Hybrid Modelling and Control of Power Electronics

Hybrid Modelling and Control of Power Electronics Matthew Senesky, Gabriel Eirea, and T. John Koo EECS Department, University of California, Berkeley {senesky,geirea,koo}@eecs.berkeley.edu

Abstract. Switched circuits in power electronics by their nature present hybrid behavior. Such circuits can be described by a set of discrete states with associated continuous dynamics. A control objective, usually regulation of the output in the face of disturbances in the continuous system, is accomplished by choosing among discrete states. We describe a hybrid systems perspective of several common tasks in the design and analysis of power electronics. A DC-DC boost converter circuit is presented as an illustrative example, and the extension of this circuit to a multiple output configuration is provided to show the favorable scaling properties and broad utility of the hybrid approach.

1

Introduction

Since their introduction in the 1950’s, power semiconductor components have steadily improved in performance, price, and convenience. Modern components like power MOSFETs and IGBTs (Insulated Gate Bipolar Transistors) offer impressive specifications for switching frequency and on-resistance, while eliminating the problems with forced commutation associated with earlier generations of power devices. As these components have become more attractive to designers, the use of switching circuits in power applications has become increasingly common. Such circuits typically employ PWM (Pulse Width Modulation) or similar switching techniques to regulate the voltage or current delivered to a load, and networks of linear circuit elements to filter the switching transients from this output. 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 supply power to the load R at a higher voltage Vout (hence the name “boost”). This is accomplished by first closing SW1 (with SW2 open) to store energy in the inductor L, and then closing SW2 (with SW1 open) to transfer that energy to the capacitor C, where it is available to the load R. For the circuit to function properly, this switching must occur continually, and its timing must be controlled. In the following, we will refer frequently to this example. As described in Section 2 below, much of the typical analysis of switching circuits relies on averaging or discretization techniques to make analysis of the

iL

+

Vin

L

SW2 SW1

+

C

−

R

Vout −

Fig. 1. The DC-DC boost converter.

circuit more tractable. While this approach is adequate in many cases, it is worthwhile and instructive to reconsider the system analysis and controller synthesis in light of hybrid systems literature. This will not only allow the exploration of a larger space of controllers, but also make available a number of hybrid analysis tools. Switching circuits are a particularly good candidate for such analysis because they are inherently hybrid in structure. Under this hybrid model the system has only discrete inputs, only continuous outputs, and disturbances that are either continuous, as in a changing load or source, or discrete, as in a fault condition for 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 power electronics circuits. Section 3 presents the formal definition of the class of systems we examine, and describes a hybrid systems approach to the analysis of power electronics circuits. In Section 4 we describe a methodology to synthesize guards that guarantee a desired safety property, and illustrate it with an example. In Section 5 we undertake a detailed design exercise by extending the example to a DC-DC converter with two outputs. Finally, we outline some conclusions and future work.

2

State of the Art: Analysis of Power Electronics

Many of the power electronics topologies currently in use predate much of the literature on nonlinear and hybrid systems. In addition, simplicity and low cost often win out over high performance in application. Thus, the most common techniques for analysis, simulation, and control synthesis involve considerable approximation, and produce results that are limited in utility for higherperformance designs. One approach to simplifying switching circuits is to obtain an averaged, continuous time model (see [1]). Under this method, switching action is replaced

by a moving average of the switched quantity, and the switching duty cycle becomes a gain in the range of [0,1]. The switching frequency does not appear in the analysis, and the system trajectories have continuous first derivatives. The model is not necessarily linear however, and in fact taking the continuous duty cycle 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. The values of quantities of interest are calculated only at discrete instants, usually synchronous with the switching frequency. Once again, the switching frequency does not appear explicitly in the analysis. As with averaging methods, discretization does not necessarily result in a linear model. It is common in either case to perform a small-signal linearization of the model about an operating point of interest by finding the Jacobian of the state– space model. Clearly, models obtained with such methods are limited in their ability to describe system dynamics. Circuit behavior between switching instances is lost, and the ability to predict important nonlinear behaviors is lost. Control synthesis is often accomplished by applying linear control techniques to a linearized averaged or discretized system as described above. The main drawback to this approach is the fact that controller performance is limited by the 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 electronics (see for example [2] and its references). Various methods of switching surface control exist, in which switching occurs when a surface in the state space is encountered. Special cases of this are sliding mode control and hysteresis control.

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 can be described as a network of electrical components selected from the following three 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 two discrete states: an open circuit and a short circuit. In a circuit with K switches, there are 2K possible discrete states. In practice however, not all of these discrete states can be visited. Some of them are not feasible because of the physical characteristics of the switches, while others are banned by the designer because of safety considerations. Because of the restricted choice of circuit elements, the resulting systems have the desirable property that the continuous dynamics of each discrete state are linear or affine. Note, however, that these dynamics can allow arbitrarily large drift of continuous states, or allow the system to relax to a trivial equilibrium point. It is by exploiting the differences among the dynamics of the various switching configurations that the desired behavior of the circuit is achieved.

Thus under the proposed definition, the only input to the system is the choice of discrete state. Discrete transitions are not necessarily under control. Some are dictated by the physical characteristics of the switching elements and the evolution of currents and voltages in the circuit. This analysis will deal only with continuous disturbances. Hence a disturbance will be considered to be a change in the value of a source or linear element over time. Switching elements are 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 of the continuous states for all q, where q ∈ Q represents the on/off configuration of all the switches in the circuit. As described above, networks are constructed from ideal sources, linear elements and ideal switches; hence for each q ∈ Q the continuous dynamics can be modelled by differential equations of the form x(t) ˙ = fq (x(t)) = Aq x(t) + bq

(3.1)

where x ∈ X, Aq ∈ Rn×n , bq ∈ Rn×1 . Furthermore, one can define I(q) ⊆ X as the subset of the continuous state space where the dynamics of f q can be applied. How and when to impose discrete transitions is a key problem in the design of power electronics circuits. We propose to address this problem with hybrid automaton theory [3, 4]. First, we introduce some useful concepts. Definition 1 (Mode). A mode, denoted Mq where q ∈ Q, is the operation of the system (3.1), i.e. x(t) ˙ = Aq x(t) + bq , while x ∈ I(q) with I(q) ⊆ X. From a given discrete state it may not be feasible to visit all other discrete states. Hence, we use E ⊆ Q × Q to define the collection of feasible discrete transitions. To each edge e = (q, q 0 ) ∈ E, the switching condition is defined by G : 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 ⊆ Rn is the continuous state space; f : Q → (X → Rn ) assigns to every discrete state a Lipschitz continuous vector field on X; I : Q → 2X assigns each q ∈ Q an invariant set; E ⊆ Q × Q is a collection of discrete transitions; G : E → 2X assigns each e = (q, q 0 ) ∈ E a guard. To simplify the notation, we will use Iq for I(q), fq for f (q), and Gqq0 for G((q, q 0 )). The task of checking if a hybrid automaton satisfies a given system property is called a verification problem. Many tools [5–9] have been developed for verifying different combinations of hybrid automata and system properties. However, we are interested in the synthesis problem, which considers the synthesis of a hybrid

automaton that satisfies given system properties. We focus on safety properties of the continuous state, which are typically encoded as subsets of the continuous state space. Let F ⊆ X be the safe set. We use 2F to denote the safety property on F , i.e., if 2F is true then ∀t x(t) ∈ F . One can manipulate the evolution of the continuous state by changing the discrete state. A guard can be specified to signal when this change occurs. Once the continuous state reaches the guard condition, a decision can be made whether to jump to one of the next possible discrete states. Since the continuous state x 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 guards between discrete states so that the system trajectories satisfy given performance criteria. Problem 1 (Synthesis Problem For a Given Safety Property). Given a collection of modes Mq for q ∈ Q, edges defined by E ⊆ Q × Q, and a safety property 2F , determine if there exist guards defined by G for all e ∈ E such that if x(t) ∈ F for t ≤ 0 then x(t) ∈ F for t ≥ 0. If so, synthesize the guards and the resulting hybrid 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 , which satisfies the safety property 2W . If x(t) ∈ W ⊆ F for t ≤ 0 then x(t) ∈ W ⊆ F for t ≥ 0. If W does exist and can be computed, one can solve the synthesis problem. In [9], an abstract algorithm is proposed to solve the synthesis problem using an iterative computation of reachable states. If the problem is feasible, a fixed point will be reached and a maximal safe set, guards and invariants will be obtained. In [10], the controller synthesis problem is formulated as a game between controller and disturbance. One can then find Hamilton-Jacobi equations whose solutions describe the boundaries of the maximal safe set, and derive 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 require that we develop more direct solution methodologies. Using the formal methods presented 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 on the 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 has been 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 collection of modes Mq for q ∈ Q, a safety property 2F , and a set point xd ∈ F , determine if there exists δ > 0 such that Bxd (δ) ⊆ F and ∀x ∈ ∂Bxd (δ) ∃q ∈ Q s.t. hx − xd , fq i ≤ 0

(3.2)

where Bxd (δ) = {x ∈ Rn : kx − xd k2 ≤ δ}. Once a “safe ball” is obtained, one can derive the guard by considering the mode that drives the continuous state inside the ball. (Note that in general, the existence of a maximal safe set does not imply the existence of a safe ball.) The ball is made controlled invariant, and thus for every starting point inside the ball the trajectory will stay in F . For points inside the ball, any discrete state is appropriate since the safety property is of concern only at the boundary of the ball. This allows hierarchical organization of a family of controllers to meet different specifications.

4

Control Synthesis

In this section, we address the synthesis problem for power electronics circuits. Our concern is to guarantee the safety property 2F , where F is called the admissible set, and represents the specification given by the designer. In a simple formulation, F is a rectangular set given by the minimum and maximum values tolerated 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 an incremental way. First, we describe the hybrid modelling of a power electronics circuit as suggested by the definitions in Section 3. 4.1

Modelling

It is a straightforward task to formulate the hybrid model for a power electronics circuit as defined in Definition 2. Note that unlike the modelling techniques discussed in Section 2, the hybrid model captures the exact behavior of the circuit, without approximation. We consider the example of the conventional DC-DC boost converter shown in 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 } and T E = {(q1 , q2 ), (q2 , q1 )}. The state of the system is defined as x = [iL vo ] , which gives the affine state equations for qi (i = 1, 2) in the form of Equation 3.1, where      vin  0 0 0 − L1 L A1 = , b = b = b = , A = 2 1 2 1 1 1 0 0 − RC C − RC and the numerical values to be used are vin = 1.5V , L = 150µH, C = 110µF and R = 6Ω. To further simplify the notation above, we use fi for fqi , Ai for Aqi and bi for bqi , and we define Λ = {1, . . . , N } and I1 = I2 = X = R2 . For implementation, in order to decouple the discrete logic with the continuous dynamics, the hybrid automaton H can be decomposed into two hybrid automata H1 and H2 . H1 is a finite state machine governing the discrete transition which depends on the continuous signal x from H2 , while H2 accepts the discrete symbol σ ∈ Σ from H1 and the continuous state x evolves accordingly. The system is shown in Figure 2.

H1

q1 û = û1

x 2 G12

x2X

x 2 G21

q2 û = û2

H2

q1 xç (t) = f q 1(x(t)) x(t) 2 Iq 1 û2Î

û = û2 û = û1

q2 xç (t) = f q 2(x(t)) x(t) 2 Iq 2

Fig. 2. A power electronics circuit modelled as the parallel composition of two hybrid automata 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 there always exists an input σ to drive the state into the ball, then we claim that it is possible to stay inside the ball B indefinitely. The only requirement is to choose the appropriate control action when the state reaches the boundary. Here, we propose a strategy for solving Problem 2 by determining the existence of the ball, and constructing the ball if it does indeed exist. The existence of such a safe 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 such that a ball Bxd (δ) = {x ∈ X : kx − xd k ≤ δ} has the following properties: 1. Bxd (δ) ⊆ F ; 2. ∀x ∈ ∂Bxd (δ), ∃i ∈ Λ s.t. hx − xd , fi (x)i ≤ 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: by construction 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 admissible set F . The set B may not be unique — there could exist a set of balls of different sizes that satisfy our requirement. If we make δ as small as possible, we get a characterization of a controller with the smallest possible deviation from the set point. If we make δ as large as possible, we find a “safety” controller, which protects the system from undesirable states. In between these extremes, it is possible 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 Ball Initialize δ = 0, largest good delta = 0; While δ < δmax δ = δ + ∆δ; is good delta = true; For all x ∈ ∂Bxd (δ) if minhx − xd , fi (x)i > 0 i∈Λ

is 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 is computed as the maximum radius of the ball contained in F ; when F is rectangular 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 the vector field pointing inwards, by computing the inner product hx − xd , fi (x)i for each 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 field variation is small between adjacent points. The radius of the ball is increased until δmax is reached. The largest δ that satisfies the invariance requirements is chosen; however if at the end of the algorithm largest good delta is 0, there is no solution. The algorithm can be solved in two ways: by setting a grid on the boundary of 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 careful choice of the grid size, while the latter can provide an answer only in limited cases. An additional degree of freedom is the value of ∆δ, which sets the grid for δ.

Returning to our DC-DC converter example, we choose to parameterize the points in the boundary of the ball as iL = iL,d + δ cos θ and vo = vo,d + δ sin θ. The inner products are  1 vo,d  vin − δ 2 sin2 θ − sin θ hx − xd , f1 (x)i = δ cos θ L RC RC    v vo,d  vo,d iL,d in + sin θ + hx − xd , f2 (x)i = δ cos θ − − L L C RC     1 1 1 δ 2 sin θ cos θ − − sin2 θ C L RC The control objective is to regulate the output voltage at vo,d = 3.3V with a tolerance of ±10%, while the current in the inductor must remain in the range [0, 2.5A]. This implies an admissible set F = {x ∈ R2 : 0 ≤ x1 ≤ 2.5, 2.97 ≤ x2 ≤ v2

o,ss 3.63}. Steady–state operation requires that iL,ss = vin ·R , where iL,ss and vo,ss are the steady–state inductor current and output voltage respectively. Imposing the condition vo,ss = vo,d we conclude that iL,ss = 1.21A, which we will also refer to 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 that largest 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 can concentrate, 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 be useful to formulate controllers that satisfy various performance criteria inside the safe set. As an example, we present two controllers for the interior of the safe set of the DC-DC converter. The first one, called “minimum ripple control”, always chooses the control whose vector field points closer to the set point xd . The control action minimizes the cosine of the angle between x − xd and fi (x) as σi = arg min i∈Λ

hx − xd , fi (x)i kfi (x)k

where we omit kx − xd k 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 driving the state inside the ball is selected and kept constant until the boundary is reached again, and so on. Notice that, by construction, such a control action always exists. The names chosen for these controllers reveal the purpose of each. In the first case, the state is expected to roam around the set point without moving too far away 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 infinite number of times in a finite time interval; in practice this is avoided by assigning a small fixed minimum time between successive switchings. In the second case the state is allowed to move away from the set point, and switches only when it is necessary for the stability of the system; the average switching is expected to be less than in the previous case, at the expense of a larger ripple of the output variable. Figure 4 shows simulated trajectories for these controllers applied to the example system.

3.5

3.5

3.4

3.4

vo(V)

3.6

vo(V)

3.6

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3

3

0.9

1

1.1

1.2 iL(A)

(a)

1.3

1.4

1.5

0.9

1

1.1

1.2 iL(A)

1.3

1.4

1.5

(b)

Fig. 4. State trajectories of the example system using (a) the minimum ripple controller, and (b) the minimum switching controller. The circle represents the initial state 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 dynamics of the system. In practice, there is always uncertainty about the values of the parameters of our model. We consider now the effects of such disturbances on the computation of the safe set, and therefore on the stability of the system. The first 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 case there is always a vector field pointing into the ball. More formally, this requires modification of Proposition 1 to accommodate the condition mini∈Λ maxd∈D hx− xd , fi (x, d)i ≤ 0, where the vector fields now depend on the disturbances. However the analysis must be modified further, because the value of the set point is also affected by the disturbances. In this case, it is not possible to specify 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 between the average voltages and currents must be maintained in the steady–state, and this relationship depends on the disturbances. Below we define a function Φ, called a “steady–state relation”, such that v = Φ(w, d) where w are independent state variables, and d are disturbances. The following proposition then formalizes the modifications needed to handle disturbances. 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 fields fi : X × D → Rn , i = 1 . . . N , which can be selected at any point in time, an admissible set F ⊆ X s.t. [Φ(yd , d) yd ]T ∈ F ∀d ∈ D, if there exists δ > 0 such that a ball Bxd (δ) = {x ∈ X : kx − xd k ≤ δ}, where xd = [Φ(yd , d0 ) yd ] is the nominal set point, s.t. 1. Bxd (δ) ⊆ F 2. ∀x ∈ ∂Bxd (δ), ∃i ∈ Λ s.t. maxd∈D hx − xd , fi (x, d)i ≤ 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 this case. First, in order to find δmax , we not only have to check that the ball remains inside 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 that mini∈Λ maxd∈D hx − xd , fi (x, d)i > 0. Considering our DC-DC converter example, the steady–state relation can be v2 written as iL,ss = Φ(xo,d , vin , R) = vinodR , where vin and R are the disturbances. Since the control objective is to regulate the output voltage, the current in the inductor has to change to accommodate changes in the disturbances. It is specified that the regulation has to be achieved under changes of +5% in the load R, and −5% in the input voltage vin . Hence the range of possible values of iL,ss is [1.15, 1.27]A, which gives us a minimum value of delta: δmin = .06. We can write the inner products as vo (vo − vo,d ) (iL − iL,d ) d2 − d1 L C (iL − iL,d ) vo (vo − vo,d ) vo (iL − iL,d ) iL (vo − vo,d ) hx − xd , f2 (x, d)i = d2 − d1 − + L C L C

hx − xd , f1 (x, d)i =

where d1 = 1/R and d2 = vin . Since the relationship is linear on d, the maximum over all possible d ∈ D is obtained by substituting d1 and d2 by their maximum or minimum value according to the sign of the corresponding coefficient: if i L > 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 , the maximum of the inner products is a function with d substituted by a constant, so we 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 for θ ∈ [π/2, π], and so on. We use the same Matlab program described in Section 4.2 with the modifications described above, and we find that δ = δmax = .33 still satisfies the conditions in Proposition 2, i.e., the safe ball is robust with respect to the disturbances specified in this problem. The computation time is almost the same, because computing the maximum over the disturbance set adds very little overhead, as described above. We can also reformulate the controllers described in the previous section to take into account disturbances. In the case of the “minimum ripple control”, we select the control by minimizing the cosine of the angle between x − x d and fi (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 action can 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 specified functions continuously, while in practice all the evaluations require sampling and finite computation time. Therefore it is necessary to take into account these limitations in our model. In this section, we describe the system with a sampled data model, i.e., using a global clock of period T , such that the evaluation of the state and the decision about the control action occur at discrete moments in time tk = kT . We assume that the computation time is zero, i.e., both the measurements and the control action occur at the same time. Under these assumptions, the conditions imposed on the safe set have to be more restrictive. It is not enough to require that a safe control action can be chosen at the points in the boundary; now we must require the same condition on 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 characterize ˜ of the set of reachable points from B in time T as included in another ball B ˜ radius δ larger than that of B. Given any point x0 ∈ ∂B, let xT,i be the state after flowing for T seconds using the control σi . Since the system is affine, then Z T Ai T eAi τ dτ b = x0 + fi (x0 )T + · · · xT,i = e x0 + 0

And we have kxT,i − xd k = kx0 − xd + fi (x0 )T + · · · k ≤ kx0 − xd k + kfi (x0 )T + · · · k ≈ δ + T kfi (x0 )k where δ is the radius of B, and we have discarded higher order terms. This expression gives an approximation of δ˜ if we find the worst case for all x0 ∈ ∂B and for all i.

˜ we have to verify that the conditions of Once we have an estimation of δ, ˜ This gives us a Proposition 1 are met for all balls with radius between δ and δ. sufficient condition for the stability of the sampled–data system. The idea can be extended in the presence of disturbances by computing the worst case for all d, i.e., δ˜ = max max max kxT,i (x0 , d) − xd k i∈Λ d∈D x0 ∈∂B

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

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vo(V)

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(b)

Fig. 5. State trajectories of the example system with sampling time T = 10µs, under the presence of disturbances, using (a) the minimum ripple controller, and (b) the minimum switching controller. The circle represents the initial state, the dashed line ˜ represents 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 a DC-DC converter with two outputs. While such circuits have been proposed (see [13]), traditional methods of analysis have not, to our knowledge, yielded a viable control scheme except for limited special cases. We apply the methodology

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

iL

+

SW3 SW2

L SW1

Vin −

CA

+

RA

VA −

+

CB

RB

VB −

Fig. 6. Double output DC-DC converter

adding another discrete state. The additional capacitor adds another continuous state, and the extra load becomes another disturbance. The task of the controller is now to independently regulate the two output voltages VA and VB by switching among 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 Equation 3.1 where     0 0 0 0 0 − L1 ,  , A2 =  C1 − R 1C 0 0 A1 =  0 − RA1CA A A A 1 1 0 0 − R B CB 0 0 − R B CB   vin   1 0 0 −L L , b =  0  0 A3 =  0 − RA1CA 1 0 0 − RB1CB CB 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 energy balance 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-state current for the given range of disturbances is [0.619, 0.684]A. Given these specifications, the admissible set is the rectangular set F = {x ∈ R3 : 0 ≤ x1 ≤ 2.5, 1.6975 ≤ x2 ≤ 2.0625, 3.375 ≤ x3 ≤ 4.125}. The set point is x = [.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 is 654s on the same computer. Figure 7 shows simulations of the “minimum ripple” and the “minimum switching” controllers, designed according to the results above.

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Fig. 7. State trajectories of the Single-Input Double-Output DC-DC converter with a sampling time T = 2.5µs, using (a) the minimum ripple controller, and (b) the minimum 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 systems framework. A general model for power electronics circuits was described. This model is superior to averaged, linearized models in that no approximation is involved, and the controller synthesis is not limited by the model. We developed a simple method for synthesizing the guards that guarantee the safety property, by constructing a ball–shaped safe set. The advantage of this method is that decisions can be made with a small computation effort (just an inner product), making it very convenient for real-time control. Although we restricted our analysis to a ball shape, it is evident that the same methodology can be extended to an ellipsoid shape. The selection of an optimal ellipsoid is an interesting problem left for future research. Implementation issues such as disturbances and non-zero switching time were addressed. We presented an algorithm to solve the safety synthesis problem for power electronics formulated in Problem 2. However, an important issue exists in the use of sampling in both spatial and temporal domains to validate the safety properties of balls of interest. The safety property is only guaranteed for the sampling points on the boundary of the safe ball at specified times. More research is needed into enhanced algorithms to ensure that the safety property is guaranteed for all points in both domains. One possible research direction is to incorporate the reachability tools developed for hybrid systems to automate the synthesis procedure, even in the presence of finite computation time and disturbances. The techniques presented in this paper may be inefficient for largedimensional state spaces. However, a large set of problems in power electronics have state spaces of small dimension. Single–output and double–output DC-DC converters were used as examples to illustrate the favorable properties of our approach. The double–output prob-

lem, when considered using hybrid techniques, was shown to be only marginally more difficult to formulate than the single–output problem. The same cannot be said of linear control methods. We conclude that hybrid systems techniques are a natural choice for power electronics circuits. In the particular case of the double–output DC-DC converter, our approach led to the design of a viable controller; to the best of our knowledge, a solution to this problem has not yet been reported in the literature.

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