem (14) is nonlinear and likely to have multiple local minimums. Hence, having a good initial point is crucial. For applications such as control of the DC-DC converter described in Section 2, stationary frequency modulation ratio d0 is related to the average output voltage as in (3). Such knowledge from the physical configuration of the system can be used to find a good initial point. Furthermore, notice that for a given d0 , there could be only one pair of (τ 0 , x0 ) satisfying (7) and (10). Should this be the case, the optimization problem is one dimensional (over d0 ) and the computational complexity of the problem is significantly less than what it appears. 5. DESIGN OF FEEDBACK REGULATOR In this section, we discuss how to obtain a discrete time feedback regulator Kd in the form of "xK [k]# AK BK1 BK2 xK [k + 1] η[k] , (15) = Kd : u[k] CK DK1 DK2 ed [k] which ensures asymptotic stability of the system and optimizes J defined in (13). Despite the nice structures, the optimization problem involving error dynamics (11) and objective function (13) is highly nonlinear and in general the cost function J is finite only when the state is restricted within some neighborhood of the origin where (11) is stabilizable. As such, finding a feedback regulator which achieves globally asymptotic stability is expected to be very difficult and perhaps intractable. Hence we consider the linearization of the state equation (11) together with the quadratic approximation of the cost function J which give rise to a linear quadratic regulation problem.
6316
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
5.2 LQ Regulator for Linearized Error Dynamics with Approximated Cost Function
5.1 Small Perturbation Model The cost function J is not a quadratic functional of ¯ u) due to the term ||Tk | − T0 |2 . To obtain a quadratic (ξ, functional, we will approximate |Tk | by its dominating term in a small neighborhood around T0 . To this end, we observe that |Tk | = (dk + 1)τk = (d0 + 1 + u[k])(τ 0 + δτk ) = T0 + (d0 + 1)δτk + τ 0 u[k] + δτk u[k]. Thus, when δτk and u[k] are small, ||Tk |−T0 | is dominated by |(d0 + 1)δτk + τ 0 u[k]|. Furthermore, according to (5), δτk satisfies the following equation ¯ + x0 + F2 (τ 0 + δτk ) = 0, F1 (τ 0 + δτk )′ ξ[k] where the left-hand side of the equation is dominated by ¯ F1 (τ 0 )′ x0 + F2 (τ 0 ) + F1 (τ 0 )′ ξ[k] dF1 0 ′ 0 dF2 0 (τ ) x + (τ ) δτk + dt dt Since the sum of the first two terms of the above quantity is equal to zero, we obtain that −1 dF1 0 ′ 0 dF2 0 ¯ δτk ≈ − (τ ) x + (τ ) F1 (τ 0 )′ ξ[k] dt dt ˜ 0 )′ ξ[k]. ¯ := F(τ (16) where dF1 ′ (t) = F2′ eA1 t A1 , dt Z t ∂F2 (t) = F2′ B1 + F2′ A1 eA1 (t−s) B1 ds + F3 , ∂t 0 We have the following lemma regarding the quadratic approximation of the cost function J. ¯ Lemma 2. Assuming ξ[k] and u[k] are small, the cost function J defined in (13) is dominated by the following quadratic function ∞ X ˜ 12 ξ[k] ˜ Q ¯ ¯ Q ξ[k] 2 , ˜ 11 J˜ := ˜ 22 u[k] + q2 |ed [k]| (17) u[k] Q′12 Q k=0
where ˜ 11 := q1 C ′ C + q3 (d0 + 1)2 F(τ ˜ 0 )F(τ ˜ 0 )′ Q ˜ 0 ), Q ˜ 12 := q3 τ 0 (d0 + 1)F(τ ˜ 22 := r + q3 (τ 0 )2 Q
We include the integrator dynamics, which generates ed , in the plant to obtain the augmented system ¯ ¯ + 1] Γ Φ 0 ξ[k] ξ[k u[k] + = 0 C 1 ed [k] ed [k + 1] ¯ M 0 ξ[k] η[k] . = 0 1 ed [k] ed [k] Hence the closed-loop dynamics is given by ¯ ¯ + 1] ξ[k] ξ[k ed [k + 1] = Φcℓ ed [k] , Φcℓ := Φa + Γa KΛa , xK [k] xK [k + 1] and the linear quadratic approximation of (13) is represented by ′ ¯ ¯ ξ[0] ξ[0] J˜ = ed [0] P ed [0] xK [0] xK [0] where " # # " # " M 0 0 Γ 0 Φ0 0 Φa := C 1 0 , Γa := 0 0 , Λa := 0 1 0 , 0 0 I 0 I 0 0 0 DK1 DK2 CK K := , BK1 BK2 AK and P is a positive definite matrix satisfying the following matrix inequality: Φ′cℓ PΦcℓ − P+ (Ψa + Θa KΛa )′ (Ψa + Θa KΛa ) < 0. (19) Here Ψa and Θa are given by Ψ √0 0 Θ 0 Ψa := , , Θa := 0 q2 0 0 0 using any factorization of matrices defined in Lemma. 2: ′ ˜ 12 ˜ 11 Q Ψ Q =: [Ψ Θ] . ˜′ Q ˜ 22 Θ′ Q 12 LMI techniques can be used to search P and K. Results are summarized in the sequel. Proofs are omitted since they are similar to ones in Fujioka et al. [2007]. Theorem 3. The following two statements are equivalent:
0
Proof 1. Approximating ||Tk | − T | by its dominating term |(d0 + 1)δτk + τ 0 u[k]|, we obtain the form of J˜ by substituting (16) into (13). Around (d0 , τ 0 ), the linearized error dynamics (11) satisfies ¯ + 1] = Φξ[k] ¯ + Γu[k] ξ[k ¯ e[k] = C ξ[k] (18) ¯ η[k] = M ξ[k] ˜ ˜ 0 )′ , Γ = ∂ Γ˜ (d0 , τ 0 ), where Φ = Φ(d0 , τ 0 ) + ∂∂tΓ (d0 , τ 0 )F(τ ∂d and ˜ ∂Γ ˇ ˇ ˇ ˇ (d, t) = [In 0] dAˇ2 eA2 dt eA1 t + eA2 dt Aˇ1 eA1 t ∂t ˜ ∂Γ ˇ ˇ (d, t) = [In 0] tAˇ2 eA2 dt eA1 t ∂d With the linearized error dynamics (18) and the cost function (17), one reaches at a discrete-time LQ control problem.
6317
(i) There exist P = P′ > 0 and K satisfying (19). n (ii) There exist P1 = P1′ ∈ S+ and Y Y n+3 , Y1 ∈ R(n+2)×(n+2) Y = 1′ 3 = Y ′ ∈ S+ Y3 Y2 satisfying the following three LMIs: ˜ 11 )(M ′ )′⊥ < 0, (M ′ ) (Φ′ P Φ − P1 + Q (20) ′ ⊥ 1 Γ Γ 0 0 Θ ΠY Π′ − Yˆ Θ ⊥ < 0, (21) ⊥ 0 I2 0 I2 P1 In 0 In Y1 Y3 ≥ 0 (22) 0 Y3′ Y2 where Y1 0 Y3 0 Φ 0 0 I 0 0 Ψ 0 . , Yˆ := ′ n Π := C √1 Y3 0 Y2 0 0 q2 0 0 0 1 The subscript (·)⊥ is used to denote the orthogonal complement of a matrix.
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Thus, existence of a stabilizing controller K is characterized in terms of linear matrix inequalities (20) to (22). Should these matrix inequalities are feasible, a stabilizing controller of dimension n + 1 can be constructed. Here due to space limitation, we omit the details. 6. EXAMPLE To illustrate the theory presented in this manuscript, we consider the twin-buck converter depicted in Fig. 2. ′ The state vector of the system is x = [IL Vout ] , where IL := iL1 + iL2 , and the system matrices are 2L + Ls 2 0 − 0 − L(L + Ls ) L A1 = 1 , 1 , A2 = 1 1 − − C CR C CR ′ ′ (2L + Ls )Vin Vin 0 , B1 = 0 , B2 = L(L + Ls ) L Ls Ls . C = M = [0 1] , F1 = F2 = 0 , F3 = −2 − L Vin The nominal values of the parameters are Vin = 50, L = 250 × 10−6 , Ls = 30 × 10−6 , C = 33 × 10−6 , R = 3.
Setting vref = 37.5, we obtain corresponding d0 = 1, ′ τ 0 = 7.0107 × 10−6, and x0 = [12.7264 37.1561] . With the values of (q1 , q2 , q3 , r) in (13) selected to be (1, 0.01, 1, 23), Theorem 3 is applied to obtain an output feedback regulator. Figure 5 illustrates responses of the closed-loop system under source (Vin ) and load (R) variations. The upper diagrams show time histories of the output voltage in two different simulations, where during the time interval [0.01, 0.02] the source voltage Vin (left diagram) and the load R (right diagram) increase 20% and 100%, respectively. The lower diagrams show time histories of current flowing through one of the control switches. The diagrams verify that the switch is always disconnected when no the current flows through it. 45
47.5 45 output voltage
output voltage
42.5 40 37.5 35
42.5 40 37.5 35 32.5
32.5
30
30 0.005
0.01
0.015 0.02 Time (Second)
0.025
27.5 0.005
0
−5
−10
−15
0.015 0.02 Time (Second)
0.025
2 current through a switch
current through a switch
5
0.01
14.96
14.98 15 15.02 Time (Millisecond)
15.04
0 −2 −4 −6 −8
14.96
14.98 15 15.02 Time (Millisecond)
15.04
Fig. 5. Illustrations of transient responses of the closedloop twin-buck converter.
7. CONCLUDING REMARKS Motivated by a practical application, we consider optimal control synthesis for a class of pulse frequency modulated systems based on a sampled-data model. The problem is highly nonlinear; to simplify the task, a linear quadratic approximation is adopted and the corresponding control design problem can be solved by computationally efficient algorithm. The proposed design method is applied to find an output feedback controller for a twin-buck converter. Simulation results are provided to verify the effectiveness of the controller. REFERENCES M. Bˆ aja, H. Cormerais, and J. Buisson. Modeling and hybrid control of a four-level three-cell DC-DC converter. In Proceedings of 34th IEEE Conference on Industrial Electronics, pages 3278– 3283, 2008. A. Beccuti, S. Mari´ ethoz, S. Cliquennois, S. Wang, and M. Morari. Explicit model predictive control of DC-DC switched mode power supplies with extended Kalman filtering. IEEE Transactions on Industrial Electronics, 56(3):1864–1874, 2009. Y.-J. Chen. An interleaved twin-buck converter with zero-voltage transition. Master’s thesis, Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, 2009. H. Fujioka, C.-Y. Kao, S. Alm´ er, and U. J¨ onsson. LQ optimal control for a class of pulse width modulated systems. Automatica, 43(6): 1009–1020, 2007. H. Fujioka, C.-Y. Kao, S. Alm´ er, and U. J¨ onsson. Robust tracking with h∞ performance for PWM systems. Automatica, 45(8):1808– 1818, 2009. T. Geyer, G. Papafotiou, R. Frasca, and M. Morari. Constrained optimal control of the step-down DC-DC converter. IEEE Transactions on Power Electronics, 23(5):2454–2464, 2008. B. M. Hasanien and K. F. A. Sayed. Current source ZCS PFM DC-DC converter for magnetron power supply. In Proceedings of 12th International Middle-East Power System Conference, pages 464–469, 2008. B. Lincoln and A. Rantzer. Relaxing dynamic programming. IEEE Transactions on Automatic Control, 51(5):1249–1260, 2006. S. Mari´ ethoz and M. Morari. Explicit model predictive control of a PWM inverter with an LCL filter. IEEE Transactions on Industrial Electronics, 56(2):388–399, 2009. S. Mari´ ethoz, S. Alm´ er, M. Bˆ aja, A. G. Beccuti, D. Patino, A. Wernrud, J. Buisson, H. Cormerais, T. Geyer, H. Fujioka, U. J¨ onsson, C.-Y. Kao, M. Morari, G. Papafotiou, A. Rantzer, and P. Riedinger. Comparison of hybrid control techniques for buck and boost DC-DC converters. IEEE Transactions on Control System Technology, 18(5):1126–1145, 2010. M. Morari, J. Buisson, B. de Schutter, and G. Papafotiou. Report on the assessment of hybrid control methods for electric energy management problems. Technical Report HYCON Deliverable, Dept. of Mathematics, Royal Inst. of Technology, 2006. [Online]. Available:http://control.ee.ethz.ch/hycon/downloads free/HYCON D4a51.pdf. D. Patino, P. Riedinger, and C. Iung. Practical optimal state feedback control law for continuous-time affine switched systems with cyclic steady state. International Journal of Control, 82: 1357–1376, 2009. J. Sun, X. Ding, M. Nakaoka, and H. Takano. Series resonant ZCS-PFM DC-DC converter with multistage rectified voltage multiplier and dual-mode PFM control scheme for medical-use high-voltage x-ray power generator. IEE Proceedings on Electric Power Applications, 147(6):527–534, 2000. A. Wernrud. Approximate dynamic programming with applications. PhD thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, 2008.
6318
LQ Optimal Control Synthesis for a Class of Pulse Frequency Modulated Systems C.-Y. Kao ∗
H. Fujioka ∗∗
∗
Dept. of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]). ∗∗ Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]) Abstract: We consider linear quadratic optimal control for a class of pulse frequency modulated systems. The problem is motivated from a practical application — digital control of a new type of twin-buck switching power converters. The pulse frequency modulated technique is deemed necessary in order to achieve the so-called “zero current switching”. The control synthesis problem is posed based on a sampled data model of the original switching dynamics and a linear quadratic criterion that takes the at-sampling behavior into account. Keywords: Control of switched systems; Switching stability and control; Control design for hybrid systems 1. INTRODUCTION In this manuscript, we consider a class of systems consisting of four affine vector fields that are switched in a given order. The system is operated in cycles. In each cycle, the duration that two of the vector fields are activated is determined by the state of the system at the beginning of the cycle, which is not directly placed under the control of the control mechanism (controller). The duration that the other two vector fields are activated, on the other hand, is determined by the controller. Thus, in each operating cycle, there are two state-dependent switches and two controlled switches. More specifically, in each operating cycle, the duty ratio defined as the faction of the cycle in which two of the vector fields are activated is determined by a certain algorithm and the duration of the cycle is adjusted accordingly so that the duty ratio is realized. As such, the length of the operating cycles is varied and as the matter of fact, the mean of our control. Such switching systems may seem peculiar and special, but is nevertheless of practical interest. One application where such systems arise is control of DC-to-DC power converters with zero current switching via the pulse frequency modulation (PFM) technique, which is precisely what motivates this study (Sun et al. [2000], Hasanien and Sayed [2008], Chen [2009]). Feedback controllers for power converters are often designed based on the averaged dynamics. The classical averaging approach is well suited for fast switching frequency but may be unsuitable if the switching frequency is relatively low, as is the case in many high-power converter applications. Even if the averaging approach were to be applied, the resulting model is still nonlinear with input/state constraints and the control design process remains complicated unless further simplification of the
model is introduced. Such simplification often leads to a conservative design which reduces the controller dynamical performance. In the recent past the advances of computing technology and the corresponding availability of computational power have made digital control as an increasing viable option for power electronics and created interest in developing alternative control methods that could overcome the limitations of the classical approach. There have been several recent studies that apply hybrid control techniques for fixed-frequency, pulse-width-modulated (PWM), DC-to-DC converters. See for example Lincoln and Rantzer [2006], Fujioka et al. [2007], Bˆaja et al. [2008], Wernrud [2008], Geyer et al. [2008], Mari´ethoz and Morari [2009], Beccuti et al. [2009], Patino et al. [2009], Fujioka et al. [2009], Mari´ethoz et al. [2010] and the technical report Morari et al. [2006] for a comprehensive survey of related works in the power electronics area. In this manuscript we consider feedback control design of PFM systems based on a sampled data model involving the lifted dynamics of the original switching system with a quadratic cost function. This is similar to our previous work on the sampled-data LQ and H∞ control for the PWM systems (Fujioka et al. [2007, 2009]). While the underlying principles of the control design we proposed for the two type of systems are similar, the PFM systems is different from the PWM systems in that the frequency modulation results in aperiodic sampling. As such, the sampled-data model for the PFM systems is distinctively more complicated compared to that of the PWM systems, and the corresponding control design problem is likely to be intractable. To simplify the problem, we consider an approximation of the cost function which only takes into account the at-sampling behavior of the system. In this way, the control design problem becomes tractable and we still take into account with high frequency behavior due to switching.
⋆ C.-Y. Kao is supported by Nation Science Council, Taiwan, under the grant 99-2628-E-110-006.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
6313
10.3182/20110828-6-IT-1002.02538
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Notations For a given continuous-time signal f , a discrete-time signal obtained by ideal sampling of f will be denoted by f¯: f¯[k] = f (tk ), where tk , k = 0, 1, 2, · · · , are the sampling instances. fˆ denotes the lifted signal of f : fˆ[k](θ) = f (tk + θ), θ ∈ [0, tk+1 − tk ].
In this paper, the standard Hilbert spaces Rn , L2 [0, T ], n and L2 [0, ∞) are considered. Symbol S+ ⊂ Rn×n denotes the set of symmetric positive definite matrices. Given M ∈ Rn×n , M ′ denotes the transpose of M . Finally, for a given interval T ⊂ R, |T | denotes the length of T . 2. DESCRIPTION OF THE SYSTEM AND MOTIVATION In this section, we describe the class of linear switched systems considered in this manuscript and the motivation of considering such systems. vref
-
K
d-
- v P
y
x(tk ) and can not be adjusted freely. On the other hand, the length of Tk,2 is determined by control algorithms. Let τk := |Tk,1 | = tˆk − tk and dk := |Tk,2 |/|Tk,1 | = (tk+1 − tˆk )/τk . By the above discussion, τk is a state (x(tk )) dependent parameter while parameter dk , which dictates the length of Tk,2 and essentially the length of sampling cycle Tk , can be assigned independently. We will referred to dk (and the sequence d := {dk }∞ k=0 ) as “frequency modulation ratio”. The frequency modulation ratio signal d is to be regarded as the control input of system P by which one can manipulate the output of the system. The discrete time controller to be designed generates the frequency modulation ratio signal d based on a constant reference input vref and the discrete-time measurement output y which is determined by usual ideal sampling at tk . The primal objective of the control design is to regulate the signal v so that the error between v and the reference vref is minimized. Note that, contrast to the systems where the sampling period is fixed and the width of certain pulses is modulated, the controller modulates the sampling period, and hence the technique is called Pulse Frequency Modulation (PFM, as opposed to Pulse Width Modulation, or PWM). L
S1
Fig. 1. Feedback Control System Ls
iL1
The switching system under consideration in this manuscript S2 L is illustrated Fig. 1. Block P represents a linear switching C system governed by + iL2 S3 Vin S4 − A1 x(t) + B1 when t ∈ Tk,1 x(t) ˙ = A2 x(t) + B2 when t ∈ Tk,2 (1) Fig. 2. A Type of Twin-Buck converter. v(t) = Cx(t) y[k] = M x(tk ) where x, v are continuous time signals which denote the Tk+1 Tk states, and the output signal to be regulated. The precise meaning of “regulation” will be clarified later. The signal S1 y is a discrete time signal (or, sampled-data signal) to be fed into a discrete time controller, which is identified in Fig. 1 as the block K. We make a standing assumption S4 ˜ such that that there exists a real matrix M ˜ M. S2 C=M This assumption implies that v(tk ) can be obtained diS3 rectly from the measurement y[k]. The discrete time components of the system are synchronized to operate in cycles, in particular, the sampling operation and the switching operation. The sampling operation occurs only once per cycle and is assumed always to take place at the beginning of each cycle. The switching operation, on the other hand, happens two times in a cycle. Let Tk := [tk , tk+1 ) denotes the k th sampling cycle. At the k th sampling cycle, the dynamics of the system is governed by the first affine vector field during time interval Tk,1 := [tk , tˆk ) and by the second affine vector field during time interval Tk,2 := [tˆk , tk + 1). The switching instance tk and tˆk must obey F1′ x(tk ) + F2′ x(tˆk ) + F3 (tˆk − tk ) = 0, (2)
where F1 , F2 ∈ Rn×1 , and F3 is a real number. Thus the length of Tk,1 , i.e., tˆk −tk , is determined by the state vector
Tk,1 tk
tˆk
Tk,2 tk+1
tˆk+1
R
Vout
tk+2
Fig. 3. Timing diagram of four switching signals S1 to S4 . The motivation of considering such control systems comes from controlling a certain type of twin-buck switching power converters (Sun et al. [2000], Hasanien and Sayed [2008], Chen [2009]). An example of the circuitry of such converters is shown in Fig. 2, while the four square-wave signals in Fig. 3 illustrates the gate signals in one period. The controlled switch is turned on (connected) when the corresponding gate signal has value 1 (at high voltage level), and is turned off (disconnected) when the gate
6314
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
signal has value 0. Let IL := iL1 + iL2 . It can be shown that the circuitry is governed by the following equations 2 1 1 1 I˙L = − Vout + Vin , V˙ out = IL − Vin L L C CR when (S1 , S2 , S3 , S4 ) is equal to (1, 0, 0, 1) and (0, 1, 1, 0) (i.e., at time intervals Tk,1 , Tk+1,1 , Tk+2,1 , · · · ), and by 2L + Ls 2L + Ls I˙L = − Vout + Vin , L(L + Ls ) L(L + Ls ) 1 1 V˙ out = IL − Vin C CR when (S1 , S2 , S3 , S4 ) is equal to (1, 0, 0, 0) and (0, 1, 0, 0) (i.e., at time intervals Tk,2 , Tk+1,2 , Tk+2,2 , · · · ). To minimize the power loss, it is desirable to disconnect gates 3 and 4 when there is no current flowing through these two gates. This technique is called “zero current switching” (ZCS). In order to achieve ZCS, it can be verified that τk must satisfy Ls Ls τk = 0. (IL (tk ) + IL (tk + τk )) − 2 + Vin L Moreover, suppose the system reaches steady state, one can also verify that 2d0 + 1 the average of Vout = 0 Vin , (3) 2d + 2 where d0 is the stationary frequency modulation ratio. For the detailed working principle of such twin-buck converter, readers are referred to Chen [2009]. Note that due to the switching nature of the system, the signal v is periodic at stationarity; therefore, the error between v and vref can never converge to 0. Thus, the primal objective of our control design task is to make the error v(t) − vref small. 3. THE LIFTING REPRESENTATION AND THE PROBLEM FORMULATION In this section, we introduce the so-called lifting representation of system (1) and formulate the control synthesis problem to be considered in this manuscript. 3.1 The Lifting Representation The lifting representation of system (1) is given by ( x ¯[k + 1] = Φ(dk , τk )¯ x[k] + Γ(dk , τk ) vˆ[k] = Ψ(dk , τk )¯ x[k] + Θ(dk , τk ) (4) y[k] = M x ¯[k] where operators Φ(d, t): Rn → Rn , Γ(d, t): R → Rn , Ψ(d, t): Rn → L2 [0, (d + 1)t), and Θ(d, t): R → L2 [0, (d + 1)t) are defined with parameters d ∈ (0, ∞) and t ∈ (0, ∞) as follows: ˇ ˇ [Φ(d, t) Γ(d, t)] := [In 0] eA2 dt eA1 t [Ψ(d, t) Θ(d, t)] := Ω(d, t) and Ω(d, t): Rn+1 → C[0, ∞) is an operator defined as (Ω1 (d, t)x)(θ), when θ ∈ T1 (Ω(d, t)x)(θ) := (Ω2 (d, t)x)(θ), when θ ∈ T2 ˇ Aˇ1 θ x, (Ω1 (d, t)x)(θ) := Ce ˇ Aˇ2 (θ−t) eAˇ1 t x. (Ω2 (d, t)x)(θ) := Ce T1 := [0, t), T2 := [t, (d + 1)t).
In the above expression, Aˇ1 , Aˇ2 , and Cˇ are matrices of the following forms A1 B1 ˇ A2 B2 Aˇ1 := , A2 := , Cˇ := [C 0] . 0 0 0 0
Moreover, the constraint (2) that τk has to obey can be expressed as F1 (τk )′ x ¯[k] + F2 (τk ) = 0 (5) where F1 (t)′ := F1′ + F2′ eA1 t Z t F2 (t) := F2′ eA1 (t−s) B1 ds + F3 t 0
The derivation of (4) is straightforward by observing that one has the following expressions for x: Z θ A1 θ eA1 (θ−s) B1 ds, θ ∈ [0, τk ) e x ˆ [k](0) + 0 Z θ x ˆ[k](θ) = eA2 (θ−τk ) x ˆ[k](τk ) + eA2 (θ−s) B2 ds, τk θ ∈ [τk , (dk + 1)τk ) and the standard formula for matrix exponentials: Z θ Aθ A(θ−τ ) A B e B dτ . θ = e exp 0 0 0 0 I We assume that the plant (1) attains at least one periodic solution x0 of period T0 := (d0 + 1)τ 0 if all dk are set to d0 and the system is initialized by x0 := x0 (0). The periodicity of x0 implies that x0 = Φ0 x0 + Γ0 (6) or equivalently 0 ˇ2 d0 τ 0 A ˇ1 τ 0 0 x A 0 , (7) (I − e e )ˇx = 0, ˇx := 1
where Φ0 := Φ(d0 , τ 0 ) and Γ0 := Γ(d0 , τ 0 ). The continuous time output v corresponding to x0 , denoted by v 0 , is also periodic and satisfies vˆ0 [k] = Ψ0 x0 + Θ0 = Ω0 ˇx0 (8) for any k, where Ψ0 := Ψ(d0 , τ 0 ), Θ0 := Θ(d0 , τ 0 ), Ω0 := Ω(d0 , τ 0 ). (9) Finally, τ 0 and x0 must obey constraint (5); i.e., F1 (τ 0 )′ x0 + F2 (τ 0 ) = 0. (10) Now, let us consider the difference of the transient response of the plant and the nominal periodic solution at the sampling instances. More specifically, let ¯ := x ξ[k] ¯[k] − x0 , ¯ e[k] := v(tk ) − v 0 (T0 ) = C ξ[k], η[k] := y[k] − M x0 , where the discrete-time signal e measures the deviation of output v at the sampling instances to the desired value v 0 (T0 ). Using (4), (6), and (8), we can derive the error dynamics at sampling instants: ¯ + 1] = Φ(dk , τk )ξ[k] ¯ + Γ(d ˜ k , τk ) ξ[k ¯ (11) e[k] = C ξ[k] ¯ η[k] = M ξ[k] ˜ t) ∈ Rn is defined as follows where Γ(d, ˜ t) := [Φ(d, t) Γ(d, t)] − Φ0 Γ0 ˇx0 . Γ(d,
6315
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
0
0
Mx
-c η −6
-
Kd
d u ? c d-
P
by O (A, C, T ). Proposition 1. For given x0 , τ 0 and d0 satisfying (7) and (10), one has 0 ′ 0 1 ˇx ˇx 0 0 Js (d0 , τ 0 , x0 ) = 0 Q (d , τ ) s vref (d + 1)τ 0 vref
- v y
where Qs (d0 , τ 0 ) has the form ˆ d0 τ 0 eAˆ1 τ 0 ˆ τ 0 + eAˆ′1 τ 0 O Aˆ2 , C, O Aˆ1 , C,
Fig. 4. Feedback Control Configuration 3.2 Problem Formulation The primal objective of our control design task is to make the error v(t) − vref small. To achieve this, we consider a two step design procedure. (1) We first design d0 and x0 such that the power of the stationary error Z 1 τ 0 |v (t) − vref |2 dt Js := lim τ →∞ τ 0 (12) Z T0 1 0 2 = 0 |ˆ v [k](θ) − vref | dθ T 0 is minimized. (2) Then we design a dynamic output feedback controller which ensure asymptotic convergence to x0 in such a way that the rate of convergence measured by the cost function J of the following form ∞ X q1 |e[k]|2 + q2 |ed [k]|2 + q3 ||Tk | − T0 |2 + r|u[k]|2 k=0
(13) is minimized, where qi ≥ 0, i = 1, 2, 3, and r > 0 are design parameters, and ed and u are respectively defined as k−1 X η1 [i], u[k] := dk − d0 . ed [k] := i=0
Note the term ||Tk | − T0 |2 is introduced to penalize the aperiodicity so that, by minimizing this term, appropriate control action is produced to drive the system to the periodic state. Also note that introducing the term ed in (13) leads to an integral action in the resultant controller.
The resultant feedback configuration is illustrated in Fig. 4. Note that in the design step (2), we only take into account the at-sampling error e[k]. Ideally one would also like to take into account the inter-sampling error. This action, however, would result in a very difficult optimization problem due to the aperiodic nature of PFM. Hence, in this manuscript we only consider the simple discrete-time cost function (13) as the first step towards establishing a more sophisticate sampled-data control paradigm for PFM systems. 4. DESIGN OF THE STATIONARY FREQUENCY MODULATION RATIO The following proposition provides a computational formula for Js in (12). To facilitate the development, let us denote the finite observability grammian by Z T ′ eA t C ′ CeAt dt 0
and Aˆi , i = 1, 2, and Cˆ Aˇi Aˆi := 0
are defined as follows 0 , Cˆ := Cˇ −1 . 0
Proof. The proof follows straightforwardly that
vˆk0 (θ) − vref
0 ˆ1 θ ˇx A ˆ , θ ∈ T10 Ce vref 0 = ˆ2 (θ−τ 0 ) A ˆ1 τ 0 ˇx A ˆ Ce e , θ ∈ T20 vref
where T10 := [0, τ 0 ), and T20 := [τ 0 , (d0 + 1)τ 0 ).
By this proposition, the minimization of the stationary error can be formulated as min Js (d0 , τ 0 , x0 ) d0 ∈R,x0 ∈Rn ,τ 0 ∈R (14) subject to (7) and (10) Optimization problem (14) is nonlinear and likely to have multiple local minimums. Hence, having a good initial point is crucial. For applications such as control of the DC-DC converter described in Section 2, stationary frequency modulation ratio d0 is related to the average output voltage as in (3). Such knowledge from the physical configuration of the system can be used to find a good initial point. Furthermore, notice that for a given d0 , there could be only one pair of (τ 0 , x0 ) satisfying (7) and (10). Should this be the case, the optimization problem is one dimensional (over d0 ) and the computational complexity of the problem is significantly less than what it appears. 5. DESIGN OF FEEDBACK REGULATOR In this section, we discuss how to obtain a discrete time feedback regulator Kd in the form of "xK [k]# AK BK1 BK2 xK [k + 1] η[k] , (15) = Kd : u[k] CK DK1 DK2 ed [k] which ensures asymptotic stability of the system and optimizes J defined in (13). Despite the nice structures, the optimization problem involving error dynamics (11) and objective function (13) is highly nonlinear and in general the cost function J is finite only when the state is restricted within some neighborhood of the origin where (11) is stabilizable. As such, finding a feedback regulator which achieves globally asymptotic stability is expected to be very difficult and perhaps intractable. Hence we consider the linearization of the state equation (11) together with the quadratic approximation of the cost function J which give rise to a linear quadratic regulation problem.
6316
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
5.2 LQ Regulator for Linearized Error Dynamics with Approximated Cost Function
5.1 Small Perturbation Model The cost function J is not a quadratic functional of ¯ u) due to the term ||Tk | − T0 |2 . To obtain a quadratic (ξ, functional, we will approximate |Tk | by its dominating term in a small neighborhood around T0 . To this end, we observe that |Tk | = (dk + 1)τk = (d0 + 1 + u[k])(τ 0 + δτk ) = T0 + (d0 + 1)δτk + τ 0 u[k] + δτk u[k]. Thus, when δτk and u[k] are small, ||Tk |−T0 | is dominated by |(d0 + 1)δτk + τ 0 u[k]|. Furthermore, according to (5), δτk satisfies the following equation ¯ + x0 + F2 (τ 0 + δτk ) = 0, F1 (τ 0 + δτk )′ ξ[k] where the left-hand side of the equation is dominated by ¯ F1 (τ 0 )′ x0 + F2 (τ 0 ) + F1 (τ 0 )′ ξ[k] dF1 0 ′ 0 dF2 0 (τ ) x + (τ ) δτk + dt dt Since the sum of the first two terms of the above quantity is equal to zero, we obtain that −1 dF1 0 ′ 0 dF2 0 ¯ δτk ≈ − (τ ) x + (τ ) F1 (τ 0 )′ ξ[k] dt dt ˜ 0 )′ ξ[k]. ¯ := F(τ (16) where dF1 ′ (t) = F2′ eA1 t A1 , dt Z t ∂F2 (t) = F2′ B1 + F2′ A1 eA1 (t−s) B1 ds + F3 , ∂t 0 We have the following lemma regarding the quadratic approximation of the cost function J. ¯ Lemma 2. Assuming ξ[k] and u[k] are small, the cost function J defined in (13) is dominated by the following quadratic function ∞ X ˜ 12 ξ[k] ˜ Q ¯ ¯ Q ξ[k] 2 , ˜ 11 J˜ := ˜ 22 u[k] + q2 |ed [k]| (17) u[k] Q′12 Q k=0
where ˜ 11 := q1 C ′ C + q3 (d0 + 1)2 F(τ ˜ 0 )F(τ ˜ 0 )′ Q ˜ 0 ), Q ˜ 12 := q3 τ 0 (d0 + 1)F(τ ˜ 22 := r + q3 (τ 0 )2 Q
We include the integrator dynamics, which generates ed , in the plant to obtain the augmented system ¯ ¯ + 1] Γ Φ 0 ξ[k] ξ[k u[k] + = 0 C 1 ed [k] ed [k + 1] ¯ M 0 ξ[k] η[k] . = 0 1 ed [k] ed [k] Hence the closed-loop dynamics is given by ¯ ¯ + 1] ξ[k] ξ[k ed [k + 1] = Φcℓ ed [k] , Φcℓ := Φa + Γa KΛa , xK [k] xK [k + 1] and the linear quadratic approximation of (13) is represented by ′ ¯ ¯ ξ[0] ξ[0] J˜ = ed [0] P ed [0] xK [0] xK [0] where " # # " # " M 0 0 Γ 0 Φ0 0 Φa := C 1 0 , Γa := 0 0 , Λa := 0 1 0 , 0 0 I 0 I 0 0 0 DK1 DK2 CK K := , BK1 BK2 AK and P is a positive definite matrix satisfying the following matrix inequality: Φ′cℓ PΦcℓ − P+ (Ψa + Θa KΛa )′ (Ψa + Θa KΛa ) < 0. (19) Here Ψa and Θa are given by Ψ √0 0 Θ 0 Ψa := , , Θa := 0 q2 0 0 0 using any factorization of matrices defined in Lemma. 2: ′ ˜ 12 ˜ 11 Q Ψ Q =: [Ψ Θ] . ˜′ Q ˜ 22 Θ′ Q 12 LMI techniques can be used to search P and K. Results are summarized in the sequel. Proofs are omitted since they are similar to ones in Fujioka et al. [2007]. Theorem 3. The following two statements are equivalent:
0
Proof 1. Approximating ||Tk | − T | by its dominating term |(d0 + 1)δτk + τ 0 u[k]|, we obtain the form of J˜ by substituting (16) into (13). Around (d0 , τ 0 ), the linearized error dynamics (11) satisfies ¯ + 1] = Φξ[k] ¯ + Γu[k] ξ[k ¯ e[k] = C ξ[k] (18) ¯ η[k] = M ξ[k] ˜ ˜ 0 )′ , Γ = ∂ Γ˜ (d0 , τ 0 ), where Φ = Φ(d0 , τ 0 ) + ∂∂tΓ (d0 , τ 0 )F(τ ∂d and ˜ ∂Γ ˇ ˇ ˇ ˇ (d, t) = [In 0] dAˇ2 eA2 dt eA1 t + eA2 dt Aˇ1 eA1 t ∂t ˜ ∂Γ ˇ ˇ (d, t) = [In 0] tAˇ2 eA2 dt eA1 t ∂d With the linearized error dynamics (18) and the cost function (17), one reaches at a discrete-time LQ control problem.
6317
(i) There exist P = P′ > 0 and K satisfying (19). n (ii) There exist P1 = P1′ ∈ S+ and Y Y n+3 , Y1 ∈ R(n+2)×(n+2) Y = 1′ 3 = Y ′ ∈ S+ Y3 Y2 satisfying the following three LMIs: ˜ 11 )(M ′ )′⊥ < 0, (M ′ ) (Φ′ P Φ − P1 + Q (20) ′ ⊥ 1 Γ Γ 0 0 Θ ΠY Π′ − Yˆ Θ ⊥ < 0, (21) ⊥ 0 I2 0 I2 P1 In 0 In Y1 Y3 ≥ 0 (22) 0 Y3′ Y2 where Y1 0 Y3 0 Φ 0 0 I 0 0 Ψ 0 . , Yˆ := ′ n Π := C √1 Y3 0 Y2 0 0 q2 0 0 0 1 The subscript (·)⊥ is used to denote the orthogonal complement of a matrix.
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Thus, existence of a stabilizing controller K is characterized in terms of linear matrix inequalities (20) to (22). Should these matrix inequalities are feasible, a stabilizing controller of dimension n + 1 can be constructed. Here due to space limitation, we omit the details. 6. EXAMPLE To illustrate the theory presented in this manuscript, we consider the twin-buck converter depicted in Fig. 2. ′ The state vector of the system is x = [IL Vout ] , where IL := iL1 + iL2 , and the system matrices are 2L + Ls 2 0 − 0 − L(L + Ls ) L A1 = 1 , 1 , A2 = 1 1 − − C CR C CR ′ ′ (2L + Ls )Vin Vin 0 , B1 = 0 , B2 = L(L + Ls ) L Ls Ls . C = M = [0 1] , F1 = F2 = 0 , F3 = −2 − L Vin The nominal values of the parameters are Vin = 50, L = 250 × 10−6 , Ls = 30 × 10−6 , C = 33 × 10−6 , R = 3.
Setting vref = 37.5, we obtain corresponding d0 = 1, ′ τ 0 = 7.0107 × 10−6, and x0 = [12.7264 37.1561] . With the values of (q1 , q2 , q3 , r) in (13) selected to be (1, 0.01, 1, 23), Theorem 3 is applied to obtain an output feedback regulator. Figure 5 illustrates responses of the closed-loop system under source (Vin ) and load (R) variations. The upper diagrams show time histories of the output voltage in two different simulations, where during the time interval [0.01, 0.02] the source voltage Vin (left diagram) and the load R (right diagram) increase 20% and 100%, respectively. The lower diagrams show time histories of current flowing through one of the control switches. The diagrams verify that the switch is always disconnected when no the current flows through it. 45
47.5 45 output voltage
output voltage
42.5 40 37.5 35
42.5 40 37.5 35 32.5
32.5
30
30 0.005
0.01
0.015 0.02 Time (Second)
0.025
27.5 0.005
0
−5
−10
−15
0.015 0.02 Time (Second)
0.025
2 current through a switch
current through a switch
5
0.01
14.96
14.98 15 15.02 Time (Millisecond)
15.04
0 −2 −4 −6 −8
14.96
14.98 15 15.02 Time (Millisecond)
15.04
Fig. 5. Illustrations of transient responses of the closedloop twin-buck converter.
7. CONCLUDING REMARKS Motivated by a practical application, we consider optimal control synthesis for a class of pulse frequency modulated systems based on a sampled-data model. The problem is highly nonlinear; to simplify the task, a linear quadratic approximation is adopted and the corresponding control design problem can be solved by computationally efficient algorithm. The proposed design method is applied to find an output feedback controller for a twin-buck converter. Simulation results are provided to verify the effectiveness of the controller. REFERENCES M. Bˆ aja, H. Cormerais, and J. Buisson. Modeling and hybrid control of a four-level three-cell DC-DC converter. In Proceedings of 34th IEEE Conference on Industrial Electronics, pages 3278– 3283, 2008. A. Beccuti, S. Mari´ ethoz, S. Cliquennois, S. Wang, and M. Morari. Explicit model predictive control of DC-DC switched mode power supplies with extended Kalman filtering. IEEE Transactions on Industrial Electronics, 56(3):1864–1874, 2009. Y.-J. Chen. An interleaved twin-buck converter with zero-voltage transition. Master’s thesis, Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, 2009. H. Fujioka, C.-Y. Kao, S. Alm´ er, and U. J¨ onsson. LQ optimal control for a class of pulse width modulated systems. Automatica, 43(6): 1009–1020, 2007. H. Fujioka, C.-Y. Kao, S. Alm´ er, and U. J¨ onsson. Robust tracking with h∞ performance for PWM systems. Automatica, 45(8):1808– 1818, 2009. T. Geyer, G. Papafotiou, R. Frasca, and M. Morari. Constrained optimal control of the step-down DC-DC converter. IEEE Transactions on Power Electronics, 23(5):2454–2464, 2008. B. M. Hasanien and K. F. A. Sayed. Current source ZCS PFM DC-DC converter for magnetron power supply. In Proceedings of 12th International Middle-East Power System Conference, pages 464–469, 2008. B. Lincoln and A. Rantzer. Relaxing dynamic programming. IEEE Transactions on Automatic Control, 51(5):1249–1260, 2006. S. Mari´ ethoz and M. Morari. Explicit model predictive control of a PWM inverter with an LCL filter. IEEE Transactions on Industrial Electronics, 56(2):388–399, 2009. S. Mari´ ethoz, S. Alm´ er, M. Bˆ aja, A. G. Beccuti, D. Patino, A. Wernrud, J. Buisson, H. Cormerais, T. Geyer, H. Fujioka, U. J¨ onsson, C.-Y. Kao, M. Morari, G. Papafotiou, A. Rantzer, and P. Riedinger. Comparison of hybrid control techniques for buck and boost DC-DC converters. IEEE Transactions on Control System Technology, 18(5):1126–1145, 2010. M. Morari, J. Buisson, B. de Schutter, and G. Papafotiou. Report on the assessment of hybrid control methods for electric energy management problems. Technical Report HYCON Deliverable, Dept. of Mathematics, Royal Inst. of Technology, 2006. [Online]. Available:http://control.ee.ethz.ch/hycon/downloads free/HYCON D4a51.pdf. D. Patino, P. Riedinger, and C. Iung. Practical optimal state feedback control law for continuous-time affine switched systems with cyclic steady state. International Journal of Control, 82: 1357–1376, 2009. J. Sun, X. Ding, M. Nakaoka, and H. Takano. Series resonant ZCS-PFM DC-DC converter with multistage rectified voltage multiplier and dual-mode PFM control scheme for medical-use high-voltage x-ray power generator. IEE Proceedings on Electric Power Applications, 147(6):527–534, 2000. A. Wernrud. Approximate dynamic programming with applications. PhD thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, 2008.
6318