[Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Modelling and Control of Power Electronics

Hybrid Modelling and Control of PowerElectronics

Matthew Senesky, Gabriel Eirea, and T. John Koo

EECS Department, University of California, Berkeleysenesky,geirea,[email protected]

Abstract. Switched circuits in power electronics by their nature presenthybrid behavior. Such circuits can be described by a set of discrete stateswith associated continuous dynamics. A control objective, usually regu-lation of the output in the face of disturbances in the continuous system,is accomplished by choosing among discrete states. We describe a hybridsystems perspective of several common tasks in the design and analysisof power electronics. A DC-DC boost converter circuit is presented asan illustrative example, and the extension of this circuit to a multipleoutput configuration is provided to show the favorable scaling propertiesand broad utility of the hybrid approach.

1 Introduction

Since their introduction in the 1950’s, power semiconductor components havesteadily improved in performance, price, and convenience. Modern componentslike power MOSFETs and IGBTs (Insulated Gate Bipolar Transistors) offer im-pressive specifications for switching frequency and on-resistance, while eliminat-ing the problems with forced commutation associated with earlier generations ofpower devices. As these components have become more attractive to designers,the use of switching circuits in power applications has become increasingly com-mon. Such circuits typically employ PWM (Pulse Width Modulation) or similarswitching techniques to regulate the voltage or current delivered to a load, andnetworks of linear circuit elements to filter the switching transients from thisoutput. Switching circuits are found in applications including power supplies,variable–speed machine drives, and DC-DC converters, just to name a few.

As a motivating example, a DC-DC “boost” converter appears in Figure 1.The purpose of the circuit is to draw power from the source Vin, and supplypower to the load R at a higher voltage Vout (hence the name “boost”). This isaccomplished by first closing SW1 (with SW2 open) to store energy in the in-ductor L, and then closing SW2 (with SW1 open) to transfer that energy to thecapacitor C, where it is available to the load R. For the circuit to function prop-erly, this switching must occur continually, and its timing must be controlled. Inthe following, we will refer frequently to this example.

As described in Section 2 below, much of the typical analysis of switchingcircuits relies on averaging or discretization techniques to make analysis of the

O. Maler and A. Pnueli (Eds.): HSCC 2003, LNCS 2623, pp. 450–465, 2003.c© Springer-Verlag Berlin Heidelberg 2003

Hybrid Modelling and Control of Power Electronics 451

+

−

+

−

iL

L

C RVin

SW2

SW1 Vout

Fig. 1. The DC-DC boost converter.

circuit more tractable. While this approach is adequate in many cases, it is worth-while and instructive to reconsider the system analysis and controller synthesisin light of hybrid systems literature. This will not only allow the exploration ofa larger space of controllers, but also make available a number of hybrid analysistools.

Switching circuits are a particularly good candidate for such analysis becausethey are inherently hybrid in structure. Under this hybrid model the system hasonly discrete inputs, only continuous outputs, and disturbances that are eithercontinuous, as in a changing load or source, or discrete, as in a fault conditionfor a particular switch. This is the class of systems that will be examined below.

1.1 Outline

The next section explores current practice in the analysis and control of powerelectronics circuits. Section 3 presents the formal definition of the class of systemswe examine, and describes a hybrid systems approach to the analysis of powerelectronics circuits. In Section 4 we describe a methodology to synthesize guardsthat guarantee a desired safety property, and illustrate it with an example. InSection 5 we undertake a detailed design exercise by extending the example toa DC-DC converter with two outputs. Finally, we outline some conclusions andfuture work.

2 State of the Art: Analysis of Power Electronics

Many of the power electronics topologies currently in use predate much of theliterature on nonlinear and hybrid systems. In addition, simplicity and lowcost often win out over high performance in application. Thus, the most com-mon techniques for analysis, simulation, and control synthesis involve consider-able approximation, and produce results that are limited in utility for higher-performance designs.

One approach to simplifying switching circuits is to obtain an averaged, con-tinuous time model (see [1]). Under this method, switching action is replaced

452 M. Senesky, G. Eirea, and T.J. Koo

by a moving average of the switched quantity, and the switching duty cycle be-comes a gain in the range of [0,1]. The switching frequency does not appear inthe analysis, and the system trajectories have continuous first derivatives. Themodel is not necessarily linear however, and in fact taking the continuous dutycycle as an input often results in a multiplicative term in the state equations.

Another approach is to develop a discrete–time or sampled–data model. Thevalues of quantities of interest are calculated only at discrete instants, usuallysynchronous with the switching frequency. Once again, the switching frequencydoes not appear explicitly in the analysis. As with averaging methods, discretiza-tion does not necessarily result in a linear model.

It is common in either case to perform a small-signal linearization of themodel about an operating point of interest by finding the Jacobian of the state–space model. Clearly, models obtained with such methods are limited in theirability to describe system dynamics. Circuit behavior between switching in-stances is lost, and the ability to predict important nonlinear behaviors is lost.

Control synthesis is often accomplished by applying linear control techniquesto a linearized averaged or discretized system as described above. The maindrawback to this approach is the fact that controller performance is limited bythe accuracy of the model; because the system dynamics are only approximated,the full space of controllers cannot be explored.

There is extensive literature on the use of nonlinear control for power elec-tronics (see for example [2] and its references). Various methods of switchingsurface control exist, in which switching occurs when a surface in the state spaceis encountered. Special cases of this are sliding mode control and hysteresis con-trol.

3 Hybrid Modelling and Analysis

Here we more formally define the class of hybrid systems proposed for study,which we refer to as “power electronics circuits”. A power electronics circuit canbe described as a network of electrical components selected from the followingthree groups: ideal voltage or current sources, linear elements (e.g. resistors,capacitors, inductors, transformers), and nonlinear elements acting as switches.At this level of abstraction, the behavior of a switch is idealized as having twodiscrete states: an open circuit and a short circuit.

In a circuit with K switches, there are 2K possible discrete states. In practicehowever, not all of these discrete states can be visited. Some of them are notfeasible because of the physical characteristics of the switches, while others arebanned by the designer because of safety considerations.

Because of the restricted choice of circuit elements, the resulting systems havethe desirable property that the continuous dynamics of each discrete state arelinear or affine. Note, however, that these dynamics can allow arbitrarily largedrift of continuous states, or allow the system to relax to a trivial equilibriumpoint. It is by exploiting the differences among the dynamics of the variousswitching configurations that the desired behavior of the circuit is achieved.


Thus under the proposed definition, the only input to the system is the choiceof discrete state. Discrete transitions are not necessarily under control. Someare dictated by the physical characteristics of the switching elements and theevolution of currents and voltages in the circuit. This analysis will deal onlywith continuous disturbances. Hence a disturbance will be considered to be achange in the value of a source or linear element over time. Switching elementsare assumed to always function correctly.

3.1 Problem Statement

Let X ⊆ Rn be a continuous state space and let Q = q1, . . . , qN be a finite

set of discrete states. The continuous state space specifies the possible values ofthe continuous states for all q, where q ∈ Q represents the on/off configurationof all the switches in the circuit. As described above, networks are constructedfrom ideal sources, linear elements and ideal switches; hence for each q ∈ Q thecontinuous dynamics can be modelled by differential equations of the form

x(t) = fq(x(t)) = Aqx(t) + bq (3.1)

where x ∈ X, Aq ∈ Rn×n, bq ∈ R

n×1. Furthermore, one can define I(q) ⊆ Xas the subset of the continuous state space where the dynamics of fq can beapplied.

How and when to impose discrete transitions is a key problem in the designof power electronics circuits. We propose to address this problem with hybridautomaton theory [3,4]. First, we introduce some useful concepts.Definition 1 (Mode). A mode, denoted Mq where q ∈ Q, is the operation ofthe system (3.1), i.e. x(t) = Aqx(t) + bq, while x ∈ I(q) with I(q) ⊆ X.

From a given discrete state it may not be feasible to visit all other discretestates. Hence, we use E ⊆ Q × Q to define the collection of feasible discretetransitions. To each edge e = (q, q′) ∈ E, the switching condition is defined byG : E → 2X which assigns each edge a guard. Given the collection of modes,edges, and guards, one can form a hybrid automaton which is defined as follows:

Definition 2 (Hybrid Automaton). A hybrid automaton is a collection H =(Q, X, f, I, E, G) where Q = q1, . . . , qN is a set of discrete states; X ⊆ R

n isthe continuous state space; f : Q → (X → R

n) assigns to every discrete statea Lipschitz continuous vector field on X; I : Q → 2X assigns each q ∈ Q aninvariant set; E ⊆ Q × Q is a collection of discrete transitions; G : E → 2X

assigns each e = (q, q′) ∈ E a guard.To simplify the notation, we will use Iq for I(q), fq for f(q), and Gqq′ forG((q, q′)).

The task of checking if a hybrid automaton satisfies a given system prop-erty is called a verification problem. Many tools [5,6,7,8,9] have been developedfor verifying different combinations of hybrid automata and system properties.However, we are interested in the synthesis problem, which considers the syn-thesis of a hybrid automaton that satisfies given system properties. We focus on


safety properties of the continuous state, which are typically encoded as subsetsof the continuous state space. Let F ⊆ X be the safe set. We use F to denotethe safety property on F , i.e., if F is true then ∀t x(t) ∈ F .

One can manipulate the evolution of the continuous state by changing thediscrete state. A guard can be specified to signal when this change occurs. Oncethe continuous state reaches the guard condition, a decision can be made whetherto jump to one of the next possible discrete states. Since the continuous statex is globally defined, there is no reset in the values of the continuous variables.The design objective for power electronics circuits is to determine the guardsbetween discrete states so that the system trajectories satisfy given performancecriteria.

Problem 1 (Synthesis Problem For a Given Safety Property). Given a collectionof modes Mq for q ∈ Q, edges defined by E ⊆ Q×Q, and a safety property F ,determine if there exist guards defined by G for all e ∈ E such that if x(t) ∈ Ffor t ≤ 0 then x(t) ∈ F for t ≥ 0. If so, synthesize the guards and the resultinghybrid automaton H.

Several approaches [9,10] have been proposed to solve the synthesis problem.The idea of these approaches is to obtain a maximal safe set, W ⊆ F , whichsatisfies the safety property W . If x(t) ∈ W ⊆ F for t ≤ 0 then x(t) ∈ W ⊆ Ffor t ≥ 0. If W does exist and can be computed, one can solve the synthesisproblem. In [9], an abstract algorithm is proposed to solve the synthesis problemusing an iterative computation of reachable states. If the problem is feasible,a fixed point will be reached and a maximal safe set, guards and invariantswill be obtained. In [10], the controller synthesis problem is formulated as agame between controller and disturbance. One can then find Hamilton-Jacobiequations whose solutions describe the boundaries of the maximal safe set, andderive an associated least restrictive controller.

We are also interested in the synthesis of guards, as specified in Problem 1.However, there are some distinct characteristics of the application which requirethat we develop more direct solution methodologies. Using the formal methodspresented in [9,10], one can obtain the maximal safe set W inside F if it exists.In general, the safe set W can have an arbitrary shape which depends heavily onthe dynamics. However, in order to precisely determine the switching conditions,we seek an explicit form for describing the boundary of the safe set. Furthermore,we require that the switching conditions can be computed effectively. Therefore,we propose to use a closed ball to specify the safe set. A similar consideration hasbeen taken by [11], where an ellipsoid is used to specify the switching conditions.We cast the synthesis problem based on a ball as follows:

Problem 2 (Safety Synthesis Problem For Power Electronics). Given a collectionof modes Mq for q ∈ Q, a safety property F , and a set point xd ∈ F , determineif there exists δ > 0 such that Bxd

(δ) ⊆ F and

∀x ∈ ∂Bxd(δ) ∃q ∈ Q s.t. 〈x − xd, fq〉 ≤ 0 (3.2)

where Bxd(δ) = x ∈ R

n : ‖x − xd‖2 ≤ δ.


Once a “safe ball” is obtained, one can derive the guard by considering themode that drives the continuous state inside the ball. (Note that in general, theexistence of a maximal safe set does not imply the existence of a safe ball.) Theball is made controlled invariant, and thus for every starting point inside theball the trajectory will stay in F . For points inside the ball, any discrete stateis appropriate since the safety property is of concern only at the boundary ofthe ball. This allows hierarchical organization of a family of controllers to meetdifferent specifications.

4 Control Synthesis

In this section, we address the synthesis problem for power electronics circuits.Our concern is to guarantee the safety property F , where F is called theadmissible set, and represents the specification given by the designer. In a simpleformulation, F is a rectangular set given by the minimum and maximum valuestolerated for each state variable. It could, however, involve a different shape.

In the remainder of this section, we address the synthesis of a controller in anincremental way. First, we describe the hybrid modelling of a power electronicscircuit as suggested by the definitions in Section 3.

4.1 Modelling

It is a straightforward task to formulate the hybrid model for a power electronicscircuit as defined in Definition 2. Note that unlike the modelling techniquesdiscussed in Section 2, the hybrid model captures the exact behavior of thecircuit, without approximation.

We consider the example of the conventional DC-DC boost converter shownin Figure 1. There are two discrete states ([SW1 on, SW2 off], and [SW1 off,SW2 on]) which we will call q1 and q2 respectively. Hence, Q = q1, q2 andE = (q1, q2), (q2, q1). The state of the system is defined as x = [iL vo]

T ,which gives the affine state equations for qi (i = 1, 2) in the form of Equation3.1, where

A1 =[

0 00 − 1

RC

], A2 =

[0 − 1

L1C − 1

RC

], b1 = b2 = b =

[vin

L0

]

and the numerical values to be used are vin = 1.5V , L = 150µH, C = 110µFand R = 6Ω. To further simplify the notation above, we use fi for fqi

, Ai forAqi and bi for bqi , and we define Λ = 1, . . . , N and I1 = I2 = X = R

2.For implementation, in order to decouple the discrete logic with the contin-

uous dynamics, the hybrid automaton H can be decomposed into two hybridautomata H1 and H2. H1 is a finite state machine governing the discrete tran-sition which depends on the continuous signal x from H2, while H2 accepts thediscrete symbol σ ∈ Σ from H1 and the continuous state x evolves accordingly.The system is shown in Figure 2.


q1

xç (t) = fq1(x(t))x(t) 2 Iq1

q2

xç (t) = fq2(x(t))x(t) 2 Iq2

û = û2

û = û1

q1

û = û1

q2

û = û2

x 2 G12

x 2 G21

û 2 Î

x 2 X

H1

H2

Fig. 2. A power electronics circuit modelled as the parallel composition of two hybridautomata where H1 governs discrete evolution and H2 governs continuous evolution.

4.2 Stability

The existence of a safe ball B is directly linked with the notion of stability,at least in a broad sense. If we can find a ball B on whose boundary therealways exists an input σ to drive the state into the ball, then we claim that it ispossible to stay inside the ball B indefinitely. The only requirement is to choosethe appropriate control action when the state reaches the boundary. Here, wepropose a strategy for solving Problem 2 by determining the existence of theball, and constructing the ball if it does indeed exist. The existence of such asafe ball B can be characterized by the following proposition.

Proposition 1. Given a continuous state space X ⊆ Rn, N continuous vector

fields fi : X → Rn, i = 1 . . . N , which can be selected at any point in time, a set

point xd ∈ X, an admissible set F ⊆ X s.t. xd ∈ F , if there exists δ > 0 suchthat a ball Bxd

(δ) = x ∈ X : ‖x − xd‖ ≤ δ has the following properties:

1. Bxd(δ) ⊆ F ;

2. ∀x ∈ ∂Bxd(δ), ∃i ∈ Λ s.t. 〈x − xd, fi(x)〉 ≤ 0,

then, Bxd(δ) is controlled invariant.

By “controlled invariant” we mean that if x(0) ∈ Bxd(δ), then there exists a

control input for t ≥ 0 such that x(t) ∈ Bxd(δ) ∀t ≥ 0. The proof is trivial: byconstruction when the flow reaches the boundary of B, the control can choose a


vector field that points into B. As a corollary, the state never leaves the admis-sible set F .

The set B may not be unique — there could exist a set of balls of differentsizes that satisfy our requirement. If we make δ as small as possible, we get acharacterization of a controller with the smallest possible deviation from the setpoint. If we make δ as large as possible, we find a “safety” controller, whichprotects the system from undesirable states. In between these extremes, it ispossible to find a collection of balls that satisfy different control objectives;clearly a trade–off exists between tight regulation and control effort.

Algorithm 1: Safe BallInitialize δ = 0, largest good delta = 0;While δ < δmax

δ = δ + ∆δ;is good delta = true;For all x ∈ ∂Bxd(δ)

if mini∈Λ

〈x − xd, fi(x)〉 > 0is good delta = false; Break;

End;End;If is good delta

largest good delta = δ;End;

End;

Fig. 3. An algorithm to find a safe ball B with maximum radius inside F

Figure 3 shows an algorithm to find the safe ball B. The value of δmax iscomputed as the maximum radius of the ball contained in F ; when F is rect-angular this computation is trivial. The algorithm starts with a ball of radius∆δ and checks if all points in the boundary have at least one element of thevector field pointing inwards, by computing the inner product 〈x−xd, fi(x)〉 foreach i. The points in the boundary of B must be parameterized in a grid over∂B, so it is important to define the size of the grid such that the vector fieldvariation is small between adjacent points. The radius of the ball is increaseduntil δmax is reached. The largest δ that satisfies the invariance requirements ischosen; however if at the end of the algorithm largest good delta is 0, thereis no solution.

The algorithm can be solved in two ways: by setting a grid on the boundaryof the ball and solving the problem numerically; or by using symbolic tools [14,15] to solve it as a Quantifier Elimination problem. The former needs a carefulchoice of the grid size, while the latter can provide an answer only in limitedcases. An additional degree of freedom is the value of ∆δ, which sets the gridfor δ.


Returning to our DC-DC converter example, we choose to parameterize thepoints in the boundary of the ball as iL = iL,d + δ cos θ and vo = vo,d + δ sin θ.The inner products are

〈x − xd, f1(x)〉 = δ(cos θ

vin

L− sin θ

vo,d

RC

)− δ2 sin2 θ

1RC

〈x − xd, f2(x)〉 = δ

(cos θ

(vin

L− vo,d

L

)+ sin θ

(iL,d

C− vo,d

RC

))+

δ2(

sin θ cos θ

(1C

− 1L

)− sin2 θ

1RC

)

The control objective is to regulate the output voltage at vo,d = 3.3V with atolerance of ±10%, while the current in the inductor must remain in the range[0, 2.5A]. This implies an admissible set F = x ∈ R

2 : 0 ≤ x1 ≤ 2.5, 2.97 ≤ x2 ≤3.63. Steady–state operation requires that iL,ss = v2

o,ss

vin·R , where iL,ss and vo,ss

are the steady–state inductor current and output voltage respectively. Imposingthe condition vo,ss = vo,d we conclude that iL,ss = 1.21A, which we will also referto as iL,d. Thus the set point is (iL,d, vo,d) = (1.21, 3.3), which gives δmax = .33.

A Matlab program using a grid size of .01 on θ and ∆δ = .01 finds thatlargest good delta = .33 (i.e., δmax). The computation time is 5s on a PIII,800MHz with 256Mb of RAM.

4.3 Regulation

Once a safe set is found, the stability of the system is guaranteed. We canconcentrate, then, on the design of controllers for the interior of the safe set.What form these might take depends on the application. In general, it may beuseful to formulate controllers that satisfy various performance criteria insidethe safe set.

As an example, we present two controllers for the interior of the safe set ofthe DC-DC converter. The first one, called “minimum ripple control”, alwayschooses the control whose vector field points closer to the set point xd. Thecontrol action minimizes the cosine of the angle between x − xd and fi(x) as

σi = arg mini∈Λ

〈x − xd, fi(x)〉‖fi(x)‖

where we omit ‖x − xd‖ in the denominator because it is independent of i.The second controller, called “minimum switching control”, keeps the control

constant until the boundary of the ball is reached. Then a new control drivingthe state inside the ball is selected and kept constant until the boundary isreached again, and so on. Notice that, by construction, such a control actionalways exists.

The names chosen for these controllers reveal the purpose of each. In the firstcase, the state is expected to roam around the set point without moving too faraway from it, at the expense of switching continually. It is reasonable to expect


that this controller might present Zeno behavior, i.e. try to switch an infinitenumber of times in a finite time interval; in practice this is avoided by assigninga small fixed minimum time between successive switchings. In the second casethe state is allowed to move away from the set point, and switches only when itis necessary for the stability of the system; the average switching is expected tobe less than in the previous case, at the expense of a larger ripple of the outputvariable. Figure 4 shows simulated trajectories for these controllers applied tothe example system.

0.9 1 1.1 1.2 1.3 1.4 1.5

3

3.1

3.2

3.3

3.4

3.5

3.6

iL(A)

v o(V)

0.9 1 1.1 1.2 1.3 1.4 1.5

3

3.1

3.2

3.3

3.4

3.5

3.6

iL(A)

v o(V)

(a) (b)

Fig. 4. State trajectories of the example system using (a) the minimum ripple con-troller, and (b) the minimum switching controller. The circle represents the initialstate and the dashed line represents the boundary of the safe set.

4.4 Disturbances

So far in our analysis we have assumed complete knowledge of the dynamicsof the system. In practice, there is always uncertainty about the values of theparameters of our model. We consider now the effects of such disturbances on thecomputation of the safe set, and therefore on the stability of the system. Thefirst natural extension of the previous result is to impose the condition that,while the disturbances d can change arbitrarily in some set D, in the worst casethere is always a vector field pointing into the ball. More formally, this requiresmodification of Proposition 1 to accommodate the condition mini∈Λ maxd∈D〈x−xd, fi(x, d)〉 ≤ 0, where the vector fields now depend on the disturbances.

However the analysis must be modified further, because the value of theset point is also affected by the disturbances. In this case, it is not possible tospecify an arbitrary set point in the state space; one can only specify a range


based on the range of the disturbances. This is because the relationship betweenthe average voltages and currents must be maintained in the steady–state, andthis relationship depends on the disturbances.

Below we define a function Φ, called a “steady–state relation”, such thatv = Φ(w, d) where w are independent state variables, and d are disturbances.The following proposition then formalizes the modifications needed to handledisturbances.

Proposition 2. Given a continuous state space X ⊆ Rn with x =

[x1:m xm+1:n]T ∈ X, an output space Y ⊆ Rn−m with y = xm+1:n, a set point

yd = xm+1:n,d ∈ Y , a disturbance set D ⊆ Rp with a nominal disturbance

d0 ∈ D, a steady state relation x1:m,ss = Φ(yd, d), N continuous vector fieldsfi : X × D → R

n, i = 1 . . . N , which can be selected at any point in time, anadmissible set F ⊆ X s.t. [Φ(yd, d) yd]T ∈ F ∀d ∈ D, if there exists δ > 0 suchthat a ball Bxd

(δ) = x ∈ X : ‖x − xd‖ ≤ δ, where xd = [Φ(yd, d0) yd] is thenominal set point, s.t.

1. Bxd(δ) ⊆ F

2. ∀x ∈ ∂Bxd(δ), ∃i ∈ Λ s.t. maxd∈D〈x − xd, fi(x, d)〉 ≤ 0

3. [Φ(yd, d) yd]T ∈ Bxd(δ) ∀d ∈ D

Then, B is controlled invariant.

The algorithm described in Figure 3 needs two modifications to work in thiscase. First, in order to find δmax, we not only have to check that the ball remainsinside F , but also that the range of possible set points remains inside the ball.Second, for every point in the boundary of B, the condition to check is thatmini∈Λ maxd∈D〈x − xd, fi(x, d)〉 > 0.

Considering our DC-DC converter example, the steady–state relation can bewritten as iL,ss = Φ(xo,d, vin, R) = v2

od

vinR , where vin and R are the disturbances.Since the control objective is to regulate the output voltage, the current inthe inductor has to change to accommodate changes in the disturbances. It isspecified that the regulation has to be achieved under changes of +5% in theload R, and −5% in the input voltage vin. Hence the range of possible values ofiL,ss is [1.15, 1.27]A, which gives us a minimum value of delta: δmin = .06. Wecan write the inner products as

〈x − xd, f1(x, d)〉 =(iL − iL,d)

Ld2 − vo(vo − vo,d)

Cd1

〈x − xd, f2(x, d)〉 =(iL − iL,d)

Ld2 − vo(vo − vo,d)

Cd1 − vo(iL − iL,d)

L+

iL(vo − vo,d)C

where d1 = 1/R and d2 = vin. Since the relationship is linear on d, the maximumover all possible d ∈ D is obtained by substituting d1 and d2 by their maximum orminimum value according to the sign of the corresponding coefficient: if iL > iL,d,substitute d2 by d2,max, and else by d2,min; if vo > vo,d, substitute d1 by d1,min,and else by d1,max. In each one of the four quadrants defined around xd, themaximum of the inner products is a function with d substituted by a constant, sowe can apply the same procedure as before. Instead of having a unique function


for θ ∈ [0, 2π], now we have four functions, one for θ ∈ [0, π/2], another forθ ∈ [π/2, π], and so on.

We use the same Matlab program described in Section 4.2 with the mod-ifications described above, and we find that δ = δmax = .33 still satisfies theconditions in Proposition 2, i.e., the safe ball is robust with respect to the dis-turbances specified in this problem. The computation time is almost the same,because computing the maximum over the disturbance set adds very little over-head, as described above.

We can also reformulate the controllers described in the previous section totake into account disturbances. In the case of the “minimum ripple control”,we select the control by minimizing the cosine of the angle between x − xd andfi(x, d) under the worst case for all d ∈ D. The “minimum switching control”can be derived in the same way.

4.5 Sampling Time

The previous results are valid under the assumption that the control actioncan be taken at any point in continuous time. This is a strong assumption,because in practice switches need a non-zero time to turn on and off. Moreover,the assumption also implies that the controller is able to evaluate the specifiedfunctions continuously, while in practice all the evaluations require samplingand finite computation time. Therefore it is necessary to take into account theselimitations in our model.

In this section, we describe the system with a sampled data model, i.e., usinga global clock of period T , such that the evaluation of the state and the decisionabout the control action occur at discrete moments in time tk = kT . We assumethat the computation time is zero, i.e., both the measurements and the controlaction occur at the same time.

Under these assumptions, the conditions imposed on the safe set have to bemore restrictive. It is not enough to require that a safe control action can bechosen at the points in the boundary; now we must require the same conditionon any point that can be reached from inside the safe set in time T .

Given a safe set described by a safe ball B as in Section 4.2, we characterizethe set of reachable points from B in time T as included in another ball B ofradius δ larger than that of B. Given any point x0 ∈ ∂B, let xT,i be the stateafter flowing for T seconds using the control σi. Since the system is affine, then

xT,i = eAiT x0 +∫ T

0eAiτdτb = x0 + fi(x0)T + · · ·

And we have

‖xT,i − xd‖ = ‖x0 − xd + fi(x0)T + · · · ‖≤ ‖x0 − xd‖ + ‖fi(x0)T + · · · ‖ ≈ δ + T‖fi(x0)‖

where δ is the radius of B, and we have discarded higher order terms. Thisexpression gives an approximation of δ if we find the worst case for all x0 ∈ ∂Band for all i.


Once we have an estimation of δ, we have to verify that the conditions ofProposition 1 are met for all balls with radius between δ and δ. This gives us asufficient condition for the stability of the sampled–data system. The idea canbe extended in the presence of disturbances by computing the worst case for alld, i.e.,

δ = maxi∈Λ

maxd∈D

maxx0∈∂B

‖xT,i(x0, d) − xd‖

To stay inside the admissible set F , we have to impose the condition thatδ ≤ δmax. This requires a modification of the algorithm in Figure 3 to compute δfor each step when is good delta is True. In our example, since δ was originallyon the edge of the admissible set, the new ball will be naturally smaller. Thevalues computed are δ = 0.18, and δ = 0.33 for a sampling period of 10µs. Thetime to compute the solution is 6s. Simulations with these values are shown inFigure 5. The state trajectories are guaranteed to stay inside B in the presenceof disturbances.

0.9 1 1.1 1.2 1.3 1.4 1.5

3

3.1

3.2

3.3

3.4

3.5

3.6

iL(A)

v o(V)

0.9 1 1.1 1.2 1.3 1.4 1.5

3

3.1

3.2

3.3

3.4

3.5

3.6

iL(A)

v o(V)

(a) (b)

Fig. 5. State trajectories of the example system with sampling time T = 10µs, underthe presence of disturbances, using (a) the minimum ripple controller, and (b) theminimum switching controller. The circle represents the initial state, the dashed linerepresents the boundary of B, and the dash-dotted line represents the boundary of B.

5 Design Example: A Double-Output DC-DC Converter

The circuit shown in Figure 6 is an extension of the previous example to aDC-DC converter with two outputs. While such circuits have been proposed(see [13]), traditional methods of analysis have not, to our knowledge, yielded aviable control scheme except for limited special cases. We apply the methodology


described above to this example to show the useful scalability properties of ourapproach. There are now three switches that operate in an exclusive fashion,

+

−

+

−

+

−

iL

L

Vin

SW2

SW1

SW3

VBVACA RA CB RB

Fig. 6. Double output DC-DC converter

adding another discrete state. The additional capacitor adds another continuousstate, and the extra load becomes another disturbance. The task of the controlleris now to independently regulate the two output voltages VA and VB by switchingamong three discrete states. If we define the state vector as x = [iL VA VB ]T ,the continuous dynamics associated with these states are governed by Equation3.1 where

A1 =

0 0 0

0 − 1RACA

00 0 − 1

RBCB

, A2 =

0 − 1

L 01

CA− 1

RACA0

0 0 − 1RBCB

,

A3 =

0 0 − 1

L0 − 1

RACA0

1CB

0 − 1RBCB

, b =

vin

L00

The circuit parameters are L = 75µH, RA = 6.25Ω, RB = 34.1Ω, CA =800µF , CB = 146.6µF , and Vin = 1.5V . The desired output voltages are VA,d =1.875V and VB,d = 3.75V . The steady-state current, computed using an energybalance equation, is iL,d = 0.65A. The output voltages are restricted to ±10%,and the current limited to the range [0, 2.5]A. The load resistors can vary by+5%, and the input voltage by −5%. The range of variation of the steady-statecurrent for the given range of disturbances is [0.619, 0.684]A.

Given these specifications, the admissible set is the rectangular set F = x ∈R

3 : 0 ≤ x1 ≤ 2.5, 1.6975 ≤ x2 ≤ 2.0625, 3.375 ≤ x3 ≤ 4.125. The set point isx = [.65 1.875 3.75]T . Then δmax = .1875.

Introducing a sampling time T = 2.5µs, we compute δ = .1 and δ = .18,using the same algorithm as in the previous section. The computation time is654s on the same computer.

Figure 7 shows simulations of the “minimum ripple” and the “minimumswitching” controllers, designed according to the results above.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

0.5

0.6

0.7

0.8i L(A

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

1.86

1.88

1.9

1.92

1.94

VA(V

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

3.7

3.8

3.9

VB(V

)

t(S)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

0.5

0.6

0.7

0.8

i L(A)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

1.86

1.88

1.9

1.92

1.94

VA(V

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

3.7

3.8

3.9

VB(V

)

t(s)

(a) (b)

Fig. 7. State trajectories of the Single-Input Double-Output DC-DC converter witha sampling time T = 2.5µs, using (a) the minimum ripple controller, and (b) theminimum switching controller. The dashed line represents the ideal steady-state values.

6 Conclusions and Future Work

We have addressed the study of power electronics circuits using a hybrid systemsframework. A general model for power electronics circuits was described. Thismodel is superior to averaged, linearized models in that no approximation isinvolved, and the controller synthesis is not limited by the model. We developeda simple method for synthesizing the guards that guarantee the safety property,by constructing a ball–shaped safe set. The advantage of this method is thatdecisions can be made with a small computation effort (just an inner product),making it very convenient for real-time control. Although we restricted our anal-ysis to a ball shape, it is evident that the same methodology can be extended toan ellipsoid shape. The selection of an optimal ellipsoid is an interesting problemleft for future research.

Implementation issues such as disturbances and non-zero switching time wereaddressed. We presented an algorithm to solve the safety synthesis problem forpower electronics formulated in Problem 2. However, an important issue existsin the use of sampling in both spatial and temporal domains to validate thesafety properties of balls of interest. The safety property is only guaranteed forthe sampling points on the boundary of the safe ball at specified times. Moreresearch is needed into enhanced algorithms to ensure that the safety propertyis guaranteed for all points in both domains. One possible research direction isto incorporate the reachability tools developed for hybrid systems to automatethe synthesis procedure, even in the presence of finite computation time anddisturbances. The techniques presented in this paper may be inefficient for large-dimensional state spaces. However, a large set of problems in power electronicshave state spaces of small dimension.

Single–output and double–output DC-DC converters were used as examplesto illustrate the favorable properties of our approach. The double–output prob-


lem, when considered using hybrid techniques, was shown to be only marginallymore difficult to formulate than the single–output problem. The same cannot besaid of linear control methods.

We conclude that hybrid systems techniques are a natural choice for powerelectronics circuits. In the particular case of the double–output DC-DC con-verter, our approach led to the design of a viable controller; to the best of ourknowledge, a solution to this problem has not yet been reported in the literature.

References

1. John G. Kassakian, Marin F. Schlecht and George C. Verghese. Principles of PowerElectronics, Addison-Wesley, 1991.

2. S. Banerjee and G.C. Verghese. Nonlinear Phenomena in Power Electronics. IEEEPress, 2001.

3. R. Alur and D. Dill. A theory of time automata. Theoretical Computer Science,126:183–235, 1994.

4. R. Alur and T.A. Henzinger. Modularity for timed and hybrid systems. In Proceed-ings of the Eighth International Conference on Concurrency Theory (CONCUR),pages 74–88, 1997.

5. J. Lygeros, C. Tomlin, S. Sastry. Controllers for Reachability Specifications forHybrid Systems, Automatica, Volume 35, Number 3, March 1999.

6. G. Lafferriere, G.J. Pappas, S. Yovine. Reachability Computation for Linear HybridSystems. In Proceedings of the 14th IFAC World Congress, volume E, pages 7–12,Beijing, 1999.

7. A.B. Kurzhanski, P.Varaiya. Ellipsoidal Techniques for Reachability Analysis, Hy-brid Systems : Computation and Control, Lecture Notes in Computer Science, 2000.

8. A. Chutinan, B.H. Krogh, Verification of polyhedral-invariant hybrid systems usingpolygonal flow pipe approximations, Hybrid Systems : Computation and Control,Lecture Notes in Computer Science, 1999.

9. E. Asarin, O. Bournez, T. Dang, O. Maler, A. Pnueli. Effective Synthesis of Switch-ing Controllers for Linear Systems, The Proceedings of IEEE, Volume 88, Number7, Pages 1011–1025, July 2000.

10. C. Tomlin, J. Lygeros, S. Sastry. A Game Theoretic Approach to Controller Designfor Hybrid Systems, The Proceedings of IEEE, Volume 88, Number 7, Pages 949–970, July 2000.

11. C. Altafini, A. Speranzon, K. H. Johansson. Hybrid Control of a Truck and TrailerVehicle, Hybrid Systems : Computation and Control, Lecture Notes in ComputerScience, 2002.

12. Ian Mitchell and Claire Tomlin. Level Set Methods for Computation in HybridSystems, Hybrid Systems : Computation and Control, LCNS series, Volume 1790,Springer-Verlag, 2000.

13. Wing-Hung Ki and Dongsheng Ma. Single-Inductor Multiple-Output SwitchingConverters, IEEE Power Electronics Specialists Conference, pp.226–231, 2001.

14. G. Collins, H. Hong. Partial Cylindrical Algebraic Decomposition for QuantifierElimination, J. Symb. Comput., 12, 199–328, 1991.

15. A. Dolzman, T. Sturm. REDLOG: Computer Algebra Meets Computer Logic.ACM SIGSAM Bulletin, 31, 2–9, 1997.

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