Pulse width modulation (PWM), used in the control of switching power-supply devices, has a main deficiency in its stressed noise at the frequency of commutations and its multiples. Harmonics in the high frequency range produce electromagnetic interference (EMI) and induce disturbances in supplied devices; these are recognized as undesired and have been made the subject of regulation. The introduction of randomness into control enables the transfer of the harmonic energy of deterministic approach into the continuum of the frequency spectrum, easing this problem of deterministic control.

        The random approach to the control of power supply devices (switching power converters) was initially developed in the 1990s, expanding basic ideas from the statistical theory of communications developed in the 1960s into the field of switching controlled devices. Results from this more recent research lie in the area of spectral density analysis of the control and other relevant variables in the device, but only for the fixed operating point in the open loop (in other words ignoring the dynamics of the closed loop in the bandwidth of the regulation). On the other hand, deterministic switching control, in the framework of the linear robust control theory, starts by establishing the nominal small-signal linear time-invariant (LTI) model of the plant, along with the bounds of maximal permitted modeling uncertainty, in which robust stability and robust performance are guaranteed for all models from the family of possible models.

        In the literature, the only results available that deal with the analytical consideration of the modeling and closed loop control of a random switching DC/DC converter (as far as the authors have been able to discern) are contained in.

        The goal of this paper is to synthesize the random switching approach to the control of the converter (aiming for harmonic amplitude and EMI reduction) on the one hand, and robust control in the closed loop (obtaining efficient disturbance elimination in the face of high level uncertainty in the modeling of the plant parameters) on the other hand.

        In the process of random modulation, the switching control variable D(t)=E(q(t)) can be generated in several ways, randomly varying one of three variables of the modulated control q(t) pulse shape ui(t-еi) (depicted in Fig. 1). In the figure Di is the duty ratio for the ith cycle of the switching, ei is the position of the start of the impulse in the ith cycle, and the period of the switching is Ti.

Fig. 1 Shape of the ith switching cycle pulse ui(t-ei) in the control signal q(t)

        Random change in variables ei, Di or Ti shapes the frequency spectrum of the output voltage; in other words it reduces the discrete component (harmonic) for the frequency of commutations.

        The randomness of ei and Di do not affect the periodicity of the sequence of the modulated control signal q(t), so Random Pulse Position Modulation (RPPM) and Random Pulse Width Modulation (RPWM) are periodic modulations, while techniques which include varying of the period of switching are called aperiodic modulations.

        The block diagram for the control structure is the same as for deterministic switching design. The signals of interest in the closed loop system are: reference r for output voltage vout, error in reference tracking e=r–vout, and plant input disturbance signal din.

        Pv and Pi are nominal converter model transfer functions from the control signal to the output voltage and inductance current, respectively, Ki is the current loop gain (for Voltage Mode Control, Ki=0), K is the voltage loop controller to be synthesized, Wp is the sensitivity weighting, used to give a preferred frequencyshaped sensitivity signal e., and lMI is the input multiplicative uncertainty bound of modeling.

        For random switching converters, the uncertainty bound is constructed with dynamics describing deterministic switching uncertainty, but with the DC value increased for the level of randomness applied – the maximal excursion of the control variable from its mean value due to its random distribution.

        The robust control theory adopts measures of robust stability (RS) and robust performance (RP) as a suitable objectives that define the performance of the closed loop system in the presence of structured uncertainty.

        Optimization can be done with respect to either reference signal r, ensuring optimal reference tracking, or input disturbance signal din, ensuring optimal input disturbance rejection.

        In the context of random switching, a suitable design procedure for the closed loop control would be:

        1. adopt control structure in the closed loop, design controller for R=0 (deterministic switching controller), and find PNB(R=0)

        2. adopt the random technique and start with a small value of random level R to be applied to the control (for example 5–10% of control variable random excursion)

        3. design optimal controller K

        4. inspect RS; if RS approached 1, go to step 7

        5. inspect PNB(R); if reduction of harmonic power with respect to deterministic switching PNB(R=0) is satisfactory, go to step 7

        6. increase random level R and go to step 3

        7. K is the optimal random switching robust controller to be applied in the closed loop; R is the random level to be used in the chosen random switching technique.

        Seven controllers are designed, resulting in l-optimal parameter tuning of fixed structure controllers or synthesized full order controllers:

        1. IMC -optimal for reference tracking, fixed structure

        2. IMC -optimal for input disturbance rejection, fixed structure

        3. the full-order H Loop Shaping Controller, is not subject to change with increase in R because its design is based on nominal performance, and robust stabilization is in the closed form of the Glover-Doyle compensator, again not dependent on R.

        4. full-order -suboptimal for reference tracking

        Вoth the zeroes and the poles of the controller transfer function do change place, but the dominant behavior of the control comes from the velocity constant change.

        5. full-order -suboptimal for input disturbance rejection.

        6. PI -optimal for reference tracking, fixed structure

        7. PI -optimal for input disturbance rejection, fixed structure.