Optimal Sampled Data Control of PWM Systems Using Piecewise ...
49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
Optimal Sampled Data Control of PWM Systems Using Piecewise Affine Approximations S. Alm´er, S. Mari´ethoz and M. Morari Automatic Control Laboratory, ETH Zurich, Switzerland [almers,mariethoz,morari]@control.ee.ethz.ch
Abstract— The paper considers optimal control of a class of pulse-width modulated systems based on sampled data modeling and piecewise affine approximations. The problem of minimizing an integral cost over a finite horizon is stated in discrete time by lifting the system dynamics. By approximating the lifted dynamics with a piecewise affine system the control problem simplifies to a mixed integer quadratic problem. The optimization problem is solved off-line and the solution is represented in a look-up table which can be implemented in a receding horizon control approach. The contribution of the paper is to introduce a sampled data model which allows to better represent the true control objective. This removes the need to filter out the ripple from the measured state and thus has potential for improved performance.
I. I NTRODUCTION In the present paper we consider a class of pulse-width modulated (PWM) systems [6] and state a constraint optimal control problem which is (approximately) solved by combining model predictive control (MPC) [9], [15] , piecewise affine (PWA) approximations [16] and sampled data modeling [3]–[5]. The control problem considered is motivated mainly by applications in power electronics such as switched mode power converters [8], [12]. Conventionally, control design for such systems is based on the so-called state space averaged model [11] which approximates the switched converter dynamics with a continuous system describing the average value of the rippling state. The averaged model approximation is usually sufficiently accurate to design high performance controllers. However, in certain applications, such as AC-DC converters, the state ripple has an adverse effect on performance and may even lead to instability. Thus, the ripple of the meassured state must be filtered out before the measurements are fed back to the controller. To compensate for the ripple of the state, multi-sampling can be applied where the state is measured several times within each switching interval. The most straight forward way of removing the ripple from the multi-sampled signal is to apply a low pass filter (such as a moving average filter) to the measurements. However, this introduces a delay in the feedback loop which reduces performance. In [14], [17], the authors propose to use various notch filters to separate the high frequency ripple from the average value of the multi-sampled signal. This method is accurate at steady state. In transient however, it need not give precise estimates. It should also be noted that multi-sampling puts
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higher requirements on hardware such as AD converters and may therefore be impractical. The present paper introduces a control design approach which takes the ripple into account and thus avoids the need for filtering. A cost criterion is defined in terms of an integral of the (continuous) state trajectory over a finite horizon. Thus, the criterion takes the full rippling state into account. By lifting the system output signal [3], [4] the (continuous time) cost criterion is equivalently formulated as a function of the sampled system state. Minimizing this function subject to the switched system dynamics is in general an intractable problem. However, by approximating the system dynamics with a discrete time PWA system which describes the state at the sampling instances [7], [16], the intractable optimization problem is simplified to a mixed integer quadratic problem (MIQP). Linear inequality constraints on the state and control input can be included in the problem without essentially changing the problem structure. The suggested controller is implemented in an MPC receding horizon framework. This means that at each sampling instant, the (approximate) finite horizon optimal control problem is solved with the current value of the state as initial condition. Only the first control move of the optimal solution is implemented and at the next time instant the state is measured again and the process is repeated. The MIQP corresponding to the approximate optimal control problem can be solved off-line [2], [18] and the solution can be represented in a look-up table. This implies that the solution can be applied on-line in application where the control frequency is very high. MPC has been applied successfully in a wide range of power electronics applications [7], [10], [13] where there is a need to handle state and input constraints while optimizing performance. The contribution of the present paper is to improve the discrete time models which are conventionally used by including a lifted output signal which describes the inter sample behavior of the system. The resulting sampled data model allows for a better representation of the true control objective and has potential for improved performance compared to conventional modeling techniques.. The paper is organized as follows. Section II introduces the class of PWM systems considered. In Section III we derive the PWA system approximation which is used in Section IV to formulate the control problem in a tractable way. Section V provides a numerical example and finally, conclusions are drawn in Section VI.
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and a lifted output signal describing the continuous time output. A. PWA approximation; continuous time model (a) Switch function
(b) PWA approximation of the switch function
Fig. 1. Above; switch function s. Below; approximation of the switch function by a piecewise affine function of the duty cycle d.
II. P ULSE - WIDTH MODULATED SYSTEMS In the present paper we consider pulse-width modulated (PWM) systems on the form x(t) ˙ = Ax(t) + s(t)Bvs y(t) = Cx(t) n
n×n
(1)
The switched system (1)-(2) is approximated with a PWA approximation using the approach in [7]. The PWA approximation is defined over a partition of the domain of the duty cycle and the switch function is approximated by a PWA function of the duty cycle (see Fig. 1(b)). Let ν ∈ N, ν ≥ 1. The domain [0, 1] of the duty cycle is partitioned into ν intervals Dj according to j−1 j , j = 1, . . . , ν. , Dj := ν ν On each piece of the partition, the system dynamics are approximated by a set of switched equations according to t ∈ Ion (k, j) Ax(t) + Bvs , x(t) ˙ = Ax(t) + Bvs (νdk + 1 − j), t ∈ Iav (k, j) Ax(t), t ∈ Ioff (k, j) (3) y(t) = Cx(t) if
n×1
where x ∈ R is the state, A ∈ R , B ∈ R and C ∈ Rp×n are constant matrices, vs is a possibly parameter (typically representing a source voltage) and s ∈ {0, 1} is the switch function which is defined as ( 1, t ∈ [kTs , (k + dk )Ts ) s(t) = (2) 0, t ∈ [(k + dk )Ts ), (k + 1)Ts ). The switch function is controlled using fixed frequency switching. This means that the time axis is partitioned into intervals [kTs , (k + 1)Ts ] where k ∈ N and Ts > 0 is the switching period. At the beginning of each switching interval (at time kTs ) a so-called duty cycle dk ∈ [0, 1] is determined. The switch function takes value one for a fraction dk of the switching interval. At time (k + dk )Ts it takes value zero and remains at zero until the beginning of the next switching interval where the procedure is repeated, see Fig. 1(a) for an illustration. It should be note that the duty cycle is by definition confined to the interval zero one and that it is the only control input. The equations (1)-(2) describe a large class of switched mode power supplies. An example can be found in Section V. Note also that the design methodology presented in this paper can be adapted to more general switch functions than (2). For example, a phase shift in the leading edge of the pulse can easily be accounted for.
d k ∈ Dj
where Ion (k, j) := [kTs , (k + (j − 1)/ν)Ts ) Iav (k, j) := [(k + (j − 1)/ν)Ts , (k + j/ν)Ts ) Ioff (k, j) := [(k + j/ν)Ts , (k + 1)Ts ). It should be noted that in the PWA system defined above, the switching instances are not determined by the duty cycle. This is key to formulating a tractable control problem. In (3), the parameter vs multiplies the control input. In order to synthesize a controller valid for a time varying parameter we remove this product by introducing the auxiliary control input uk := dk · vs,k where vs,k is the value of vs at time kTs . Using the new control input, the system (3) can be equivalently written t ∈ Ion (k, j) Ax(t) + Bvs,k x(t) ˙ = Ax(t) + Bvs,k (1 − j) + Bνuk t ∈ Iav (k, j) Ax(t) t ∈ Ioff (k, j) y(t) = Cx(t) j−1 j if vs ≤ uk ≤ vs ν ν
(4)
III. C ONTROL MODEL To minimize some cost criterion subject to the switched equations (1)-(2) will in general be an intractable problem. To obtain a tractable control problem we therefore approximate (1)-(2) with a sampled data model comprising a discrete time PWA system describing the state at the sampling instances
B. PWA approximation; sampled data model The continuous time PWA system (4) can be represented equivalently as a discrete time PWA system with a lifted output signal describing the continuous time state trajectory between sampling times. To derive the lifted system we first
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introduce some notation. Let ei be the ith unit vector in Rn+2 and let " # " # A B 0n×1 A 0n×2 Aon := , Aoff := 02×n 02×1 02×1 02×n 02×2 " # i h In 0n,2 I := In 0n,2 , E := 02×n 02×2 h i E2 := 0(n+2)×n en+1 0(n+2)×1 i h E3 := 0(n+2)×(n+1) en+1 Mj := eAon
Ts ν
The output signal y in (4) is equivalently represented by a lifted signal yˆk (θ) obtained by multiplying x ˆk (θ) by the output matrix C. Thus, by introducing the notation Mon,j (θ) := CIeAon θ jTs Mav,j (θ) := CI eAon θ + EeAon (θ− ν ) Mj jTs jTs Moff,j (θ) := CIeAoff (θ− ν ) eAon ν + EMj
the sampled data model is
xe,k+1 = Aj xe,k + Bj uk " # x e,k Mon,j (θ) , θ ∈ Ion (0, j) uk " # xe,k yˆk (θ) = Mav,j (θ) , θ ∈ Iav (0, j) uk " # xe,k , θ ∈ Ioff (0, j). Moff,j (θ) u k j j−1 vs,k ≤ uk ≤ vs,k if ν ν
(νE3 − jE2 ).
Furthermore we introduce the auxiliary vector h i′ zk := x′k vs,k uk
where xk := x(kTs ) and vs,k := vs (kTs ) are the state and parameter at the sampling times kTs . Using the relation " # ! Rθ A(θ−τ ) Aθ e Bdτ 0 A B 0 e exp θ = 0 0 0 0 0 I
it can be shown that the state x on the time interval [kTs , (k+ 1)Ts ] can be represented as x(kTs + θ) = x ˆk (θ), θ ∈ [0, Ts ] where x ˆk is the lifted signal IeAon θ zk , θ ∈ Ion (0, j) s) Aon θ Aon (θ− jT ν M j zk , θ ∈ Iav (0, j) + Ee x ˆk (θ) = I e jTs jTs IeAoff (θ− ν ) eAon ν + EM z , θ ∈ I (0, j). j
k
off
(5)
By setting θ = Ts in (5) we get a map from zk to xk+1 . Thus, by extending the state and introducing " # xk xe,k := vs,k
(7)
IV. C ONTROL PROBLEM As mentioned above, the only control input in the system (1)-(2) is the duty cycle. When the duty cycle is constant (dk = d ∀k) the system will ideally attain a periodic solution. The control objective is to steer the system state to a reference periodic solution. In the present section we state this problem in a receding horizon framework and suggest an approximate solution based on the PWA sampled data model derived in Section III above. A. Problem statement Given the sampled state xk := x(kTs ) at time kTs we consider the finite horizon optimal control problem of determining the duty cycles dk , . . . , dk+N −1 which minimize the cost
we can define the discrete time PWA system xe,k+1 = Aj xe,k + Bj uk
1 J := N Ts
j j−1 vs,k ≤ uk ≤ vs,k ν ν (6)
describing the state of (4) at the sampling instances kTs . The matrices of the PWA system are " # " # ¯j A¯j f¯j B Aj := , Bj := 0 1 0 where A¯j := eATs ¯j := IeAoff (1− νj )Ts EMj en+2 B j j f¯j := IeAoff (1− ν )Ts eAon ν Ts + EMj en+1 .
(k+N Z )Ts
y(t)′ Qy(t)dt,
Q = Q′ ≥ 0
(8)
kTs
subject to the dynamic constraints (1)-(2) and constraints on the duty cycle and state. We note that a reference trajectory can be included in the state of the system (1) and can thus appear in the output y above. From the optimal solution, only the first control move d∗k is implemented. At the next time instant (k + 1)Ts , the state xk+1 is measured and the procedure is repeated. Minimizing (8) subject to the switched dynamics (1)-(2) is in general an intractable problem. In the section below we use the PWA sampled data system derived in Section III to reformulate the control problem as a tractable optimization problem.
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B. Approximate solution Consider the cost criterion (8) and assume that the state trajectory is described by the PWA sampled data model (7). In this case, the cost criterion can be equivalently reformulated as 1 J= N
k+N X−1 m=k
(m+1)T Z s
1 Ts
Fig. 2.
y(t)′ Qy(t)dt
mTs
ZTs
k+N X−1
1 yˆm (θ)′ Qˆ ym (θ)dθ Ts m=k 0 " " #′ # k+N −1 X xe,m xe,m 1 = Qjm N um um m=k =
1 N
(9)
where jm denotes the index of the region of the mth duty cycle dm . In terms of the auxiliary control um , the index jm is such that jm − 1 jm vs,m ≤ um ≤ vs,m . ν ν The matrices Qj above are defined as Qj = Ξ(eAon θ , (j − 1)Ts /ν) Ts
where
+ Ξ(eAon ((j−1)Ts /ν+θ) + EeAon (θ− ν ) Mj , Ts /ν) + Ξ(eAoff θ eAon jTs /ν + EMj , (1 − j/ν)Ts ) 1 Ξ(f (θ), t) := Ts
Z
t
′
f (θ)′ C QCf (θ)dθ.
0
In conclusion, the problem of minimizing the cost criterion (8) subject to the sampled data dynamics (7) can be formulated as the MIQP " " #′ # k+N X−1 xe,m xe,m min Qjm um um m=k s.t. xe,m+1 = Ajm xe,m + Bjm um jm jm − 1 vs,m ≤ um ≤ vs,m ν ν jm ∈ {1, . . . , ν}
where linear inequality constraints on the state may also be included. V. E XAMPLE To verify the control approach outlined in this paper we consider a simple numerical example; the synchronous stepdown (buck) DC-DC converter depicted in Fig. 2. Defining the state as x := [vc , iℓ ]′ where vc is the capacitor voltage and iℓ is the inductor current, the system dynamics are x(t) ˙ = Ax(t) + s(s)Bvs (t) + F io (t)
(10)
where "
0 A= − x1ℓ
1 xc rℓ +rc − xℓ
#
Synchronous step down converter.
,B =
"
0 1 xℓ
#
,F =
"
− x1c rc xℓ
#
where vs is the source voltage and io is the load current. The system parameters are taken from [1]. The system dynamics have been scaled to obtain switching period Ts = 1 and all parameters are expressed in the per-unit system. They are xℓ = 0.095 p.u., xc = 2.228 p.u., rℓ = 0.05 p.u., rc = 0.005 p.u. The nominal value of the source voltage is vs = 1.8 p.u. and the nominal load current is io = 1 p.u. The state of the system (10) is extended to include the load current io . The system can then be written on the form (1)-(2) considered in the present paper. The domain of the duty cycle was partitioned into ν = 4 pieces "and an # explicit solution 104 0 was derived using cost matrix Q = and prediction 0 1 horizon N = 2. In the optimal control problem we also included the state constraint iℓ,k ≤ iℓ,max where iℓ,max = 3. It should be noted that the penalty on the voltage (state x1 ) is considerably larger than the penalty on the inductor current. This is because the peak-to-peak ripple of the voltage is only one percent of the ripple magnitude of the current. This difference is made even greater by the fact that the cost is quadratic. The controller was verified in simulation where a number of scenarios of practical interest were considered. The results are shown in Fig. 3-6 below. All scenarios include a step response from zero initial condition. In Fig. 3, the start-up transient is followed by a step in the load current from the nominal value io = 1 up to io = 2 and then back again to io = 1. In Fig. 4, the load current decreases form the nominal value io = 1 to io = 0.5 and then returns to the original value. In Fig. 5 the source voltage steps up from nominal value vs = 1.8 to vs = 3 and then back again to the nominal value. Finally, in Fig.. 6 the source voltage steps down to vs = 1.1 and then returns to the nominal value. In all scenarios the step is taken immediately after the duty cycle has been determined. In other words, the load/source changes at time kTs , but the new value is not available form measurement until time (k + 1)Ts The capacitor voltage reaches the nominal value with small overshoot and a marginal violation of the current constraint. In fact, it can be seen that during the start up transient, the lower edge of the rippling current is exactly at the limit iℓ,max . This is because the constraint on the current is imposed on the sampled state and the sampled values of the current correspond to the lower peak of the triangular current waveform. The controller efficiently mitigates all three disturbances without steady state error.
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Fig. 3. Step response from zero initial condition followed by a step in the load current form io = 1 p.u. up to io = 2 p.u. and back.
Fig. 4. Step response from zero initial condition followed by a step in the load current form io = 1 p.u. down to io = 0.5 p.u. and back.
VI. CONCLUSIONS
the need for filtering out the ripple from the state measurements and thus has potential for improved performance. Future work includes applying the sampled data modeling strategy to AC-DC converters where the negative impact of the state ripple is greater than in DC-DC applications and thus, potential for improved performance is greater.
The problem of optimizing an integral cost criterion subject to the switched dynamics of a PWM system is approximated and stated as a tractable optimization problem by using MPC and sampled data modeling. The PWM system is approximated by a PWA sampled data model which describes the system state at the sampling instances and provides a sequence of lifted signals describing the inter sample behavior. Minimizing the integral cost subject to the dynamics of the sampled data model can be stated as a mixed integer quadratic problem which can be solved off-line and implemented using a look-up table. The contribution of the paper is to extend the discrete time models used for MPC in power electronics applications. By including a lifted output signal and thus considering a sampled data model, the true control objective can be better represented in the MPC problem formulation. This removes
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R EFERENCES [1] S. Alm´er, H. Fujioka, U. J¨onsson, C.-Y. Kao, D. Patino, P. Riedinger, T. Geyer, A. Beccuti, G. Papafotiou, M. Morari, A. Wernrud, and A. Rantzer. Hybrid Control Techniques for Synchronous DC-DC Converters, Part I: The Step-Down Topology. In Proc. American Control Conference, pages 5450–5457, 2007. [2] F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems. Springer, 2003. [3] T. Chen and B. A. Francis. Optimal sampled-data control systems. Springer, London, 1995. [4] 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.
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Fig. 5. Step response from zero initial condition followed by a step in the source voltage form vs = 1.8 p.u. up to vs = 3 p.u. and back.
Fig. 6. Step response from zero initial condition followed by a step in the source voltage form vs = 1.8 p.u. down to vs = 1.1 p.u. and back.
[5] 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. [6] A. Kh. Gelig and A. N. Churilov. Stability and Oscillations of Nonlinear Pulse-Modulated Systems. Birkh¨auser, Berlin, 1998. [7] T. Geyer, G. Papafotiou, and M. Morari. Hybrid Model Predictive control of the Step-Down DC-DC Converter. IEEE Transactions on Control Systems Technology, 16(6):1112–1124, 2008. [8] M. F. Schlecht J. G. Kassakian and G. C. Verghese. Principles of Power Electronics. Addison-Wesley, Reading, Mass., 1991. [9] J.M. Maciejowski. Predictive Control. Prentice Hall, 2002. [10] S. Mari´ethoz and M. Morari. Explicit model predictive control of a PWM inverter with an LCL filter. IEEE Trans. on Ind. El., 56(1), 2009. ´ [11] R. D. Middlebrook and S. Cuk. A general unified approach to modeling switching-converter power stages. In Proc. IEEE Power Electronics Specialists Conference, pages 18–34, 1976. [12] Ned Mohan, Tore M. Undeland, and William P. Robbins. Power Electronics. Converters Applications and Design. John Wiley and Sons Inc, New York, USA, 3rd edition, 2003. [13] G. Papafotiou, J. Kley, K. Papadopoulos, P. Bohren, and M. Morari. Model Predictive Direct Torque Control - Part II: Implementation
and Experimental Evaluation. IEEE Trans. on Industrial Electronics, 56(6):1906–1915, 2009. A. Prodi´c, J. Chen, D. Maksimovi´c, and R. W. Erickson. Self-Tuning Digitally Controlled Low-Harmonic Rectifier Having Fast Dynamic Response. IEEE Transactions on Power Electronics, 18(1):420–428, 2003. R. Soeterboek. Predictive Control: A Unified Approach. Prentice Hall international, 1999. E.D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Transactions on Automatic Control, 26(2):346–358, April 1981. E. Tedeschi, P. Mattavelli, D. Trevisan, and L. Corradini. Repetitive Ripple Estimation in Multi-sampling Digitally Controlled dc-dc Converters. In Proc. IEEE Conference on Industrial Electronics, pages 1685–1690, 2006. P. Tondel, T. A. Johansen, and A. Bemporad. An algorithm for multi-parametric quadratic programming and explicit mpc solutions. Automatica, 39(3):489–497, 2003.
[14]
[15] [16] [17]
[18]
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Optimal Sampled Data Control of PWM Systems Using Piecewise Affine Approximations S. Alm´er, S. Mari´ethoz and M. Morari Automatic Control Laboratory, ETH Zurich, Switzerland [almers,mariethoz,morari]@control.ee.ethz.ch
Abstract— The paper considers optimal control of a class of pulse-width modulated systems based on sampled data modeling and piecewise affine approximations. The problem of minimizing an integral cost over a finite horizon is stated in discrete time by lifting the system dynamics. By approximating the lifted dynamics with a piecewise affine system the control problem simplifies to a mixed integer quadratic problem. The optimization problem is solved off-line and the solution is represented in a look-up table which can be implemented in a receding horizon control approach. The contribution of the paper is to introduce a sampled data model which allows to better represent the true control objective. This removes the need to filter out the ripple from the measured state and thus has potential for improved performance.
I. I NTRODUCTION In the present paper we consider a class of pulse-width modulated (PWM) systems [6] and state a constraint optimal control problem which is (approximately) solved by combining model predictive control (MPC) [9], [15] , piecewise affine (PWA) approximations [16] and sampled data modeling [3]–[5]. The control problem considered is motivated mainly by applications in power electronics such as switched mode power converters [8], [12]. Conventionally, control design for such systems is based on the so-called state space averaged model [11] which approximates the switched converter dynamics with a continuous system describing the average value of the rippling state. The averaged model approximation is usually sufficiently accurate to design high performance controllers. However, in certain applications, such as AC-DC converters, the state ripple has an adverse effect on performance and may even lead to instability. Thus, the ripple of the meassured state must be filtered out before the measurements are fed back to the controller. To compensate for the ripple of the state, multi-sampling can be applied where the state is measured several times within each switching interval. The most straight forward way of removing the ripple from the multi-sampled signal is to apply a low pass filter (such as a moving average filter) to the measurements. However, this introduces a delay in the feedback loop which reduces performance. In [14], [17], the authors propose to use various notch filters to separate the high frequency ripple from the average value of the multi-sampled signal. This method is accurate at steady state. In transient however, it need not give precise estimates. It should also be noted that multi-sampling puts
978-1-4244-7746-3/10/$26.00 ©2010 IEEE
higher requirements on hardware such as AD converters and may therefore be impractical. The present paper introduces a control design approach which takes the ripple into account and thus avoids the need for filtering. A cost criterion is defined in terms of an integral of the (continuous) state trajectory over a finite horizon. Thus, the criterion takes the full rippling state into account. By lifting the system output signal [3], [4] the (continuous time) cost criterion is equivalently formulated as a function of the sampled system state. Minimizing this function subject to the switched system dynamics is in general an intractable problem. However, by approximating the system dynamics with a discrete time PWA system which describes the state at the sampling instances [7], [16], the intractable optimization problem is simplified to a mixed integer quadratic problem (MIQP). Linear inequality constraints on the state and control input can be included in the problem without essentially changing the problem structure. The suggested controller is implemented in an MPC receding horizon framework. This means that at each sampling instant, the (approximate) finite horizon optimal control problem is solved with the current value of the state as initial condition. Only the first control move of the optimal solution is implemented and at the next time instant the state is measured again and the process is repeated. The MIQP corresponding to the approximate optimal control problem can be solved off-line [2], [18] and the solution can be represented in a look-up table. This implies that the solution can be applied on-line in application where the control frequency is very high. MPC has been applied successfully in a wide range of power electronics applications [7], [10], [13] where there is a need to handle state and input constraints while optimizing performance. The contribution of the present paper is to improve the discrete time models which are conventionally used by including a lifted output signal which describes the inter sample behavior of the system. The resulting sampled data model allows for a better representation of the true control objective and has potential for improved performance compared to conventional modeling techniques.. The paper is organized as follows. Section II introduces the class of PWM systems considered. In Section III we derive the PWA system approximation which is used in Section IV to formulate the control problem in a tractable way. Section V provides a numerical example and finally, conclusions are drawn in Section VI.
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and a lifted output signal describing the continuous time output. A. PWA approximation; continuous time model (a) Switch function
(b) PWA approximation of the switch function
Fig. 1. Above; switch function s. Below; approximation of the switch function by a piecewise affine function of the duty cycle d.
II. P ULSE - WIDTH MODULATED SYSTEMS In the present paper we consider pulse-width modulated (PWM) systems on the form x(t) ˙ = Ax(t) + s(t)Bvs y(t) = Cx(t) n
n×n
(1)
The switched system (1)-(2) is approximated with a PWA approximation using the approach in [7]. The PWA approximation is defined over a partition of the domain of the duty cycle and the switch function is approximated by a PWA function of the duty cycle (see Fig. 1(b)). Let ν ∈ N, ν ≥ 1. The domain [0, 1] of the duty cycle is partitioned into ν intervals Dj according to j−1 j , j = 1, . . . , ν. , Dj := ν ν On each piece of the partition, the system dynamics are approximated by a set of switched equations according to t ∈ Ion (k, j) Ax(t) + Bvs , x(t) ˙ = Ax(t) + Bvs (νdk + 1 − j), t ∈ Iav (k, j) Ax(t), t ∈ Ioff (k, j) (3) y(t) = Cx(t) if
n×1
where x ∈ R is the state, A ∈ R , B ∈ R and C ∈ Rp×n are constant matrices, vs is a possibly parameter (typically representing a source voltage) and s ∈ {0, 1} is the switch function which is defined as ( 1, t ∈ [kTs , (k + dk )Ts ) s(t) = (2) 0, t ∈ [(k + dk )Ts ), (k + 1)Ts ). The switch function is controlled using fixed frequency switching. This means that the time axis is partitioned into intervals [kTs , (k + 1)Ts ] where k ∈ N and Ts > 0 is the switching period. At the beginning of each switching interval (at time kTs ) a so-called duty cycle dk ∈ [0, 1] is determined. The switch function takes value one for a fraction dk of the switching interval. At time (k + dk )Ts it takes value zero and remains at zero until the beginning of the next switching interval where the procedure is repeated, see Fig. 1(a) for an illustration. It should be note that the duty cycle is by definition confined to the interval zero one and that it is the only control input. The equations (1)-(2) describe a large class of switched mode power supplies. An example can be found in Section V. Note also that the design methodology presented in this paper can be adapted to more general switch functions than (2). For example, a phase shift in the leading edge of the pulse can easily be accounted for.
d k ∈ Dj
where Ion (k, j) := [kTs , (k + (j − 1)/ν)Ts ) Iav (k, j) := [(k + (j − 1)/ν)Ts , (k + j/ν)Ts ) Ioff (k, j) := [(k + j/ν)Ts , (k + 1)Ts ). It should be noted that in the PWA system defined above, the switching instances are not determined by the duty cycle. This is key to formulating a tractable control problem. In (3), the parameter vs multiplies the control input. In order to synthesize a controller valid for a time varying parameter we remove this product by introducing the auxiliary control input uk := dk · vs,k where vs,k is the value of vs at time kTs . Using the new control input, the system (3) can be equivalently written t ∈ Ion (k, j) Ax(t) + Bvs,k x(t) ˙ = Ax(t) + Bvs,k (1 − j) + Bνuk t ∈ Iav (k, j) Ax(t) t ∈ Ioff (k, j) y(t) = Cx(t) j−1 j if vs ≤ uk ≤ vs ν ν
(4)
III. C ONTROL MODEL To minimize some cost criterion subject to the switched equations (1)-(2) will in general be an intractable problem. To obtain a tractable control problem we therefore approximate (1)-(2) with a sampled data model comprising a discrete time PWA system describing the state at the sampling instances
B. PWA approximation; sampled data model The continuous time PWA system (4) can be represented equivalently as a discrete time PWA system with a lifted output signal describing the continuous time state trajectory between sampling times. To derive the lifted system we first
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introduce some notation. Let ei be the ith unit vector in Rn+2 and let " # " # A B 0n×1 A 0n×2 Aon := , Aoff := 02×n 02×1 02×1 02×n 02×2 " # i h In 0n,2 I := In 0n,2 , E := 02×n 02×2 h i E2 := 0(n+2)×n en+1 0(n+2)×1 i h E3 := 0(n+2)×(n+1) en+1 Mj := eAon
Ts ν
The output signal y in (4) is equivalently represented by a lifted signal yˆk (θ) obtained by multiplying x ˆk (θ) by the output matrix C. Thus, by introducing the notation Mon,j (θ) := CIeAon θ jTs Mav,j (θ) := CI eAon θ + EeAon (θ− ν ) Mj jTs jTs Moff,j (θ) := CIeAoff (θ− ν ) eAon ν + EMj
the sampled data model is
xe,k+1 = Aj xe,k + Bj uk " # x e,k Mon,j (θ) , θ ∈ Ion (0, j) uk " # xe,k yˆk (θ) = Mav,j (θ) , θ ∈ Iav (0, j) uk " # xe,k , θ ∈ Ioff (0, j). Moff,j (θ) u k j j−1 vs,k ≤ uk ≤ vs,k if ν ν
(νE3 − jE2 ).
Furthermore we introduce the auxiliary vector h i′ zk := x′k vs,k uk
where xk := x(kTs ) and vs,k := vs (kTs ) are the state and parameter at the sampling times kTs . Using the relation " # ! Rθ A(θ−τ ) Aθ e Bdτ 0 A B 0 e exp θ = 0 0 0 0 0 I
it can be shown that the state x on the time interval [kTs , (k+ 1)Ts ] can be represented as x(kTs + θ) = x ˆk (θ), θ ∈ [0, Ts ] where x ˆk is the lifted signal IeAon θ zk , θ ∈ Ion (0, j) s) Aon θ Aon (θ− jT ν M j zk , θ ∈ Iav (0, j) + Ee x ˆk (θ) = I e jTs jTs IeAoff (θ− ν ) eAon ν + EM z , θ ∈ I (0, j). j
k
off
(5)
By setting θ = Ts in (5) we get a map from zk to xk+1 . Thus, by extending the state and introducing " # xk xe,k := vs,k
(7)
IV. C ONTROL PROBLEM As mentioned above, the only control input in the system (1)-(2) is the duty cycle. When the duty cycle is constant (dk = d ∀k) the system will ideally attain a periodic solution. The control objective is to steer the system state to a reference periodic solution. In the present section we state this problem in a receding horizon framework and suggest an approximate solution based on the PWA sampled data model derived in Section III above. A. Problem statement Given the sampled state xk := x(kTs ) at time kTs we consider the finite horizon optimal control problem of determining the duty cycles dk , . . . , dk+N −1 which minimize the cost
we can define the discrete time PWA system xe,k+1 = Aj xe,k + Bj uk
1 J := N Ts
j j−1 vs,k ≤ uk ≤ vs,k ν ν (6)
describing the state of (4) at the sampling instances kTs . The matrices of the PWA system are " # " # ¯j A¯j f¯j B Aj := , Bj := 0 1 0 where A¯j := eATs ¯j := IeAoff (1− νj )Ts EMj en+2 B j j f¯j := IeAoff (1− ν )Ts eAon ν Ts + EMj en+1 .
(k+N Z )Ts
y(t)′ Qy(t)dt,
Q = Q′ ≥ 0
(8)
kTs
subject to the dynamic constraints (1)-(2) and constraints on the duty cycle and state. We note that a reference trajectory can be included in the state of the system (1) and can thus appear in the output y above. From the optimal solution, only the first control move d∗k is implemented. At the next time instant (k + 1)Ts , the state xk+1 is measured and the procedure is repeated. Minimizing (8) subject to the switched dynamics (1)-(2) is in general an intractable problem. In the section below we use the PWA sampled data system derived in Section III to reformulate the control problem as a tractable optimization problem.
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B. Approximate solution Consider the cost criterion (8) and assume that the state trajectory is described by the PWA sampled data model (7). In this case, the cost criterion can be equivalently reformulated as 1 J= N
k+N X−1 m=k
(m+1)T Z s
1 Ts
Fig. 2.
y(t)′ Qy(t)dt
mTs
ZTs
k+N X−1
1 yˆm (θ)′ Qˆ ym (θ)dθ Ts m=k 0 " " #′ # k+N −1 X xe,m xe,m 1 = Qjm N um um m=k =
1 N
(9)
where jm denotes the index of the region of the mth duty cycle dm . In terms of the auxiliary control um , the index jm is such that jm − 1 jm vs,m ≤ um ≤ vs,m . ν ν The matrices Qj above are defined as Qj = Ξ(eAon θ , (j − 1)Ts /ν) Ts
where
+ Ξ(eAon ((j−1)Ts /ν+θ) + EeAon (θ− ν ) Mj , Ts /ν) + Ξ(eAoff θ eAon jTs /ν + EMj , (1 − j/ν)Ts ) 1 Ξ(f (θ), t) := Ts
Z
t
′
f (θ)′ C QCf (θ)dθ.
0
In conclusion, the problem of minimizing the cost criterion (8) subject to the sampled data dynamics (7) can be formulated as the MIQP " " #′ # k+N X−1 xe,m xe,m min Qjm um um m=k s.t. xe,m+1 = Ajm xe,m + Bjm um jm jm − 1 vs,m ≤ um ≤ vs,m ν ν jm ∈ {1, . . . , ν}
where linear inequality constraints on the state may also be included. V. E XAMPLE To verify the control approach outlined in this paper we consider a simple numerical example; the synchronous stepdown (buck) DC-DC converter depicted in Fig. 2. Defining the state as x := [vc , iℓ ]′ where vc is the capacitor voltage and iℓ is the inductor current, the system dynamics are x(t) ˙ = Ax(t) + s(s)Bvs (t) + F io (t)
(10)
where "
0 A= − x1ℓ
1 xc rℓ +rc − xℓ
#
Synchronous step down converter.
,B =
"
0 1 xℓ
#
,F =
"
− x1c rc xℓ
#
where vs is the source voltage and io is the load current. The system parameters are taken from [1]. The system dynamics have been scaled to obtain switching period Ts = 1 and all parameters are expressed in the per-unit system. They are xℓ = 0.095 p.u., xc = 2.228 p.u., rℓ = 0.05 p.u., rc = 0.005 p.u. The nominal value of the source voltage is vs = 1.8 p.u. and the nominal load current is io = 1 p.u. The state of the system (10) is extended to include the load current io . The system can then be written on the form (1)-(2) considered in the present paper. The domain of the duty cycle was partitioned into ν = 4 pieces "and an # explicit solution 104 0 was derived using cost matrix Q = and prediction 0 1 horizon N = 2. In the optimal control problem we also included the state constraint iℓ,k ≤ iℓ,max where iℓ,max = 3. It should be noted that the penalty on the voltage (state x1 ) is considerably larger than the penalty on the inductor current. This is because the peak-to-peak ripple of the voltage is only one percent of the ripple magnitude of the current. This difference is made even greater by the fact that the cost is quadratic. The controller was verified in simulation where a number of scenarios of practical interest were considered. The results are shown in Fig. 3-6 below. All scenarios include a step response from zero initial condition. In Fig. 3, the start-up transient is followed by a step in the load current from the nominal value io = 1 up to io = 2 and then back again to io = 1. In Fig. 4, the load current decreases form the nominal value io = 1 to io = 0.5 and then returns to the original value. In Fig. 5 the source voltage steps up from nominal value vs = 1.8 to vs = 3 and then back again to the nominal value. Finally, in Fig.. 6 the source voltage steps down to vs = 1.1 and then returns to the nominal value. In all scenarios the step is taken immediately after the duty cycle has been determined. In other words, the load/source changes at time kTs , but the new value is not available form measurement until time (k + 1)Ts The capacitor voltage reaches the nominal value with small overshoot and a marginal violation of the current constraint. In fact, it can be seen that during the start up transient, the lower edge of the rippling current is exactly at the limit iℓ,max . This is because the constraint on the current is imposed on the sampled state and the sampled values of the current correspond to the lower peak of the triangular current waveform. The controller efficiently mitigates all three disturbances without steady state error.
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1.5
1
1 vc p.u.
vc p.u.
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0.5 0
0
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il p.u.
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0
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io p.u.
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0
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60
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Fig. 3. Step response from zero initial condition followed by a step in the load current form io = 1 p.u. up to io = 2 p.u. and back.
Fig. 4. Step response from zero initial condition followed by a step in the load current form io = 1 p.u. down to io = 0.5 p.u. and back.
VI. CONCLUSIONS
the need for filtering out the ripple from the state measurements and thus has potential for improved performance. Future work includes applying the sampled data modeling strategy to AC-DC converters where the negative impact of the state ripple is greater than in DC-DC applications and thus, potential for improved performance is greater.
The problem of optimizing an integral cost criterion subject to the switched dynamics of a PWM system is approximated and stated as a tractable optimization problem by using MPC and sampled data modeling. The PWM system is approximated by a PWA sampled data model which describes the system state at the sampling instances and provides a sequence of lifted signals describing the inter sample behavior. Minimizing the integral cost subject to the dynamics of the sampled data model can be stated as a mixed integer quadratic problem which can be solved off-line and implemented using a look-up table. The contribution of the paper is to extend the discrete time models used for MPC in power electronics applications. By including a lifted output signal and thus considering a sampled data model, the true control objective can be better represented in the MPC problem formulation. This removes
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R EFERENCES [1] S. Alm´er, H. Fujioka, U. J¨onsson, C.-Y. Kao, D. Patino, P. Riedinger, T. Geyer, A. Beccuti, G. Papafotiou, M. Morari, A. Wernrud, and A. Rantzer. Hybrid Control Techniques for Synchronous DC-DC Converters, Part I: The Step-Down Topology. In Proc. American Control Conference, pages 5450–5457, 2007. [2] F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems. Springer, 2003. [3] T. Chen and B. A. Francis. Optimal sampled-data control systems. Springer, London, 1995. [4] 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.
1.5
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vc p.u.
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0.5 0
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−0.5 10
20
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40 50 switch period
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90
0
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vc p.u. (zoom)
vc p.u. (zoom)
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40 50 switch period
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3.5 3 vs p.u.
3 vs p.u.
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il p.u.
il p.u.
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Fig. 5. Step response from zero initial condition followed by a step in the source voltage form vs = 1.8 p.u. up to vs = 3 p.u. and back.
Fig. 6. Step response from zero initial condition followed by a step in the source voltage form vs = 1.8 p.u. down to vs = 1.1 p.u. and back.
[5] 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. [6] A. Kh. Gelig and A. N. Churilov. Stability and Oscillations of Nonlinear Pulse-Modulated Systems. Birkh¨auser, Berlin, 1998. [7] T. Geyer, G. Papafotiou, and M. Morari. Hybrid Model Predictive control of the Step-Down DC-DC Converter. IEEE Transactions on Control Systems Technology, 16(6):1112–1124, 2008. [8] M. F. Schlecht J. G. Kassakian and G. C. Verghese. Principles of Power Electronics. Addison-Wesley, Reading, Mass., 1991. [9] J.M. Maciejowski. Predictive Control. Prentice Hall, 2002. [10] S. Mari´ethoz and M. Morari. Explicit model predictive control of a PWM inverter with an LCL filter. IEEE Trans. on Ind. El., 56(1), 2009. ´ [11] R. D. Middlebrook and S. Cuk. A general unified approach to modeling switching-converter power stages. In Proc. IEEE Power Electronics Specialists Conference, pages 18–34, 1976. [12] Ned Mohan, Tore M. Undeland, and William P. Robbins. Power Electronics. Converters Applications and Design. John Wiley and Sons Inc, New York, USA, 3rd edition, 2003. [13] G. Papafotiou, J. Kley, K. Papadopoulos, P. Bohren, and M. Morari. Model Predictive Direct Torque Control - Part II: Implementation
and Experimental Evaluation. IEEE Trans. on Industrial Electronics, 56(6):1906–1915, 2009. A. Prodi´c, J. Chen, D. Maksimovi´c, and R. W. Erickson. Self-Tuning Digitally Controlled Low-Harmonic Rectifier Having Fast Dynamic Response. IEEE Transactions on Power Electronics, 18(1):420–428, 2003. R. Soeterboek. Predictive Control: A Unified Approach. Prentice Hall international, 1999. E.D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Transactions on Automatic Control, 26(2):346–358, April 1981. E. Tedeschi, P. Mattavelli, D. Trevisan, and L. Corradini. Repetitive Ripple Estimation in Multi-sampling Digitally Controlled dc-dc Converters. In Proc. IEEE Conference on Industrial Electronics, pages 1685–1690, 2006. P. Tondel, T. A. Johansen, and A. Bemporad. An algorithm for multi-parametric quadratic programming and explicit mpc solutions. Automatica, 39(3):489–497, 2003.
[14]
[15] [16] [17]
[18]
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