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Regulation Theory

This paper reviews the design of regulation loops for power converters. Power converter control being a vast domain, it does not aim to be exhaustive. The objective is to give a rapid overview of the main synthesis methods in both continuous- and discrete-time domains.


Introduction
Power converters convert electric energy from one form to another which is optimally tailored for user loads. Therefore regulation is an important part of the design and construction of any power converter.
Owing to the fast development of digital electronics, power converter control uses more and more embedded computing systems like microcontrollers or Digital Signal Processors (DSPs). These devices replace progressively analogue controllers that have dominated the control of power electronics systems for many years.
After a period during which the discrete-time control laws were directly obtained from a discretization of the continuous-time control laws in order to ensure continuity with the existing methods and equipment, automation has refined its tools and methodologies: new approaches based on discrete-time models of the processes to be controlled have been proposed. With these new techniques, more powerful synthesis methods have been developed. Alternatives to the still widely used PID-type controllers have emerged, like RST controllers or more exotic ones such as dead-beat or fuzzy controllers. All these alternatives offer potential improvements in performance.
In the first part of this paper, the main concepts of continuous-time control are reviewed. Only the control of single-input single-output linear and time-invariant systems is addressed here. Digital control is presented in the second part. After a few reminders about Z-transforms, the concept of the discrete-time model is introduced. The main digital controller synthesis methods are then described and, depending on the chosen method, the choice of the sampling frequency is discussed.
If the plant is a single-input single-output linear and time-invariant system (time-invariant systems are systems whose characteristics do not change with time), then its input u(t) and output y(t) are related by the following differential equation with constant coefficients: where is the Laplace operator.
Some basic but important definitions are recalled: − The poles are the roots of the denominator polynomial of the system transfer function H(s). The zeros are the roots of the numerator polynomial.
− The system stability is determined by the location of the poles. If their real part is strictly negative, then the system is stable.
− If the transfer function behaves like ⁄ for → 0, then α is called the system class. The static gain is obtained for equal to zero.
− If there is a period of time during which the output does not react to the input, the system is defined as a system with delay. The transfer function d ( ) of a system having a time delay of 0 can be expressed as where ( ) is the transfer function of the system without delay.

Importance of feedback control
Feedback is a very powerful mechanism. Why is it necessary? An intuitive strategy would be to use a feedforward controller and to make the product of the plant and this feedforward controller unity, hence to find a controller that equals the plant inverse ( = −1 in Fig. 1, assuming a unit gain for the actuator transfer function). Then in theory the system output Y would be equal to the controller input Yref. This works well as long as the feedforward controller approximates the plant inverse sufficiently well. In practice, uncertainties and unmodelled dynamics make this impossible. To fulfil the requirements, a feedback correction is necessary to guarantee performance, even with model uncertainties, to reduce sensitivity to parameter variations, to reject disturbances, and obviously to stabilize unstable open-loop systems. Thus a typical control structure combines feedback and feedforward controls (see Fig. 2). The feedback controller guards robust stability and improves static and dynamic precision whereas the feedforward controller improves tracking behaviour (via compensation of the input) or regulation behaviour (via compensation of the perturbations).
It is important to understand that the performance requirements (static, dynamic behaviour, stability, and robustness) of the closed loop translate into constraints on the compensated system in open loop. We will see later in this paper how to make use of the open-loop frequency response to design controllers. The typical control structure for current regulated power supplies is represented in Fig. 3. This so-called 'cascade' structure is based on a nested connection of a fast inner voltage loop and a slower outer current loop. The fast inner voltage loop acts as an active filter to reject the output voltage ripple and the output voltage fluctuations due to float of input mains, and it also simplifies the design of the current loop. The outer current loop ensures the overall stability of the power supply and it provides an inherent overvoltage protection.

Definition of open-loop and closed-loop transfer functions
In this section, we focus on the closed-loop system shown in Fig. 4. To study it, we use the open-loop transfer function obtained by breaking the feedback loop, the expression for which is given by The open-loop bandwidth is defined as the interval of angular frequencies for which the open-loop gain | ( ) • ( ) • ( )| is higher than unity. The closed-loop transfer function is The transfer functions of the error versus input and perturbation can be calculated as (6)

Static error
In steady state, the error versus the input can be derived from Eq. (6). Using the 'final value' theorem, we have − For a step input ( To have a perfect rejection, the open loop must contain the classes of the perturbations. Thus to reject disturbances of class N, at least N integrators are needed in the open-loop transfer function.

Dynamic error
Let us assume that the velocity ν and the acceleration γ of the input are limited. The input signal is then defined by the following constraints: Maintaining the dynamic error d ε below a given limit In general, this constraint results in imposing a finite open-loop bandwidth.

Stability and robustness of closed-loop systems
A closed-loop system is stable if the poles of its transfer function have a negative real part.
The stability of a feedback system can also be determined through the simple knowledge of the open-loop frequency response, using the 'Nyquist criterion'. This is of great practical importance for controller design as it easily demonstrates how to modify the controller to make an unstable system stable. The closed-loop system is stable if the critical point −1 is on the left-hand side of the open-loop Nyquist curve, for w increasing. Two quantitative measures are commonly used to determine how stable (robust) the system is. The phase margin is the amount of phase lag required to reach the stability limit. The gain margin, similarly, states how much the controller gain can be increased before reaching the stability limit.
Phase margin: where cr w is such that Gain margin: where π w is such that There is an obvious contradiction between precision and stability specifications. Thus the controller design will be the result of a trade-off.

Influence of the poles on the transient behaviour
The contribution of real and complex poles on the transient behaviour of a system is represented in the complex plane in Fig. 6. An important observation is that the most distant poles to the left yield a faster transient regime. So the poles closest to the imaginary axis are the ones that tend to dominate the response since their contribution takes a longer time to die out. These poles are called dominant poles if the ratio of their real part to that of any other poles is typically lower than 1/5. A transfer function can be simplified by keeping only the dominant poles (and the static gain unchanged). This makes control loop analysis and design easier. Also, lower-order controllers can be obtained.

Particular case: second-order systems with complex conjugate poles
The search for a good compromise between speed and stability generally leads to the choice of a firstor second-order behaviour for the closed loop. For second-order system behaviour, the design specifications imply constraints on the cut-off frequency wn and the damping ratio ζ of the targeted transfer function Eq. (12). Figure 7 shows the parameters usually used to characterize the step response, and Eqs. (13)-(15) link cut-off frequency and damping ratio to these parameters.

Influence of a zero on the system behaviour
In the case of a second-order system, the corresponding transfer function is ( ) ( where z0 is assumed to be a strictly negative real number.
So the unit step response of this system can be written as which gives in the time domain 2nd order 2nd order 0 where 2nd order ( ) y t is the step response of the system Eq. (16) without zeros.
The second term of Eq. (18) reveals that the additional zero makes the system faster and more oscillatory, and all the more so as its value is close to zero. So special care should be taken to design appropriate zeros in a closed-loop transfer function.

Controller design process
In the design of a controller, the first step is to get the dynamic model of the system to be controlled. The principal difficulty in modelling power converters is that they are inherently non-linear and present several distinct electric configurations during a switching period. By constructing equivalent averaged circuit models, large-signal average models of the power converter can be determined. Then linearization about a quiescent operating point may be necessary to obtain linear small-signal transfer functions [1][2][3][4]. To elaborate power converter models, an identification processes from experimental data can also be used.
Once the model is known, the specifications of the desired closed-loop performance have to be defined. As explained in the preceding section, the choice of these specifications results from a tradeoff between speed and robustness. This choice is obviously linked to the plant dynamics but also to the power availability of the power converter during the transient: the acceleration of the plant natural response requires control peaks that are greater than the steady-state values. If the control variable comes to saturation, the feedback loop is broken and the actuator remains at its limit independently of the plant output. To prevent actuator saturation, the closed-loop performance has to be chosen accordingly.
The last step is to choose the controller type and its design method.

Introducing PID control
The PID controller is by far the most dominant form of feedback in use today [5][6][7], especially for analogue control. PID feedback is simple and intuitive: it involves only three separate constant parameters to tune the control loop. As shown in Eqs. (19) and (20), it is based on the past control error (integral term), the present control error (proportional term), and the future control error (derivative term). Many controllers (PI) do not even use derivative action. This kind of controller is well suited for systems exhibiting dominant first-or second-order behaviour, for which the desired performance of the closed-loop as compared to the open-loop response of the system is not too demanding.
PID algorithm: Controller transfer function: As a pure derivative amplifies noise, the transfer function standard form includes a low-pass filter on the derivative term.
A number of alternative approaches for PID tuning are available, including the following ones:

Putting it into practice
Let us focus on two design methods-loop shaping and pole placement-in the following example where controllers have to be designed for the current and voltage control loops of a Buck converter feeding an inductive (R-L) load. However, with this setting the phase margin requirement cannot be reached. A derivative action is then added to correct the phase margin and meet the stability requirement.
where ) (s CL V is the closed voltage loop transfer function. If the current loop bandwidth is low compared to the voltage loop bandwidth, Eq. (21) can be simplified as follows: The system pole can be cancelled by setting i The closed-loop transfer function then shrinks down to a first-order transfer function: To make the closed-loop behave like a first-order system with a rise time equal to r t , the closed-loop pole has to be placed at This results in the following choice for the parameter p k : The pole cancellation technique requires a good knowledge of the process: if 1 a is likely to vary ( load ( ) L f I = ), this method can bring about poor results (especially when the pole is close to the origin). In that case, system pole cancellation should be avoided. The closed loop transfer function is then of second order. The PI parameters can be computed by identifying the coefficients of the closedloop transfer function denominator with those of the polynomial 2 2 (2 ) 1 n n s w w s ζ + ⋅ ⋅ + , where n w is the desired cut-off frequency (which can be related via Eqs. (13) and (15) to rise time and settling time) and ζ is the damping ratio (which has to be chosen greater than 1 for aperiodic behaviour). We obtain p 1 0 This controller setting, however, gives rise to a zero i k − in the closed-loop transfer function, which may affect the transient response (see Section 2.3.6). A solution to this issue is to cancel this zero by filtering the reference, as shown in Fig. 10. Another issue is the so-called integral windup phenomenon: when the output of either the current or the voltage controller saturates, the integral term of the controller increases and may become very large. At the end of the saturated mode of operation, a negative error is needed to remove the accumulated positive error, which may give large transients. Many anti-windup methods have been proposed [8][9][10]. A simple way to implement anti-windup is to switch off the integrator (by zeroing its input) in case of control output saturation. Other schemes, like the one represented in Fig. 11, enable the integral part of the controller to be dynamically saturated. Discrete-time control The development of digital technology over the past two decades along with increased performance demands have led to a growing use of digital control in power converters [11,12]. The benefits of using digital over analogue control are numerous: performance enhancement (as digital control allows more complex regulation schemes), improved flexibility, system monitoring and archiving capabilities, better noise immunity, to name a few. One major issue is the delay introduced with a digital controller. Other issues, such as aliasing, quantization errors, or limit cycling, may also affect the system operation [13,14].

Introducing the Z-transform
The Z-transform is the discrete-time counterpart of the Laplace transform. It is an essential tool for the analysis and design of discrete-time systems. Let us consider a sequence x(k) of sampled signals, where x(k) takes the value of the analogue signal x(t) at the kth sampling instant (the sampling period Ts is assumed constant). The Z-transform of this sequence is defined as where z is a complex variable which is related to the Laplace operator by e s Ts z ⋅ = .
From the Laplace time-shifting property, we know that e s Ts − shifting property: − convolution: − final value: Some examples of Z-transforms: Tables of commonly encountered Z-transforms can be found in the literature (see the references).

Relation between Laplace transforms and Z-transforms
Let x(t) be a causal continuous-time signal whose Laplace transform is X(s), and let x(k) be the discrete-time signal obtained by sampling x(t) at a uniform rate 1/Ts.

Case of signals having only simple poles
If X(s) has only simple poles, it takes the following form: The expression of the corresponding Z-transform is This relation enables one to establish the correspondence between the pole location in the sdomain and that in the z-domain. An important observation is that poles in the s-plane whose imaginary parts differ by a multiple of Ts π ⋅ 2 belong to the same locations in the z-plane: the unit disc can only represent signals of frequency up to the half the sampling frequency (the Nyquist frequency). The condition which has to be satisfied so that two different poles in the s-plane do not correspond to the same point in the z-plane is

General case
In the general case, the relation between X(s) and X(z) is given by The residue at j s , a pole of multiplicity m, can be calculated as follows:

Modelling of digitally controlled continuous-time systems
Let us focus on the physical process represented in Fig. 13, which is digitally controlled at the period Ts, and whose output is sampled at the same period. The structure of this hybrid system consists of a sampler (the analog-to-digital converter, or ADC), a digital controller, the continuous-time process, and a so-called hold-device (here assumed to be a digital-to-analog converter, or DAC). The sampler is preceded by an anti-aliasing filter (not represented in Fig. 13), which should preferably be taken into account for modelling. (36) From an external point of view, the set {zero-order hold + plant + sampler} is a discrete system, whose transfer function can be derived from the plant transfer function according to The open-loop transfer function is then and the closed-loop transfer function is (39) The use of the operator 1 − z is often preferred as it refers to the past and not the future. The transfer function of the digital controller can then be written as Using the Z-transform shifting property, Eq. (28), the following difference equation is obtained, where the present output is dependent on the present input and the past inputs and outputs: This is the controller algorithm.

Stability and robustness of closed-loop discrete systems
As seen in Section 3.2, a discrete system is stable if all the poles of its transfer function have a modulus strictly inferior to 1. So the closed-loop discrete system Eq. (39)

Influence of the poles on the system behaviour
The influence of the system poles on its transient behaviour is summarized in Fig. 15. Two interesting remarks are as follows: − poles closer to the origin lead to a faster transient regime; − the impulse response caused by poles having negative real part cycles back and forth between positive and negative deviations from the steady-state value. Fig. 15: Influence of the poles on the system transient behaviour

Particular case: second-order systems
As seen previously, a common controller design method consists in deriving the controller parameters from a pole placement, such that the dominant closed-loop dynamics is of first or second order. If a second-order system behaviour is chosen, this strategy consists in identifying the coefficients of the closed-loop transfer function denominator with those of the following reference polynomial:

Precision of closed-loop discrete systems
The conclusions in the continuous-time domain remain valid: to achieve zero steady-state error, the open-loop transfer function must contain the internal model of the reference (for example, at least one integrators (pole at z = 1) is required for a step input). To reject perturbations of class N in steady state, at least N integrators are needed in the open-loop transfer function.

Discrete-time controller synthesis
There are two main strategies employed to design digital controllers: − Emulation design, which consists in designing first a controller in the continuous-time domain. Afterwards, the controller is discretized such that the overall system preserves its characteristics.
− Direct discrete-time design, which consists in discretizing the plant at first using an appropriate hold device. Analysis and design can then be carried out in the discrete-time domain.

Principle of emulation design
The first step is the continuous-time controller design. At this stage the sampling is ignored. The next step is the discretization of the controller by using an approximation (emulation) method enabling the approximation at best of the continuous-time controller. The usual approximation methods are the Euler method and the Tustin method (also called bilinear transformation), which consist in substituting s by Eqs. (47) and (48). The controller algorithm (difference equation) can then be derived.
Backward Euler transformation: Tustin transformation: As an example, let us discretize a PI controller using the Tustin transformation:

Comparison between Euler and Tustin approximation methods
Both approximation methods preserve the stability, as shown in Fig. 16.
Eqs. (49) and (50) constitute, respectively, a first-order and a second-order approximation. The higher the sampling frequency, the smaller the approximation error.
The poles (and the zeros) being not preserved, the system frequency response is changed. By way of example, the frequency responses of the derivative terms obtained through both transformations are shown in Fig. 17. It can be seen that there is an inherent filtering effect at high frequencies with the Euler transformation, whereas with the Tustin transformation the magnitude increases with the frequency, resulting in noise amplification at high frequencies. Thus the Euler transformation is more appropriate for the discretization of high-pass filters, and the Tustin approximation is more appropriate for the discretization of low-pass filters. To convert continuous-time controllers to discrete-time ones, the Tustin method is the approximation of choice, as it provides better accuracy.
To conclude, an emulation controller always suffers performance degradation as compared to its continuous-time counterpart. To make the set {ADC -digital controller -DAC} behave the same as the continuous-time controller, a very fast sampling frequency may be needed. Special attention should be paid to the impact on the loop phase margin of the control delay (ADC and DAC delays + delay due to the computation of control algorithm by the processor) and the anti-aliasing filter.

Direct discrete-time design
Working entirely with discrete systems, using plant discrete-time modelling, is another approach. This design method enables us to relax the constraint on the sampling period. − The plant discretization may lead to the introduction of unstable zeros, which are difficult to compensate for.
On the contrary, a too large sampling period results in the following.
− Loss of information and aliasing, in the case of the violation of the Nyquist-Shannon sampling theorem.
− The regulation may not react readily enough to disturbances affecting the system.
− Plant discretization may yield poles having a negative real part: this is not desirable as the step response caused by such poles cycles back and forth between positive and negative deviations from the steady-state value. To prevent this, the following condition must be satisfied: A suitable choice for the sampling period is where CL B F is the desired closed-loop bandwidth [2,14].

RST-controller design
The general structure of a discrete controller for linear single-input single-output systems is represented in Fig. 18. This tri-branched structure is known as the RST structure: R, S, and T are three z −1 polynomials to be determined, usually by pole-zero placement. The control signal is calculated as This is a combination of feedforward (first term) and feedback (second term), which can be tuned separately. The general approach with RST is to decouple the tracking behaviour (reference following) from the regulation behaviour (disturbance rejection); R and S give the regulation behaviour, and T gives the tracking behaviour [15][16][17][18].
The tracking transfer function is and the regulation transfer function is (55) Note that a PID controller is a special case of RST controller, where The RST control law can be synthesized using the following design method: 1. Choice of the desired closed-loop transfer function: If a second-order system behaviour is chosen, m A has the following form (see Section 3.4.3): If a first-order system behaviour is desired, dom P is of the form contains the poles which will not be compensated, namely poles lying outside the unit circle (called unstable poles), poorly damped poles, and poles with negative real part. (60) The pole-zero cancellation yields 3. Perturbation rejection in steady state: an appropriate number of integral terms is introduced into the loop by means of the polynomial S: where n and l are, respectively, the perturbation and plant classes.  The controller sampling period is chosen according to Eq. (52).
Let us assume that oversampling is used to improve resolution and reduce noise of the current measure, and that the cut-off frequency of the anti-aliasing filter is high compared to (73) Then the tracking transfer function becomes