Abstract
Due to its order reduction property, good dynamic performance and low sensitivity to disturbances and plant parameter variations, sliding mode control (SMC) has been the method of choice for handling nonlinear systems with uncertain dynamics and disturbances.
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18.1 Introduction
Due to its order reduction property, good dynamic performance and low sensitivity to disturbances and plant parameter variations, sliding mode control (SMC) has been the method of choice for handling nonlinear systems with uncertain dynamics and disturbances. Moreover, this control methodology reduces the complexity of feedback control design since the system is decoupled into independent lower dimensional subsystems [38, 39]. Because of these properties, sliding mode control has a wide range of applications in the areas of electric motors, power systems, power electronics, robotics, aviation and automotive control [1, 38,39,40].
Power electronic converters belong to the category of circuits controlled by switching devices with “ON/OFF” as the only admissible operating states [22, 28]. Therefore, control variables can take only values from a discrete set. For this type of circuits, SMC is preferred not only from a theoretical view but also a technological one. In the past, State Space Averaging approach was the method of choice to analyze some of the power conversion circuits [26, 35]. Using this technique, state space models and equations are obtained for each possible switch’s position. The equations are then averaged over a switching cycle to obtain a low frequency averaged model. Linear (after linearization if needed) or nonlinear control theory are then applied to design a feedback compensator. SMC provides a powerful tool to control power converters such that linearization is not needed and analysis and control design methods are developed in the framework of the nonlinear models [26, 33, 35, 39].
The main drawback of any system with pulse-width modulation (PWM) control approaches, including SMC, is the appearance of undesirable oscillations having finite amplitude and frequency due to the presence of unmodeled dynamics or discrete time implementation. This phenomenon, called “Ripple” in power electronics literature or “Chattering” in control theory, may lower control accuracy or incur unwanted wear of mechanical components [24, 39, 40]. An additional obstruction of SMC is the fact that it results in frequency variations, which is not acceptable in many applications [24].
Although the methods outlined in literature [4, 9, 10, 12,13,14, 36] showed efficient suppression of chattering, these methods are disadvantageous or not even applicable when dealing with power electronics systems controlled by switches with “ON/OFF” as the only admissible operating states. For example, gain-dependent and equivalent-control based methods are not applicable to reduce chattering since the output of power converters can only take two (or finite number of) values. Equivalent control method of SMC also yields the continuous motion equation directly. This motion, called “slow motion,” exits along with high frequency components [40]. The boundary layer approach, on the other hand, avoids generating sliding mode by replacing the discontinuous control action with a continuous saturation function [36, 38,39,40]. However, control discontinuities are inherent to these power electronics systems. When implementing a continuous controller, techniques such as PWM have to be exploited to adopt the continuous control law and feed it to the discontinuous system inputs. Therefore, it seems unjustified to bypass the inherent discontinuities in the system by converting a continuous control law to a discontinuous one by means of PWM. As an alternative, discontinuous control inputs should be used directly in control, and new methods should be investigated to reduce chattering under these operating conditions [38,39,40].
The most natural and straightforward way to reduce chattering in power converters is to increase the switching frequency. As technology progresses, switching devices are manufactured with enhanced switching frequency (up to 100 s KHz) and high power rating. However, heat power losses enforce a new restriction. Even though switching is possible with high switching frequency; it is limited to the maximum allowable heat losses caused by switching.
The objective of this chapter is to provide a comprehensive sliding mode control design procedure for power electronic converters that are used in energy conversion systems. This includes the DC/DC, AC/DC and DC/AC power electronic converters.
First, a comprehensive control design procedure for DC/DC power converters based on sliding mode control methodology. The input inductor current control and subsequently output capacitor voltage regulation for different types of DC/DC are investigated. Sliding mode control algorithms for Buck and Boost converters with incomplete information about system states are developed by designing state observers. Chattering issues are also discussed and a chattering reduction design approach for power converters is proposed.
Second, a new sliding model control strategy for AC/DC power converters is presented. The basic idea is to apply a feedback implementation of Pulsewidth Modulation (PWM). The proposed control algorithm exhibits low sensitivity to disturbances and fast dynamic performance in addition to the main converter properties. This includes unity power factor, sinusoidal input currents, and low level of DC output voltage ripple.
Third, Sliding Mode Pulsewidth Modulation (SMPWM) control methodologies for current-controlled inverters are proposed. Two novel approaches based on the sliding mode concept are presented to make the system tracking reference inputs. The proposed approaches control the phase currents and the neutral point voltage simultaneously. Optimization of different operational criteria is also presented.
18.2 Sliding Mode Control of DC/DC Power Converters
The input inductor current and the output capacitor voltage are normally selected as state variables for DC/DC converters. For most converters used in practice, the motion rate of the current is much faster than the motion rate of the output voltage. Calling upon the Singular Perturbation theory [19, 20],Footnote 1 the control problems of the DC/DC converters can be solved by using a cascaded control structure with two control loops: an inner current loop and an outer voltage loop. The current control loop is based on PWM that can be implemented using SMC or hysteresis control while output voltage regulation is normally a result of the current control loop such that output voltage converges to the desired reference value if designing the ideal current tracking system is accomplished [32, 34, 39].
18.2.1 Direct SMC of DC/DC Power Converters
(A) Buck-Type DC/DC Converters
The circuit structure of a Buck-type DC/DC converter is shown in Fig. 18.1, where E is the source voltage, C is the storage capacitor, L is the loop inductor, R is the load resistance, I is the input current and V is the output voltage.
Based on circuit analysis, the dynamic model of the Buck converter is given by:
where \(x_{1}=I\) and \(x_{2}=V.\)
Buck-type DC/DC converter
The switching function of the switch can be defined as:
The control objective is to achieve a constant output voltage denoted by\(\ V_{d}\). In other words, the steady state behavior of the Buck converter should be given by:
The proposed design methodology follows a two-step procedure known as integrator backstepping or regular form control [23, 25, 43]. First, it is assumed that \(x_{1}\) in Eq. (18.2) can be handled as a control input. However, since \( x_{1}\) is also the output of the current loop, given by Eq. (18.1), this first design step leads to the desired current \({x^{*}}_{1}\). After substituting Eqs. (18.4) and (18.5) into (18.2), the desired current is given by:
The goal of the second step of the proposed design procedure is to ensure that the actual current \(x_{1}\) tracks the desired current \({x^{*}}_{1}\) exactly. Due to its ideal tracking properties, sliding mode approach is an efficient tool for this task. If sliding mode is enforced in:
then, \(x_{1}=V_{d}/R\). In order to enforce sliding mode in the manifold \(s=0\), control u is defined as:
The condition for sliding mode to exist is derived from \(\left( \mathop {\mathrm {lim}}_{t\rightarrow +0}\dot{s}<0\right. \), \(\left. \mathop {\mathrm {lim}}_{t\rightarrow -0}\dot{s}>0\right) \). As a result, sliding mode is enforced if:
Equation (18.9) defines an attraction domain of the sliding manifold. Since the control\(\ u\), given by Eq. (18.8), does not contain a control gain that needs to be adjusted, the domain of attraction, given by Eq. (18.9) is predetermined by the system architecture. In steady state conditions, Eq. (18.9) is fulfilled by the definition of a Buck converter where the output voltage is smaller than the source voltage.
After the state of the inner current loop has reached the sliding manifold, i.e. converged to \(s=0\ \)at time\(\ t=t_{h}\), \({{x_{1}=x}^{*}} _{1}=V_{d}/R \) holds for \(t>t_{h}\) and the outer voltage loop is governed by:
The solution of Eq. (18.10) is given by:
where \(x_{2}\) tends to \(V_{d}\) exponentially. Hence, the goal of the proposed control design is achieved.
Boost-type DC/DC converter
(B) Boost-Type DC/DC Converters
The circuit structure of a Boost-type DC/DC converter is shown in Fig. 18.2, where circuit variables are similar to those defined for the Buck converter. Based on circuit analysis, the dynamic model of the Boost converter is given by:
where \(x_{1}=I\) and \(x_{2}=V\) and switching function u is given by Eq. (18.3).
Boost converters are more difficult to control than Buck converters. This is because of the fact that the control u appears in the voltage and current equations, and both in a bilinear fashion. Such a configuration implies a highly nonlinear system with a difficult control design. It also leads to unstable dynamics with respect to current when voltage is the only variable to be controlled [32, 34].
Since Boost converters satisfy the Motion-Separation principle that is based on the Singular Perturbation theory [19, 20], their control problems can also be solved by using two cascaded control loops: an inner current and an outer voltage control loops. Sliding Mode control techniques will be used again to design the current control loop.
Similar to the proposed control design for the Buck converter, the desired current is obtained from the outer voltage loop as:
where \(V_d\) is the desired output capacitor voltage. The switching function of the inner current control is defined as:
In order to enforce the current \(x_1\) to track the desired current \({x^*}_1\), control u can be designed as:
Under the above control scheme, the equivalent control is formally derived by substituting \(\dot{s}=\dot{x_{1}}=0\) in (18.12) and solve for the control input u as:
where \(x_{\mathrm {2}}\) is the output voltage of the slow voltage loop. The motion equation of the outer voltage loop during sliding mode is obtained by substituting the equivalent control, given by Eq. (18.17), into (18.13).
Equation (18.18) can be solved explicitly as:
where \(t_{h}\) represents the reaching instant of the sliding manifold \(s=0\) and\(\ x_{2}\left( t_{h}\right) \ \)is the output voltage at time \(t_{h}\). Apparently, \(x_{2}\) tends to \(V_{d}\) asymptotically as time goes to infinity.
The attraction domain of the sliding manifold \(s=0\) is found by applying the convergence condition \(s\dot{s}<0\) to the Eqs. (18.12) and (18.13), yielding:
Condition (18.20) implies that sliding mode can be enforced as long as the output voltage is higher than the source voltage. This requirement is essential for a Boost-type DC/DC converter. A careful consideration of the initial conditions is also required to guarantee the convergence to \(s=0\).
18.2.2 Observer-Based Sliding Mode Control of DC/DC Converters
In recent years, research on sliding mode control theory has revealed great advantages of incorporating certain dynamics into the sliding mode controllers [38,39,40]. This new approach, called Observer-based Sliding Mode Control, fall into the category of dynamic feedback control where the sliding mode controller simulates the ideal plant model in parallel with the real one [25, 38, 42]. This helps to reduce the number of states that need to be measured since the real plant states are substituted by observer states. The mismatch between the measurable state(s) and the observed one is the ‘bridge’ that establishes coupling between the plant system space and the controller space and keeps both systems operating close to each other. It can also be been utilized in different ways to improve the control performance. Overall, an observer can be viewed as an artificially introduced auxiliary dynamic system in order to improve the control performance [11, 16].
This subsection investigates the input inductor current control and subsequently output capacitor voltage regulation for different types of DC/DC. It develops sliding mode control algorithms for Buck and Boost converters with incomplete information about system states by designing state observers. For Buck converters governed by linear equations, conventional observer design methods could be applied directly while non-linear observers should be designed for Boost converters. It is assumed that the variable under control is measured only for both converters.
(A) Observer-Based Control of Buck-Type DC/DC Converters
The observer equations, or the auxiliary system, of the buck converter are designed as:
where \(l_1,l_2\) are constant scalar observer gains and \({{\hat{x}}_1,\ \hat{x }}_2\) are the estimates of the inductor current and output voltage. Note that it is only required to measure the inductor current \(x_1\).
Equations with respect to the mismatches \({\overline{x}}_1,\ {\overline{x}} _2 \) are defined as:
The error dynamics can be derived by subtracting Eqs. (18.1) and (18.2) from (18.21) and (18.22):
The characteristic polynomial of the above linear system is defined as:
For the second order error system given in Eq. (18.27), stability and desired rate of convergence can be provided by the proper choice of \(l_1,l_2.\ \)This implies that the observer errors \({\overline{x}}_{\mathrm {1}}\) and \({\overline{x}}_{\mathrm {2}}\) tend to zero asymptotically.
The switching function for the sliding mode current controller will be designed based on the observed current \({\hat{x}}_{\mathrm {1}}\) instead of the measured current \(x_{\mathrm {1}}\) as done in Eq. (18.7):
The control u that is applied to the real plant and the observer system is of the same form as in the case of the control scheme without an observer.
Assuming that sliding mode can be enforced in the vicinity of the sliding manifold \(\hat{s}=0\), \({\hat{x}}_{1}\)can be calculated as:
where \(t_{h}\) denotes the reaching instant of the sliding manifold \(\hat{s}\). The equivalent control of u can be obtained by solving \(\dot{\hat{s}}=0\) with respect to control:
After substituting \(u_{eq}\) into the real plant model, given by Eqs. (18.1) and (18.2), and considering the observer model, given by Eqs. (18.21) and (18.22), the motion in sliding mode can be represented as a reduced order system, comprising the motion of the real plant and the slow dynamics (output voltage loop) of the observer, as shown in the following equations:
Further analysis of Eqs. (18.32)–(18.34) results in:
The remaining task is to find the condition under which the occurrence of the sliding mode can be guaranteed. Applying the existence condition of sliding mode to Eq. (18.21) along with the substitution of Eqs. (18.28) and (18.29) yields:
Since \(s\left( 0\right) =0\) with zero initial conditions for the power converter and E is high enough, sliding mode existence condition (18.38) is fulfilled.
(B) Observer-Based Control of Boost-Type DC/DC Converters
In order to simplify the derivation of the observer-based control scheme of Boost converters, a new control input is introduced and defined as \(v=(1-u)\). The observer dynamics that are designed for a Boost converter are governed by:
where \({\hat{x}}_{\mathrm {1}},{\hat{x}}_{\mathrm {2}}\) are the observed inductor current and output voltage, i.e. outputs of the observer; and \(l_{ \mathrm {1}},l_{\mathrm {2}}\) are positive scalar observer gains.
Equations with respect to the mismatches \({\overline{x}}_1,\ {\overline{x}} _2 \) are defined as:
The error dynamics can be derived by subtracting Eqs. (18.12) and (18.13) from (18.41) and (18.42):
Equations (18.43) and (18.44) are nonlinear, since the system states are multiplied by the control input v. For the convergence proof, select a Lyapunov function candidate as:
The time derivative \(\dot{V}\) on the system trajectory can be easily calculated as:
Therefore, the convergence rate can be selected can be selected by proper choice of \(l_1,l_2\) under Sylvester condition \(\left( \frac{Ll_1}{R}>\frac{{({ Cl}_2)}^2}{4}\right) \) for quadratic form (18.46). As a result, observer errors \({\overline{x}}_1\) and \({\overline{x}}_2\) tend to zero asymptotically. The convergence rate of the inductor current estimation can be adjusted by the observer gains \(l_1,l_2\).
The switching function of the sliding mode current controller will be designed based on the observed current \({\hat{x}}_1\) instead of measured current \(x_1\) as done in (18.15):
The control u that is applied to the real plant and the observer system is of the same form as in the case of the control scheme without an observer.
Equation (18.48) can be represented in terms of the new control input \(v=(1-u)\) as:
Assuming that sliding mode can be enforced in the vicinity of the sliding manifold \(\hat{s}=0\), \({\hat{x}}_1\)can be calculated as:
where \(t_h\) denotes the reaching instant of the sliding manifold \(\hat{s}\). The equivalent control of u can be obtained by solving \(\dot{\hat{s}}=0\):
After substituting \(v_{eq}\) into the real plant model, given by Eqs. (18.12) and (18.13), and considering the observer model, given by Eqs. (18.39) and (18.40), the motion in sliding mode can be represented as a reduced order system, comprising the motion of the real plant and the slow dynamics (output voltage loop) of the observer.
By defining the errors \({\overline{x}}^{\mathrm {*}}_{\mathrm {1}}\,\mathrm {=}\, \frac{V^{\mathrm {2}}_d}{RE}\,\mathrm {-}x_{\mathrm {1}}\) and \({\overline{x}}_{ \mathrm {2}}\,\mathrm {=}\,{\hat{x}}_{\mathrm {2}}\,\mathrm {-}x_{\mathrm {2}}\) and since \({\overline{x}}_{\mathrm {1}}\) tends to zero at the desired rate, the above equations can be transformed into a second order error system as:
Further analysis of Eqs. (18.55) and (18.56) results in:
This means that \({\overline{x}}^*_1\) converges to zero asymptotically at a rate that depends on \(l_1\). As a result, \({\overline{x}}_2\) and \(x_2\) tend respectively to zero and \(V_d\). The remaining task is to find the condition under which the occurrence of the sliding mode can be guaranteed. Applying the existence condition of sliding mode to (18.47) along with the substitution of Eqs. (18.39), (18.40) and (18.49) yields:
Equation (18.60) can be represented in terms of the new control input \(v= (1-u)\) as:
Since \(x_1\) is measured and \({\hat{x}}_1,\ {\hat{x}}_2\) are state variables in the controller space, the initial conditions of the observer, \({\hat{x}} _1(0)\) and \({\hat{x}}_2\left( 0\right) \), can be designed such that the occurrence of the sliding mode is always guaranteed.
18.2.3 Simulation Results
In order to evaluate the proposed sliding mode control strategies for DC/DC power converters, several computer simulations have been conducted using MATLAB/Simulink software.
(A) Direct Sliding Mode Control of Boost Converters
Parameters of the simulated Buck converter are listed in Table 18.1. Figures 18.3 and 18.4 show the simulation results of the proposed control algorithm for Boost converters. As it can be seen from the figures, both the inductor current and the output capacitor voltage converge rapidly to their reference values.
Current response of a sliding mode-controlled Buck DC/DC power converter
Voltage response of a sliding mode-controlled Buck DC/DC power converter
(B) Direct Sliding Mode Control of Boost Converters
Parameters of the simulated Boost converter are listed in Table 18.2. Figures 18.5 and 18.6 show the simulation results of the proposed control algorithm for Boost converters. As it can be seen from the figures, both the inductor current and the output capacitor voltage converge rapidly to their reference values.
Current response of a sliding mode-controlled Boost DC/DC power converter
Voltage response of a sliding mode-controlled Boost DC/DC power converter
(C) Observer-Based Sliding Mode Control of Buck Converters
Parameters of the simulated Buck converter are listed in Table 18.1. The observer gain is selected as \(l=200.\) Two sets of the observer initial conditions, listed in Table 18.3, were selected in order to compare the influence of observer initial conditions on the system response.
Figures 18.7 and 18.8 show the response of real and estimated inductor current and output capacitor voltage under different initial conditions. Note that both the inductor current and the output capacitor voltage converge rapidly to their reference values and the system response can be influenced by the design of the observer initial conditions.
(D) Observer-Based Sliding Mode Control of Boost Converters
Parameters of the simulated Boost converter are listed in Table 18.2. The observer gain is selected as \(l=200.\) Two sets of the observer initial conditions, listed in Table 18.4, were selected in order to compare the influence of observer initial conditions on the system response.
Figures 18.9 and 18.10 show the response of real and estimated inductor current and output capacitor voltage under different initial conditions. Note that both the inductor current and the output capacitor voltage converge rapidly to their reference values and the system response can be influenced by the design of the observer initial conditions.
18.2.4 Sliding Mode Control of Multi-phase DC/DC Power Converters
This section presents a new chattering reduction approach for power converters. The proposed scheme is based on the idea of designing a multi-phase converter system that operates at a given and fixed switching frequency without any additional dynamic elements. This may be possible by providing a desired phase shift between phases for any loads or frequencies in order to implement the “Harmonic Cancellation” method [24]. Several attempts have been made to apply this idea to PWM such that phase shifts are interconnected and can be controlled using a transformer with primary and secondary coils in different phases. On the other hand, the phase shift was obtained using delays, filters, or set of triangular inputs with selected delays [27, 45, 48].
Response of real (curves start from zero) and estimated current of a buck converter under different initial conditions. Solid line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=0.12}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=5.0}\); dashed line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=0.07}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=2.5}\)
Response of real (curves start from zero) and estimated voltage of a buck converter under different initial conditions. Solid line \({\hat{x}}_1\left( 0\right) =0.12\), \({\hat{x}}_2\left( 0\right) =5.0\); dashed line \({\hat{x}}_1\left( 0\right) =0.07\), \({\hat{x}}_2\left( 0\right) =2.5\)
Response of real (curves start from zero) and estimated current of a Boost converter under different initial conditions. Solid line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=0}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=0}\); dashed line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=1.95}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=38.5}\)
Response of real (curves start from zero) and estimated voltage of a Boost converter under different initial conditions. Solid line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=0}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=0}\); dashed line \({\hat{x}}_{\mathrm {1}}\left( 0\right) \mathrm {=1.95}\), \({\hat{x}}_{\mathrm {2}}\left( 0\right) \mathrm {=38.5}\)
18.2.4.1 Frequency Control
Chattering frequency normally depends on the width of the hysteresis loop in the switching element which is usually selected such that the frequency does not exceed the maximum admissible level \(f_{des}\ \)for all operation modes. This can be achieved by varying the width of hysteresis loop [9, 29].
It order to formulate the problem statement, the following system with scalar control is considered.
It is assumed that control u should be designed as a continuous function of state variables \({(u}_{0}\left( x\right) )\). This situation is usually assumed with the “Cascade Control” approach that is used for electric motors with the current as a control input [23, 39]. Power converters often utilize PWM as a principle operation mode in order to implement the desired control algorithm [22, 28]. Sliding mode is one of the tools to implement this operation mode based on the feedback approach as shown in Fig. 18.11, which illustrates that output u tracks the reference input \(u_{0}(x)\) in sliding mode.
Sliding mode control for a simple power converter model structure
Select the sliding surface as:
It is evident that sliding mode, in the surface \(s=0\) with \(u\equiv u_0\), exists if\(\ M>\left| g(x)\right| \). If the control is implemented with a hysteresis loop, the switching frequency can be maintained at the desired level \(f_{des}\) by varying the width of the hysteresis loop \(\varDelta \) [24].
18.2.4.2 Chattering Suppression
Suppose that the desired control is implemented by m power converters with \(s_i\mathrm {=}\frac{u_0}{m}\mathrm {-}u_i\mathrm {,\ (}i\mathrm {=1,2,\dots }m \mathrm {)}\) and \(\frac{u_0}{m}\) as reference inputs as shown in Fig. 18.12. If each power converter operates properly, the output is equal to the desired control \(u_0\mathrm {(}x\mathrm {)}\). Amplitude and frequency of chattering in each converter can be found as:
The chattering amplitude of output u depends on the oscillation in each phase. Its maximum value is m times higher than that of each converter. For the system shown in Fig. 18.12, phases depend on initial conditions and cannot be controlled.
m-phase converter with evenly distributed reference
First, it should be demonstrated that, by varying phases, the output oscillation amplitude can be controlled. Suppose that a multiphase converter with m phases is to be designed such that the period of chattering T is the same in each phase, and two subsequent phases have phase shift T/m. Since chattering is a periodic time function, it can be represented as Fourier series with frequencies:
The effect of the k-th harmonic in the output signal is the sum of individual outputs from all phases and can be easily calculated as:
To find Z, consider the following equation:
Since
then,
The function \(e^{\mathrm {-}j\frac{\mathrm {2}\pi k}{m}}\) is equal to 1 only if \(k\mathrm {/}m\) is an integer or\(\mathrm {\ }k = m\mathrm {,2}m\mathrm { ,\ldots }\), which means that \(Z=0\) for all other cases.
The above analysis shows that all harmonics except for \(lm\left( l=1,2,\dots \right) \) are suppressed in the output signal. As a result, the amplitude of chattering can be reduced to the desired level by increasing the number of phases and providing a desired phase shift between each two subsequent phases.
18.2.4.3 Design Principle
In order to describe the design idea of the proposed chattering suppression approach, suppose that two power converters are implemented as shown in Fig. 18.13.
A controller model with two interconnected phases
The switching function for the second converter is selected as:
where,
The time derivatives of \(s_{1},\ s_{2}^{*}\) are:
Figures 18.14 and 18.15 analyze the system behavior in the plane \(s_{1}\) and \(s_{2}^{*}\) with \(a=0\), \(\propto =1.\) \( \varDelta \) and \(\alpha \) are the widths of the hysteresis loops for the two sliding surfaces. T can be easily calculated as:
The system behavior in s-plane
Control of the phase between \(v_{\mathrm {1}}\mathrm {,\ }v_{\mathrm {2}}\)
By analyzing Fig. 18.14, it can be seen that the phase shift becomes:
where \(\varPhi \) is equal to the time for changing \(s^{*}_{ \mathrm {2}}\) from \(\frac{\alpha {\varDelta }}{\mathrm {2}}\) to \(\mathrm {-} \frac{\alpha {\varDelta }}{\mathrm {2}}\) or vice versa.
The previous analysis demonstrates that the phase shift between oscillations can be selected by proper choice of \(\mathrm {\propto }\) for any switching frequency without using dynamic elements. The value \(\mathrm {\propto }\) is calculated to provide the desired phase shift. Since \(\varPhi \) must be equal to T/m, \(\propto \)can be found as:
For \(a>0,\ \propto \ \)can be selected such that phase shift is equal to what we need. Finally, the proposed chattering suppression design procedure for multiphase converters can be summarized as follows:
-
1.
Select the width of the hysteresis loop as a state function such that the switching frequency in the first phase is maintained at the desired level.
-
2.
Determine the number of phases for a given range of \(\alpha \) variation.
-
3.
Find the parameter \(\propto \) as a function of \(\alpha \) such that the phase shift between two subsequent phases is equal to \(\frac{1}{m}\) of the oscillation period of the first phase.
Finally, another approach, called “master-slave” method of multiphase converters can be used for chattering reduction even if for a given number of phases m, parameter a is beyond the admissible domain and the desired phase shift cannot be guaranteed by varying the width of the hysteresis loop. For the master-slave implementation, each phase can be complimented by several sequentially connected “slaves”, as illustrated in Fig. 18.16 for a two-phase converter, such that the total phase shift is equal to the desired value.
A modified master-slave mode schematic with two more additional systems. \(v_{\mathrm {2,3}}\) is the switching command for the second channel
18.2.4.4 Simulation Results
In order to evaluate the proposed sliding mode chattering suppression control strategies for multi-phase DC/DC power converters, computer simulations have been conducted using MATLAB/Simulink software.
Multi-phase DC/DC Converter Topology
The chattering suppression approach developed in the previous section is applied to DC/DC multi-phase buck converters. The objective is to demonstrate, via simulations, the effectiveness of the proposed design methodology in multiphase power converters. Simulations also check the range of the function “\(\varvec{a}\)” for which the chattering suppression takes place. The “master-slave” method is utilized in all simulations. A two-phase DC/DC converter is shown in Fig. 18.17. Simulation parameters are listed in Table 18.5.
A two-phase DC/DC converter
For simulation purposes, the governing equations of the m-phase converter are assumed as follows:
The following control law is used for the two-phase power converter (\(m =2\)) represented in Eqs. (18.81) and (18.82):
Hysteresis loop gain \(K_h\)
where \(V_{ref}\) and \(I_{ref}\) are the reference input voltage and the corresponding reference load current, respectively. \(a=\frac{V_{s}}{2}-\frac{ I_{ref}R_{a}}{m}-V_{L}\), \(M=\frac{V_{s}}{2L}\) and \(b=\frac{R_{a}}{L}\). The desired phase shift T / 2 is obtained by using two additional blocks with a phase shift of T / 4 in each of them. The four-phase converter (\(M=4\)) is simulated with switching frequency control of the first phase by appropriate choice of hysteresis width or hysteresis loop gain \(K_{h}\) in order to maintain the switching frequency at 50 Hz. The selected function \(K_{h}\left( V_{ref}\right) =-0.0013V_{ref}^{2}+0.0127V_{ref}-0.0007\) is shown in Fig. 18.18.
Figures 18.19, 18.20 and 18.21 demonstrates the effectiveness of the proposed chattering suppression algorithm on a 4-phase DC/DC power converter with three different reference inputs, \(v_{ref1}=3\) V, \(v_{ref2}=6\) V, and \(v_{ref3}=8\) V, respectively. Note that the inductance is relatively small in order to have fast converter dynamics. This leads to a high level of chattering in each phase, but it is practically suppressed in the output signal.
Finally, it is demonstrated that chattering can be reduced considerably following the “master-slave method” even if for a given number of phases m, parameter a is beyond the admissible domain and the desired phase shift cannot be guaranteed by varying the width of the hysteresis loop. For the master-slave implementation, each phase can be complimented by several sequentially connected “slaves”, as illustrated in Fig. 18.16 for two-phase converter, such that the total phase shift is equal to the desired value. Of course, chattering can be suppressed by increasing the number of phases preserving the “one slave in one phase” approach, as shown in Fig. 18.22 for an 8-phase converter.
Simulation results for the four-phase DC/DC converter, \(V_{ref}=3\) V
Simulation results for the four-phase DC/DC converter, \(V_{ref}=6\) V
Simulation results for the four-phase DC/DC converter, \(V_{ref}=8\) V
Simulation results for the eight-phase DC/DC converter, \(V_{ref}=1.5\) V
Experimental Setup of a 4-phase DC/DC buck converter [2]
18.2.4.5 Experimental Results
Experimental results were conducted and discussed in [2] in order to demonstrate the effectiveness of the proposed sliding mode chattering suppression approach for multi-phase converters and also verify the simulation results.
Figure 18.23 shows the experimental setup. The system consists of a 4-phase DC/DC buck converter that is controlled using two loops: A current inner loop that is sliding mode controlled with a hysteresis band, and a voltage outer loop that defines the current reference through a PI controller. Table 18.6 shows the converter parameters.
Figure 18.24 shows the experimental waveforms of the input current, output voltage and control command for a half- bridge, 1-phase DC/DC converter. The reference output voltage is 5 V. The average input current value is 2.44 A and the chattering amplitude is 0.47 A. The average output voltage is 4.96 V.
Figures 18.25 and 18.26 show the four shifted input currents and the MOSFET drain voltages of a 4-phases DC/DC converter, respectively. Both Figures demonstrate that the master-slave algorithm works properly, with respect to the current shifting. The current values of each phase are \(i_{L1}=0.62\mp 0.43\) A, \(i_{L2}=0.6\mp 0.42\) A, \( i_{L3}=0.63\mp 0.43\) A, and \(i_{L4}=0.66\mp 0.42\) A.
The effectiveness of the proposed chattering suppression algorithm is demonstrated in Fig. 18.27, which represents the input current, output voltage and control command of a 4-phase DC/DC converter, respectively. The reference output voltage is 5 V. As it can be seen, the current chattering amplitude is 0.095, which is five times less than the current chattering amplitude of the 1-phase DC/DC converter.
Input current, output voltage and command control of a 1-phase DC/DC converter, \(v_{dc\_ref}=5\) V
Input phase currents a 4-phase DC/DC converter
MOSFET drain voltages of a 4-phases DC/DC converter
Input current, output voltage and command control of a 4-phase DC/DC converter, \(v_{dc\_ref}=5\) V
Input current, output voltage and command control of a 4-phase DC/DC converter, \(v_{dc\_ref}=4.59\) V
Figure 18.28, shows the input current, output voltage and control command of a 4-phase DC/DC converted, where the reference output voltage is 4.59 V. It can be noticed that the algorithm almost cancelled the chattering completely. The average value of the output current is 2.28 A, with a chattering width of 0.033, which is about 14 times less than the current chattering amplitude of the 1-phase DC/DC converter. This is due to the fact that the four duty cycles are equal to 0.5, and the corresponding Fourier harmonic coefficients cancel each other.
18.3 Sliding Mode Control of AC/DC Power Converters
Recently, a variety of power electronic devices, including three-phase AC/DC power converters, has been widely used in energy conversion systems [28]. One of their disadvantages is that they cause inherent problems of reactive power generation and higher harmonic content in the input current. These practical problems have become more serious as the AC/DC converter capacity becomes much larger [8, 30, 44].
The ideal AC/DC converter has a constant dc output voltage (or current) and a pure sinusoidal input current at unity power factor from the ac line. Conventional thyristor phase-controlled converters have an inherent drawback which is that the power factor decreases as the firing angle increases and that harmonics of the line current are relatively high [21, 46]. In recent years, there has been a tendency to operate AC/DC converters with PWM switching patterns to improve the input and output performance of the converter. PWM AC/DC converters offer distinct advantages over the conventional rectifiers [15, 47]. These advantages include unity power factor, capability of bidirectional power flow, low harmonic components in input current and low ripple in output voltage. All of these features simplify filtering problems on both ac and dc sides of the converter [8, 39, 47].
The objective of this section is to develop a feedback control algorithm for AC/DC power converters such that the output voltage is maintained at the desired level with zero steady state error, input currents are free of higher harmonics and the reactive power is equal to zero. This mission will be accomplished using the framework of SMC methodology implying that the order of the motion is reduced with state trajectories in the pre-selected manifold with the system state space. Furthermore, the control system is designed in two reference frames: the abc natural reference frame, and the dq synchronous reference frame.
18.3.1 Circuit Model and Design Methodology
The circuit model and analysis of the three-phase PWM AC/DC voltage source converter in the natural abc and the rotating dq reference frames is presented next. First, consider the three-phase PWM AC/DC voltage source converter shown in Fig. 18.29, where \( e_{a},e_{b},e_{c}\) are the balanced three-phase AC voltages representing the infinite bus, \(R_{g}\)and \(L_{g}\ \)represent the grid-side resistance and inductance, respectively, \(C_{dc}\) is the dc-link capacitance, \( i_{a},i_{b},i_{c}\) are the three-phase AC input currents, \(i_{dc}\) is dc-link current, \(v_{dc}\) is dc-link voltage and \(i_{L}\) is the load current.
Three-phase PWM AC/DC voltage source converter scheme
Based on circuit analysis, the AC input current equations are given by:
where \(v_{an},v_{bn},v_{cn}\) are the voltages from the AC side of the converter to the power neutral point n. The balanced three-phase AC voltages are expressed as:
where \(E_0\) is the amplitude of the phase voltage and \(\omega \) is the ac power source angular frequency. If we assume that \(i=\left[ \begin{array}{c} i_a \\ i_b \\ i_c \end{array} \right] \), \(e=\left[ \begin{array}{c} e_a \\ e_b \\ e_c \end{array} \right] \),\(\ v_s=\left[ \begin{array}{c} v_{an} \\ v_{bn} \\ v_{cn} \end{array} \right] \), then equations in (18.89)–(18.91) can be re-written in a compact form as:
The switching function S of each switch can be defined as:
The voltage vector \(v_{s}\ \)be expressed in terms of the switching functions \(S\mathrm {=}\left[ \begin{array}{c} S_{a} \\ S_{b} \\ S_{c} \end{array} \right] \) as:
By substituting Eq. (18.97) into (18.95), the AC input current equations can be expressed as:
Finally, the output voltage equation can be given by:
The operating principle of the three phase PWM grid-side voltage source converter can also be analyzed by the classical rotating field theory with the well-known Park and Clarke transformations. The dynamic model of three-phase AC/DC voltage source converter in the d-q rotating reference frame can be described as [28]:
where, \(v_d=v_{dc}s_d\) and \(v_q=v_{dc}s_q\). \(s_d\) and \(s_q\) are the switching functions in the d-q synchronous reference frame. \(i_d\), \( i_q\), \(e_d\) and \(e_q\) are the input currents of the converter and grid voltages in the synchronous d-q reference frame, respectively. Note that \(v_{dc}=v_{dc\_d}\) due to the field orientation.
The design methodology proposed in this chapter is different compared to the one usually used for AC/DC power converters. It is common to start with solving the main problem of maintaining the DC output voltage at the desired level and then minimizing high order harmonics and reactive power at the input. In this chapter, the inverse sequence of actions is offered. First, a tracking system is designed with sinusoidal current references proportional to the input AC voltages. This means that the reactive power is automatically equal to zero. If the tracking system is ideal, the input current will be free of higher harmonics. The second phase of the research work is the output voltage control. It will be shown that the output voltage will be constant and depends only on the amplitude of the reference input if designing the ideal tracking system is accomplished.
Furthermore, the proposed strategy does not utilize conventional SMC. As previously mentioned, the objective is to control three variables: two phase current and output voltage, using three dimensional control vectors. Generally, this can be done by enforcing sliding mode at the intersection of three sliding surfaces that depend on the mismatches between the reference and real values. However, the following analysis will show that matrix multiplying control is singular and conventional SMC methodology is not applicable.
18.3.2 Proposed Control Scheme in the \(\varvec{abc}\) Natural Reference Frame
(A) Sliding Mode Current Tracking Control
According to the proposed design procedure, phase currents should track sinusoidal reference inputs proportional to input ac voltages. The tracking problem is solved in the framework of sliding mode methodology. Substituting Eq. (18.100) into (18.99) results in:
where, matrix \({\varGamma }_0\) is given by:
Since the sum of the three-phase currents is equal to zero, three state variables should only be controlled: output voltage and two phase currents in a system with a three dimensional control vector S. But, unfortunately the conventional sliding mode method cannot be applied directly since matrix \({\varGamma }_0\) is singular. Because of that, we will start with designing a tracking system for two phase currents only. It means that sliding mode should be enforced on the intersection of two surfaces \({\sigma } _a=L\left( i_{aref}-i_a\right) \) and \({\sigma }_b=L\left( i_{bref}-i_b\right) \) or in vector form:
Excluding the phase current \(i_c=-i_a-i_b\) yields to:
where \({\Gamma }\) is a 2\(\,\times \,\)3 matrix given by:
The ideal tracking system is based on Lyapunov function:
Discontinuous control should be selected such that time derivative of V is negative definite. The time derivative \(\dot{V}\) on the system trajectory can be easily calculated as:
where \(F\left( .\right) \) is state function, which does not depend on control. \(\alpha ,\beta \) and \(\gamma \) are given by:
If \({v_{dc}}/{L}\) is large enough, \(F\left( .\right) \) can be suppressed with control S given by:
where \(\mathrm {\upalpha }, \mathrm {\upbeta }\) and \(\mathrm {\upgamma }\) are selected such that \(\dot{V}\mathrm {<0}\). This means that \({\sigma }_{ab}\) tends to zero. Even more, \({\sigma }_{ab}\) becomes zero after a finite time interval \({t}_{s}\) [10]. As a result, sliding mode occurs with \( i_{ab}=i_{abref}\). If a tracking system is designed with sinusoidal current references proportional to the input AC voltages as:
where K is constant. Then, the first problem of having zero reactive power and phase currents free of higher harmonics is solved.
(B) Sliding Mode Output Voltage Control
The main objective is to maintain the DC output voltage at the desired level. The first question to be answered is what is the output voltage after sliding mode occurs? Sliding mode equation can be derived using the so called “equilibrium control” procedure [39]. Step 1: time derivative of vector \({\sigma } _{ab}\) on the state trajectory is made equal to zero. Step 2: the obtained algebraic equation should be solved with respect to control S and then substituted in the original system.
The equivalent control:
is the solution of the following equation:
The three phase input currents can also be written as:
After substituting Eqs. (18.114) and (18.115) into (18.107), the sliding mode equation can be obtained as:
For the given balanced three phase input voltages given by Eqs. (18.88)–(18.91), and after sliding mode occurs \((i_{ab}=Ke_a)\), it can be noticed that:
After substituting Eq. (18.118) into (18.117), the sliding mode equation is given by:
The last equation has only one asymptotically stable equilibrium point that can be described by:
In order to prove the stability of the equilibrium point, represent the right hand side of Eq. (18.119) in the form:
The derivative of \(f\left( v_{dc}\right) \) is negative and is given by:
Therefore, \(g\left( u_c\right) >0\) and \(C\frac{d\varDelta u_c}{dt}=-g\varDelta u_c\), \( \varDelta u_c\rightarrow 0.\ \)This proves that the equilibrium point of (18.119) is asymptotically stable. It can be seen that the output voltage tends to a constant value which depends on the value of constant K in the reference current input equation. If \(R_L\) is known, K can be easily assigned. Otherwise, K should be varied such that the equilibrium point is equal to desired value \(u_{cref}\). As in any boost converter, the output voltage should be greater than some minimum value. Correspondingly, \(K>K_{min}\) and \(K_{min}>0\). Selected K as an integral function of the mismatch as:
where \(\alpha \) is constant. The last term of (18.123) is added to prevent K from being less than \(K_{min}\). The only steady state of the system is \(u_c=u_{cref}\), and the value of K is given by:
If \(v_{dc\_ref}>0\), \(K\left( 0\right) >0\), and \(v_{dc}\left( 0\right) >0\), then, according to (18.120) and (18.123), \( K\left( t\right) >0\) and \(v_{dc}\left( t\right) >0\) for any \(t>0\). Substituting \( u_c\ =u_{cref}+x>0\) in Eq. (18.120) results in:
Differentiating Eq. (18.125) results in:
Select the Lyapunov function:
The time derivative \(\dot{V}\) on the system trajectory can be easily calculated as:
Since \(\dot{x}(t)\equiv 0\), this means that \(x(t)\equiv 0\) and the equilibrium point is globally asymptotically stable.
18.3.3 Proposed Control Scheme in the dq Synchronous Reference Frame
(A) Sliding Mode Current Tracking Control
Similar to the control design for electric motors, the current control of a Boost-type AC/DC power converter can be designed either in phase coordinates or in the dq coordinate frame. Since the control criteria (as listed in the performance characteristics) are normally given in the dq coordinate frame, it is more convenient to design the current control in the dq coordinate frame than in phase coordinates. Represent the current equations in (18.100) and (18.101) in a vector form as:
where,
The switching functions for the current control are designed as:
where \(i^*_d\) and \(i^*_q\) are the desired values of the currents in the (d, q) coordinate frame. The next task is to find the condition under which sliding mode can be enforced. It can be noticed that no control gain can be adjusted for the control design of AC/DC converters. The solution is to find a domain in the system space from which any state trajectory converges to the sliding manifold defined by \({\sigma }_d=0\), \({\sigma }_q=0\). Defining \({\sigma }_{dq}={[{\sigma }_d,{\sigma }_q]}^T\) and taking the time derivative of \({\sigma }_{dq}\) results in:
where \(I^*_{dq}={[i^*_d,i^*_q]}^T\), \(F_{dq}={\dot{I}}^*_{dq}-f_{dq} \left( I_{dq},e_{dq},{\omega }_g\right) \), \(D=bA^{\alpha ,\beta }_{d,q}A^{a,b,c}_{\alpha ,\beta }\), \(A^{\alpha ,\beta }_{d,q}A^{a,b,c}_{\alpha ,\beta }\) are the Park and Clarke transformation matrices, respectively, S is the vector of transformed switching functions defined in (18.8).
Design the matrix of switching functions S as:
where \(S^*={[S^*_a,S^*_b,S^*_c]}^T\) is a vector of transformed switching functions and \(sign\left( S^*\right) ={[{sign(S}^*_a),sign(S^*_b),{sign(S} ^*_c)]}^T\). The transformed vector \(S^*\) should be designed such that, \( s_{fd}\) and \(s_{fq}\) disappear in finite time. Vector \(S^*\) is selected as:
Design a Lyapunov function candidate as:
The time derivative of the Lyapunov function on the system trajectory can be easily calculated as:
where \(F^*={[ \begin{array}{ccc} F^*_a&F^*_b&F^*_c \end{array} ]}^T\) \(=D^TF_{dq}\). Substituting control equation (18.134) into (18.137) results in:
where \(D^TD\) is a singular matrix that can be calculated as:
Equation (18.138) can be expanded as:
where \(l\ne m\ne n\ and\ l,m,n\ \in \{a,b,c\}\). Equation (18.140) can be further represented as:
This means that the sufficient condition for \(\dot{V}<0\) can be given by:
Equation (18.142) defines a subspace in the system space in which the state trajectories converges to the sliding manifold defined by \({\sigma }_{dq}=0\) in finite time. This is to show that the attraction domain in the sliding manifold is bounded in the state space. It is also important to notice that parameter \(b=\frac{v_{dc}}{2L}\) should be high enough at the initial time instant. The output voltage is not zero at the initial time instant since \(v_{dc\_d}=v_{dc}\). In critical applications, this can be achieved by starting the converter operation with an open-loop control. The last step of the current control design is that the resulting controls \(S_a,\ S_b\ \mathrm {and}\ S_c\) should be mapped into the switching patterns that can be applied to the power converter. This can be done using the following system of equations:
(B) Sliding Mode Output Voltage Control
It is necessary to determine the reference currents feeding to the current controller, \(i^{*}_{d}\) and \(i^{*}_{q}\), in order to ensure asymptotic stability of the output voltage regulation. Neglecting the voltage drop over the phase resistance \(R_g\), Eqs. (18.100)–(18.102) can be simplified to:
In general, the value of the inductance satisfies L \(\ll \) 1, while the right-hand sides of equations in (18.144)–(18.146) have the values of the same order. Therefore, \(di_d/dt\), \(di_q/dt \gg du_{cd}/dt\) indicating that the dynamics of \( i_d \) and \(i_q\) are much faster than those of \(v_{dc}\). In case that the fast dynamics are stable, the output voltage control can be simplified considerably. Based on the Singular Perturbation Theory [19, 20], the left-hand sides of Eqs. (18.144)–(18.146) can be formally equal to zero, and the algebraic equations for \(s_d\) and \(s_q\) can be solved. Therefore, the following equation system is valid for control design of the slow-manifold:
where \(i^{*}_{d}\) and \(i^{*}_{q}\) are the reference values of \(i_d\) and \(i_q\), respectively. The real currents are replaced with their reference values in Eqs. (18.147) and (18.149), since we assume that the inner current control loop is in sliding mode with \({ \sigma }_d=i^{*}_{d}-i_{d}\), \({\sigma }_{q}=i^{*}_{q}-i_{q}\).
As mentioned in the introduction, the objective is to control the three-phase AC/DC power converter such that the output voltage is maintained at the desired level with zero steady state error, input currents are free of higher harmonics and the reactive power is equal to zero. The demand of sinusoidal input currents has been fulfilled automatically by involving the (d,q) transformation. The following characteristics will cover the remaining major requirements of a well-controlled three-phase AC/DC power converter:
-
1.
The output voltage should converge to its reference value \(u^*_c\).
-
2.
The input current phase-angle, \({\rho }^{\mathrm {*}}\mathrm {=arctan} \mathrm {(}i^{\mathrm {*}}_q\mathrm {/}i^{\mathrm {*}}_d\mathrm {)}\), should trace its reference value.
-
3.
The power-balance condition should be satisfied, i.e. \(e_di^*_d+\ e_qi^*_q=u^*_ci_l=v^*_{dc}i_L\).
The reference currents \(i^*_d \) and \( i^*_q\) should be calculated satisfying these requirements. Substitution of (18.147) and (18.148) into (18.149) yields:
Taking into account the power-balance condition, the above equation can be simplified to:
For a system with a pure load resistance \(R_L\), \(i_L=v_{dc}/R_L\). As a result, linear dynamics for output voltage \(u_{cd}\) can be calculated as:
Define the voltage regulation error as \({\overline{v}}_{dc}=v^*_{dc}-v_{dc}\) with a constant desired voltage\(\ {\dot{v}}^*_{dc}=0\) such that:
Equation (18.153) shows that the voltage error tends to zero asymptotically with the time-constant \(R_LC\). It also means that the output voltage converges to its reference value automatically if the power-balance condition is fulfilled. Finally, \(i^*_d \) and \( i^*_q\) can be calculated as:
where \({\rho }^*\) is the input reference current phase angle. It is usually determined by the control designer.
18.3.4 Simulation Results
In order to evaluate the proposed sliding mode control strategies for AC/DC power converters, several computer simulations have been conducted using MATLAB/Simulink software. Parameters of the simulated three-phase AC/DC converter are listed in Table 18.7. All of the reported results were obtained under the assumption that ideal sliding modes are enforced in the systems. Figure 18.30 shows the DC output voltage using the proposed SMC algorithm. As it can be seen, the output voltage is maintained at the desired level with zero steady state error. \(v_{dc\_ref}=250\) V in this case,
DC-link output voltage of the AC/DC power converter
Results of the current tracking control system are shown in Fig. 18.31. After the transient stage, phase currents track references very well. This proves that the proposed sliding mode control strategy ensures that the input currents are free of higher order harmonics.
Three-phase current tracking using the proposed SMC strategy
Input phase voltage and phase current
Figure 18.32 shows the source voltage and current. It illustrates the ability of the proposed control algorithm to reduce the reactive power to zero and maintain a unity power factor.
Figure 18.33 illustrates the ability of the input current and output voltage control systems under the proposed strategy to quickly follow a sudden change in the dc link voltage reference (\(v_{dc\_ref1}=300\) V, \(v_{dc\_ref2}=250\) V). In this case, K should be varied such that the equilibrium point is equal to desired value \({\mathrm {u}}_{\mathrm {cref}}\). For this purpose, K is selected as an integral function of the mismatch \(v_{dc\_ref}-v_{dc}\).
System response to a sudden change of the dc link voltage and input current
18.4 Sliding Mode Control of DC/AC Power Electronic Inverters
PWM originally refers to an approach used to realize or amplify a continuous function by using on/off implementation. In power electronics industry, the term PWM also refers to the control of semiconductor switches in converter circuits. With the development of new control algorithms for power converters, several PWM techniques have been proposed and implemented in the control system of DC/AC inverters, like harmonic elimination PWM [5, 6] space vector PWM [17, 18, 41], sine-triangle PWM [28], hybrid PWM [3], hysteresis band control, etc.
PWM control approaches can be classified into two main categories: open-loop control using a pre-calculated switching pattern, such as harmonic elimination PWM, and close-loop control whose switching actions depend on feedback information, such as hysteresis band PWM approach. Some of them can be used in both open-loop and close-loop schemes, such as space vector PWM [41]. Comparative studies of different PWM techniques can be found in [31, 37], and other literature.
Performance of power electronic inverters is evaluated in different aspects, for instance harmonic loss factor, reference tracking ability and robustness against variation of circuit parameters and disturbances. Therefore, Sliding mode control (SMC) is very natural to be used in the control of power inverters [1, 38,39,40], since the switching between two discrete values can be used directly as gating signals to the semiconductor switching devices in power converter circuits. SMC has excellent reference tracking ability and robust against parameter variations of the inverter.
This section presents a new PWM approach for power electronic inverters called Sliding Mode PWM (SMPWM). It is a close-loop control method based on the sliding mode concept. The control objective is to track proper reference inputs using a three-phase full-bridge inverter. It considers the fact that three arbitrary phase current references cannot be tracked for three-phase three-wire systems.
18.4.1 Circuit Model and Design Methodology
Consider the system in Fig. 18.34. The three-phase full-bridge inverter under control is to provide desired currents to the load, taking into account that they can be dependent.
Three-phase PWM DC/AC inverter scheme
Based on circuit analysis, the system equations are:
where \(v_{n}\ \)is the voltage at neutral point. \(E_{a},\ E_{b},\ E_{c}\ \)are balanced three phase voltage sources which represent three back EMF in ac motors. \(s_{a},\ s_{b},\ s_{c}\in \left\{ \pm 1\right\} \) represent the on/off control signals for six switching devices of the three-phase full-bridge converter. If a switching device is conducting, value “1” is assigned to it, otherwise “−1” is assigned. Since sum of the currents is equal to zero, \(v_{n}\ \)of the three-phase system can be found as:
It looks natural that \(v_n\ \)is discontinuous, but its average value changes continuously.
18.4.2 Sliding Mode Pulse Width Modulation (SMPWM)
The objective of the inverter’s control system is to track proper selected voltage or current references. Some existing PWM techniques use three-phase current errors as inputs to its controller, for instance hysteresis band PWM. It controls the three phases independently based on corresponding current error. However, as it was mentioned, the three currents cannot be independent.
Denote
Substituting (18.157) into (18.156) results in:
where,
Notice that matrix \(D\ \)is singular, therefore among \(i_a,i_b,i_c\), only two phase currents can be controlled independently. If the sum of reference currents \(\sum {i^*_z}\ne ~0,\ \mathrm {where}\ z=a,b,c\), then problem can not be solved. If \(\sum {i^*_z}=0\), only two of the reference inputs can be given independently, then we have one superfluous degree of freedom since the dimension of control equals to three. As result, although three currents are equal to the desired values, the motion of system is not unique. For example the voltage \(v_n\) in (18.157) can be equal to different values. In future, we assume that reference inputs and are given and discuss how to utilize the additional degree of freedom.
It should be noted from (18.156) that, since the equation for each phase current depends on all three discontinuous functions \(s_a{,s} _b,s_c\), the frequency analysis cannot be performed independently. Thus, although hysteresis band PWM works in practical implementation, its analysis needs to be carefully checked. Control algorithm should be developed to control three-phase currents while only two of them can be controlled independently. The extra one degree of freedom can be used to control one more variable.
The proposed control algorithm is based on complementing the original system by the first order equation:
where \(v_n\ \)is defined in Eq. (18.157).
The sliding surfaces \(s={\left[ s_{a},s_{b},s_{3}\right] }^{T}\) can be defined as:
where \(i_{a}^{*},i_{b}^{*},v_{n}^{*}\) are reference signals. The derivative of vector s can be easily derived as:
Three-dimensional discontinuous control \(J\ \)can be designed such that sliding mode is enforced on \(s=0\). When sliding mode occurs on \(s=0\), all three components of are equal to zero and the tracking problem is solved. Since motion equation in sliding mode depends on parameter \(v^*_n\), it can be selected in correspondence with some performance criterion. Different control methods can be used to enforce sliding mode.
Having proposed sliding surface s, the proposed control algorithm should be designed such that vector s is reduced to zero after finite time. Select the Lyapunov function:
Its time derivative is given by:
where,
If \({u_{c}}/{L}\) is large enough, \(F\left( .\right) \) can be suppressed with control S given by:
The proposed SMPWM control design for AC/DC inverters has the following advantages:
-
1.
SMPWM with Lyapunov approach takes into account all sliding surfaces directly for each phase’s control.
-
2.
SMPWM does not require the dc voltage source to have constant value.
-
3.
SMPWM can tolerate significant amount of disturbances or fluctuations from voltage source. This is an attractive advantage of SMPWM.
18.4.3 SMPWM Using Sliding Surface Decoupling Approach
Although the SMPWM method proposed in Sect. 4.2 is simple, it does not provide the capability to analyze the motion in the vicinity of the sliding manifold, since the three equations in (18.162) are interconnected. Therefore, this section presents another SMPWM technique that takes into account all sliding surfaces indirectly for each phase’s switching control.
Equation (18.162) can be written in matrix form
where the vector \(f_2=\left[ \frac{di^{*}_{a}}{dt}-K_a,\frac{di^{*}_{b}}{dt} -K_b,v^{*}_{n}\right] \) includes terms without control variables, and \(B=\left[ \begin{array}{ccc} 2 &{} -1 &{} -1 \\ -1 &{} 2 &{} -1 \\ L &{} L &{} L \end{array} \right] \). Since B is a non-singular matrix, it can be transformed into a diagonal matrix, and control variables can be decoupled for each sliding surface. Introduce new switching manifold \(\overline{s}=B^{-1}s=0\). In this case,
Differentiation of (18.169) gives \(\dot{{\overline{s}}_1} =B^{-1}\dot{s}\) and \(\dot{s}\) can be as:
where \(f_3={\left[ f_{31},f_{32},f_{33}\right] }^T\) is a 3 * 1 vector whose elements have bounded values as:
\(f_{3}\ \)does not include control variables, \(s_{a},\ s_{b}\ \)and \(s_{c}\). Equation (18.170) shows that the overall motion can be decoupled into three individual ones (with respect to control). Select the control logic for inverter switches as follows:
If \(\frac{{\mathrm {U}}_{\mathrm {dc}}}{\mathrm {3L}}\mathrm {>sup}\left\| { \mathrm {f}}_{\mathrm {3}}\right\| \), the sliding mode existence conditions \({ \overline{\mathrm {s}}}_{\mathrm {1}}\dot{{\overline{\mathrm {s}}}_{\mathrm {1}}} \mathrm {<0}\), \({\overline{\mathrm {s}}}_{\mathrm {2}}\dot{{\overline{\mathrm {s} }}_{\mathrm {2}}}\mathrm {<0}\), \({\overline{\mathrm {s}}}_{\mathrm {3}}\dot{{ \overline{\mathrm {s}}}_{\mathrm {3}}}\mathrm {<0}\) hold. Sliding manifold is reached after finite time. Reference input \({\mathrm {v}}^{\mathrm {*}}_{ \mathrm {n}}\mathrm {\ }\)can be selected depending on some operation criteria: control of switching frequency, minimization of this frequency with given accuracy, and so on.
18.4.4 Sliding Mode Control of Neutral Point Voltage \(\varvec{(}{ \varvec{v}}_{\varvec{n}}\varvec{)}\)
This section explains the physical meaning of the proposed control methodology. As it follows from (18.162), the time derivative of \(s_3\ \)depends on varying at high frequency in sliding mode. It means high frequency will be rejected on sliding surface and will depend on average value of \(v_n\).
Sliding manifold of SMPWM
Section 4.3 shows that SMPWM can track reference currents and control the average value of \(v_{n}\) at the same time. As a result, proper \(v_{n}^{*}\ \)can be found to minimize some performance criterion. As an example, the switching frequency is to be minimized in order to reduce switching losses with a given accuracy. Ideally, sliding mode is a mathematical abstraction where the sliding motion trajectories are strictly on discontinuous surfaces. However, sliding mode in a real life system occurs not strictly on its discontinuous surfaces. Instead, it appears within some boundary layer. Analysis of sliding motion in boundary layer can be found in [40]. Assume that switching devices have hysteresis loop \(\varDelta \). Then \(\varDelta \) defines the accuracy of the system, as shown in Fig. 18.35. In our system, the switching frequency depends on state velocities. Moreover, the switching frequency is not a constant value. We consider the system using surface decoupling approach shown in Sect. 4.3. Then each phase can be handled independently. The switching frequency is determined by two time intervals with \(\dot{s}>0\) and \(\dot{s} <0 \). Consider the switching behavior of the i-th motion in (18.170) where \(i\in \{1,2,3\}\). For each motion in (18.170), the time duration of “switch on” and the time duration of “switch off” can be written as functions of \(v_{n}^{*}\).
The function \(f_{3}\) in (18.170) depends on \(v_{n}^{*} \), as shown in (18.171). For each motion in (18.170), the time duration of “switch on” and the time duration of “switch off” can be written as functions of \(v_{n}^{*}\)
It means that the switching frequency of the i-th motion is a function of \( v_{n}^{*}\) as well.
Current tracking using SMPWM
Taking into account all three motions in (18.170), the variable \(f_{switch\ }\)defined in (18.87) can be used as a measurement of the overall switching frequency of the system:
Let \({f_{switch\_i}(v}^*_n)\ \)be the function to be minimized under the constraint \(\frac{U_{dc}}{3L}>sup\left\| f_3\right\| \). Then, the optimal \( v^*_n\ \)can be found from (18.172) and (18.173), such that \({f_{switch\_i}(v}^*_n)\ \)is minimized. In this particular example, the performance criterion is a function of \(v^*_n\). In other problems, criterion could be a function of \( v_n \). Then let \(v^*_n\ \)equal to the optimal value of \(v_n\), and the tracking \(v_n=v^*_n\) is provided by SMPWM.
18.4.5 Simulation Results
In order to evaluate the proposed control algorithm, computer simulations have been conducted using MATLAB/SIMULAINK software. Parameters used in simulation are listed in Table 18.8. This simulation used sliding surface decoupling approach. Results of three-phase current tracking (zoomed-in) are shown in Fig. 18.35. After the transient stage, phase currents track the references very well. Discontinuous output voltages \(v_a,\ v_b,\ v_{c\ }\)are shown in Fig. 18.36. In Fig. 18.37, the average value of \(v_n\ \)tracks a time varying reference. This average value is obtained by a first order dynamic \(\mu \dot{ x}=-x+v_n\ \)and\(\ \mu =0.001\). This low pass filter is used to calculate the average value of \(v_n\ \)in order to illustrate how close is this value to the \(v^*_n\). Of course this filter is not needed for implementation of the system. In this simulation, the reference \(v^*_n\) is a randomly selected time varying function. It is not optimized according to some criterion. The only objective of Fig. 18.37 is to show the ability to track a time varying.
Three-phase output voltages using SMPWM
Tracking a time varying \(v^*_n\left( t\right) \) using SMPWM
It is worth mentioning that SMPWM (see Fig. 18.38) is suitable for both analog and digital implementations. This due to the fact that SMPWM can be implemented with or without timing functionality, while other PWM approaches like Space Vector PWM require the DSP controller to have the ability to handle timing [7, 41].
Notes
- 1.
Formally, conventional Singular Perturbation Theory is not applicable for differential equations with discontinuous right-hand side. But, if sliding mode appears, sliding mode equations are in compliance with the conditions of the theory.
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Alsmadi, Y.M., Utkin, V., Xu, L. (2018). Sliding Mode Control Design Procedure for Power Electronic Converters Used in Energy Conversion Systems. In: Clempner, J., Yu, W. (eds) New Perspectives and Applications of Modern Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-62464-8_18
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