A Robust Sliding Mode and PI-Based Tracking Control for the MIMO “DC/DC Buck Converter–Inverter–DC Motor” System

This work addresses the control problem related to the bidirectional velocity of a DC motor fed by a DC/DC Buck converter–inverter as power amplifier. While various power electronic topologies have been proposed in the literature, this research focuses on one that was previously disregarded due to its delivery of pulse width modulated voltage signals instead of smooth voltage signals to the DC motor. However, in this paper it is demonstrated that by appropriately controlling the power electronic converter it is indeed possible to deliver the desired smooth voltage signals to the motor. Moreover, the proposed control strategy is robust, employing multiple proportional-integral loops, and a sliding mode controller. A formal proof of asymptotic stability is provided and the robustness is verified through experimental validation.


I. INTRODUCTION
PWM, or pulse width modulation, is currently the prevailing method employed for providing power to closed-loop controlled electromechanical systems.This technique allows to provide power just by switching power transistors and it tries to exploit the fact that the power dissipated by the transistor is very small because it ideally operates only at saturation (maximal current and zero voltage) or at cutoff (maximal voltage and zero current).Nevertheless, the process of switching transistors in a hard manner places significant strain on the electrical subsystem of the plant due The associate editor coordinating the review of this manuscript and approving it for publication was Jiann-Jong Chen .
to the sudden fluctuations related to electric currents and voltages [1].
To address the aforementioned issue, DC/DC power electronic converters as power amplifiers has been suggested [2].These converters have the capability to generate voltages and electric currents smooth enough due to the presence of capacitors and inductors within their design.By employing these converters, the noise commonly generated by the abrupt switching in PWM-based power amplifiers is reduced.
This constraint prompted further investigation, wherein a full H bridge inverter was introduced connecting the DC/DC power electronic converter and the DC motor.Thus, for these new connections, the topology utilizing a DC/DC Buck converter was studied in [35], [36], [37], and [38].Similarly, the configuration involving a DC/DC Boost converter was presented in [39], [40], and [41].Furthermore, in [42], [43], [44], and [45], the connection employing a DC/DC Buck-Boost converter was utilized.And, in [46] the DC/DC Sepic converter-inverter-DC motor system was exposed.Although it was demonstrated theoretically and experimentally that bipolar voltage can be delivered to DC motor, the main drawback in [35], [39], [42], and [46] is that this voltage is, however, still a PWM signal.As a result of this drawback, research was redirected towards a different power converter topology where the DC/DC part of the power converter is placed between the full H bridge inverter and DC motor (see the series of works [47], [48], [49]).This topology has been so successful that it has been applied to control permanent magnet syncnronous motors (PMSM) [50], induction motors (IM) [51], and DC motor-actuated wheeled mobile robots (WMR) [52].Lastly, alongside the state-of-the-art in DC motors driven by DC/DC converters, interesting papers have recently been published addressing the control of DC/DC converters and the control of DC motors as separate problems.In this direction, on the one hand, contributions related to the Buck converter have been analyzed in [53], [54], and [55], while new control schemes for the Boost and Buck-Boost topologies were presented in [56], [57], [58], [59], [60], and [61], respectively.On the other hand, regarding DC motor control new designs have recently been introduced in [62], [63], [64], and [65].
In the present paper, the converter topology introduced in [35] is considered.Theoretical and experimental analyses reveal that the proposed topology in [35] enables the delivery of smooth voltage and electric current signals, as opposed to their pulse width modulation (PWM) counterparts.This capability is achieved through effective control strategies implemented in the topology.The control strategy presented in this study, designed to address the aforementioned issue, consists of multiple proportional-integral (PI) loops and a sliding mode loop specifically designed to regulate the electric current passing through the converter inductor.As demonstrated through experimental validation, this approach yields a robust control scheme that effectively handles both uncertainties in plant parameters and external disturbances.Although this research only prove asymptotic stability at local level of the equilibrium of interest, the merit of this result is simplicity of the developed control scheme despite the complexity of the plant.In this respect, it is well known in the control community that the control problem complicates in nonlinear systems when considering additional electrical dynamics (i.e., the electronic power converter) between controller and DC motor.Moreover, stability proof of simple PI loops is also complicated in nonlinear electromechanical control systems.See [66] for a recent solution to such a control problem.As will be shown in the next section, plant under study in the present paper is nonlinear because of inverter.The aforementioned features outlined in this paragraph constitute the primary contributions of this study.
Finally, a few comments regarding notation.Let h ∈ R n be a vector, where ∥h∥ 1 = n i=1 |h i | is the 1-norm, and | • | is the absolute value function.Also, ∥h∥ = n i=1 h 2 i corresponds to the Euclidean norm.Given a symmetric matrix A of n × n, all of its eigenvalues are real and λ min (A) stands for its minimum eigenvalue.

II. DYNAMIC MODEL
Fig. 1 depicts the MIMO ''DC/DC Buck converter-inverter-DC motor'' system.The following dynamic behavior was found in [35] by applying Kirchhoff's Laws to Fig. 1 when modeling transistors and diode by means of ideal switches: Variable i is the electric current passing through the inductance L of the converter, while υ is the voltage across the output capacitance C of the converter (this is, the voltage provided to the input terminals of the inverter).The output terminals of the inverter are connected to the DC motor's terminals.The voltage at the DC motor terminals is denoted as ϑ = υu 2 .The electric current passing through the DC motor armature and the velocity of the motor shaft are denoted as i a and ω, respectively.Variables u 1 and u 2 are the two system inputs and represent the positions of switches that model transistor Q 1 , diode and transistors Q 2 , respectively.Variable ū2 , that models transistors Q2 , is the complement of u 2 .Variable u 1 only takes the discrete values 0 or 1, whereas variable u 2 only takes the discrete values −1 or 1. See [35] for further details.Notice that mathematical model in (1)-( 4) is nonlinear because of products i a u 2 and υu 2 .
The constants E, R, R a , L a , k e , k m , J , and B, are all positive and represent the power supply voltage, a fixed resistance at the output of the converter, the resistance and the inductance, both of the armature, the motor back electromotive force, the motor torque, motor inertia, and the viscous friction coefficient, respectively.Additionally, T L represents an unvarying and unidentified load torque applied to the motor shaft.

III. CONTROL OF THE MIMO ''DC/DC-BUCK CONVERTER-INVERTER-DC MOTOR'' SYSTEM
The main result is outlined in the following proposition.Proposition 3.1: Consider the dynamics of the MIMO ''DC/DC Buck converter-inverter-DC motor'' system (1)-( 4) with the following controller in closed-loop: where functions sign(s) and zat(υ) are defined in ( 9) and Definition 2.1, for M = 1, respectively.The desired velocity, denoted as ω d (t), is a rest-to-rest time-varying function such that ωd (t) is bounded.Moreover, ω d (t i ) = ω di at some t = t i ≥ 0 for some constant ω di and ω d (t) = ω d for some finite t f such that t ≥ t f > t i and ω d is another finite constant.We assume that |ω d | > 0 for all t ≥ 0, excepting, perhaps, some isolated points of time.
To achieve asymptotic stability within the closed-loop system, values k p1 , k i1 , k p2 , k i2 , f , r a , and γ greater than zero must exist and need to satisfy the following condition: 119398 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Remark 3.1: According to Section II, u 2 represents commutation of transistors Q 2 and such a commutation is ideally modeled as a discontinuous sign function.Hence, u 2 is restricted to assume exclusively discrete values −1 and +1.On the other hand, in real power electronic devices the actual commutation represented by variable u 2 is likely to behave as the function zat( ῡ) when M = 1 and β 0 → 0, i.e., when α 1 → 0 in Fig. 2(b).
Thus, the intention to define u 2 = zat(υ), in Proposition 3.1, as a continuously differentiable function is fourfold: a) it accurately represents a variable taking only two discrete values −1 and +1, b) it accurately represents the actual commutation that is present in practical power electronic devices, c) the derivative of u 2 is well defined and this allows the mathematical procedure presented below in the proof of Proposition 3.1, and d) for practical purposes, i.e., in experiments, u 2 = sign(υ) can be implemented as a variable taking only the discrete values −1 and +1 using the sign(•) function defined in (9).
Remark 3.2: The controller in Proposition 3.1, i.e., ( 10)-( 14), contains four main loops.See, for instance, block ''Bidirectional tracking robust control'' in Fig. 3. a) There is a sliding modes controller in the most internal loop, which aims to regulate the electric current i to achieve the desired profile i * of the electric current passing through the converter inductor.b) A linear proportional-integral (PI) controller computes the desired electric current i * based on the error between the converter output voltage v and its desired profile v. c) Another linear PI controller, augmented with terms involving velocity error and its integral, generates the desired converter output voltage v based on the error related to the motor's armature electric current.d) The desired electric current īa , associated with motor's armature, is derived by integrating the velocity error.This control strategy incorporates proportional and integral actions, fundamental components commonly used in industrial DC motor control, on 1) the armature electric current error and 2) velocity error.The utilization of sliding modes control for managing the electric current passing through the converter inductor, along with the integration of proportional and integral actions to address the converter output voltage error, enhances the anticipated robustness of the closed-loop system.The experimental verification of robustness is presented in Section IV.

A. REACHING THE SLIDING SURFACE
Considering the positive definite scalar and radially unbounded function V (s) = 1 2 s 2 and its first derivative with respect to time along the trajectories of (1), yields: where ( 1) and ( 10) have been used, if By taking into account the two alternatives 16) implies (15).It is stressed that (15) can always be satisfied in practice by selecting E accordingly to the task to be performed.With the sliding condition ṡ = 0, the dynamics (1), and condition (15); it is found that the following bound is satisfied by the equivalent control: which indicates that the system is capable of reaching the sliding regime.Additionally, equation ( 16) guarantees the achievement of the sliding surface s = i − i * = 0, meaning that i = i * is attained.Therefore, the focus is on analyzing that dynamics (2)-( 4) be stable when the closed-loop system, governed by ( 11)- (14), is evaluated at i = i * .

B. CLOSED-LOOP DYNAMICS ON THE SLIDING SURFACE
Using i = i * , u 2 = zat(υ), and ( 11) in ( 2), introducing and subtracting the terms i a zat(υ), C d dt [υzat(υ)], and 1 k m (Bω d + T L )zat(υ), and using the expression for i a from equation ( 13), the constants can be redefined as k p2 = k p /k m and k i2 = k i /k m .This leads to the following modified equations: where: and ( 5), ( 6), M = 1 and the chain rule have been employed.After adding and subtracting terms υ, k e ω d , L a di a dt in (3), using (13), , and defining r = R a + r a , e a = ρ + σ it is possible to write: After adding and subtracting the terms J ωd , k m i a , Bω d in (4), using i a defined in (13), , and e a = ρ + σ the following can be written: with ξ defined in (19).Thus, the dynamics of the closed-loop system on the sliding surface s = 0 can be described by equations ( 17)-( 21), and the state variables of such dynamics can be represented as y s = [ ω, ξ, σ, ρ, z 2 , z 1 , e, ζ ] T .

C. STABILITY ANALYSIS ON THE SLIDING SURFACE
The scalar function: is positive definite and radially unbounded if and, according to (18), ( 5), and (6): Consequently, after performing some cancelations and considering that ±qw ≤ |q| |w|, ∀q, w ∈ R, ∥h∥ 1 ≤ √ n∥h∥ 2 , ∀h ∈ R n , and |zat( ῡ)| ≤ M = 1, it is observed that the first derivative of W with respect to time, previously given, along the trajectories associated with the closed-loop system over the sliding surface s = 0, i.e., equations ( 17)-( 21), can be bounded as: Ẇ ≤ −y T Qy + ∥y∥ |x| + (δC|e| x is a scalar and bounded function of ωd and is equal to zero when ωd = 0, whereas Q represents a symmetric matrix of dimension 8 × 8 whose entries are defined as: The positivity of the eight principal minors of matrix Q are achieved through the following choices of parameters.The corresponding first principal minor can be made positive by selecting α > 0 sufficiently small due to the positivity of B. Whereas, the second principal minor can be made positive by choosing a large enough k i > 0. To make the third principal minor positive, r > 0 must be chosen sufficiently large and β > 0 must be chosen small enough.Similarly, the fourth principal minor can be made positive by selecting a large enough value r > 0 and a small enough value p > 0. For the fifth and sixth principal minors, choosing a sufficiently large γ > 0 with any f 1 > 0 is sufficient.The seventh principal minor remains consistently positive when a sufficiently large k p1 > 0 is selected, while the eighth principal minor can be rendered positive by choosing k i1 > 0 large enough and δ > 0 small enough.Therefore, guaranteeing λ min (Q) > 0 is consistently achievable.
On the other hand, note that the terms: in ( 28) might complicate the stability analysis when | ῡ| < β 0 + µ.However, from the definition of ῡ and īa in ( 13) and e a and ω in ( 14), it can be concluded that | ῡ| ≥ β 0 + µ will become true for all t > t T for some finite t T ≥ 0, if |ω d | > 0 is large enough to demand a steady state value for ῡ such that | ῡ| ≥ β 0 + µ.Thus, it can be assumed that ( 27) is the only case to be analyzed which, for some 0 < < 1, can be written as: D. PROOF OF PROPOSITION 3.1 Note that functions W j (•, •), for j = 1, . . ., 4, given in (22), are quadratic forms.Hence, there exist two class K ∞ functions α 1 (∥y s ∥), α 2 (∥y s ∥), that satisfy α 1 (∥y s ∥) ≤ W (y s ) ≤ α 2 (∥y s ∥).This and (31), after using the Theorem 4.18 in [67] (pp.172), indicate the boundedness and convergence of y s ∈ R 8 to a ball, the size of which relies on the upper limit of |x|, a scalar function of time.Furthermore, since ω d (t f ) = ω d for all t ≥ t f , where t f > 0 is a finite value, means that ω * d (t) = 0 and ωd (t) = 0 for all t ≥ t f .Consequently, |x(t)| = 0 for all t ≥ t f , as mentioned before equation ( 29).This guarantees that y s → 0 as t approaches infinity.Thus, the proof of proposition 3.1 is complete.
Remark 3.3: It is important to emphasize that despite the equalities k p2 = k p /k m and k i2 = k i /k m introduced earlier in equation ( 17), it is not necessary to have precise knowledge of the exact value of k m .However, k p and k i must satisfy the conditions of stability previously mentioned; yet, this does not demand a precise value for k m .Therefore, both expressions k p2 = k p /k m and k i2 = k i /k m can be satisfied by utilizing sufficiently large values for k p2 , k i2 , and the other gains of the controller.Furthermore, regarding the first equality in equation ( 13), it is stated that the value of the armature resistance R a needs to be precisely known.Nonetheless, this necessity can be relaxed by selecting a large enough controller gain r a such that r a ≫ R a , meaning that r ≈ r a .Analogously, the requirement for exact knowledge of the Buck converter output resistance R, as stated in equation ( 11), can be eased by choosing a large controller gain k p1 .Moreover, as discussed in Remark 3.2, this control is expected to exhibit robustness, allowing for compensation of uncertainties in both R a and R via the PI controllers, which operate based on the error in converter output voltage and the electric current of the motor armature.Experimental results presented in Section IV provide confirmation of these observations.Remark 3.4: Although it may initially appear cumbersome to verify the positivity of all the principal minors of matrix Q (as shown in equation ( 29)), it must be noted that the process is actually straightforward.The positiveness of each principal minor of Q is verified by the diagonal elements of Q.Therefore, ensuring the positivity of the (i − 1)-th principal minor (keeping in mind that the first principal minor, Q 11 , is a scalar) and selecting a sufficiently large value for the controller gain associated with Q ii can render Q ii positive.This procedure is iterated until the eighth principal minor is also established as positive.

IV. EXPERIMENTAL RESULTS
Here, experimental tests are conducted to provide an understanding of the performance achievable with the proposed approach presented in Proposition 3.1.These experiments were conducted using a MIMO ''DC/DC Buck converterinverter-DC motor'' system, which was constructed at the Laboratorio de Mecatrónica y Energía Renovable at CIDETEC-IPN, México.
Fig. 3 depicts a block diagram showing relationship among hardware and software components employed to perform experimental tests.This block diagram is made up of three parts which are detailed in the following.
• MIMO ''DC/DC Buck converter-inverter-DC motor'' system.This part is composed by the plant to be controlled, i.e., the MIMO ''DC/DC Buck converterinverter-DC motor'' system.The measure of currents i and i a are realized through Tektronix A622 current probes.Voltage signals υ and ϑ are acquired by using two Tektronix P5200A voltage probes.Velocity ω measurements are performed using an Omron E6B2-CWZ6C encoder.A permanent magnet brushed DC motor model GNM 5440E-G3.1 from Engel was employed.The nominal parameters of the MIMO system prototype under study are the following: • Bidirectional tracking robust control.Control strategy presented in Proposition 3.1 is implemented in this part using Matlab-Simulink.The desired velocity trajectory ω d is also computed here.
• Controller board and signal conditioning.Here, the connections between the prototype and the DS1104 board are shown.Control signal ū1 is computed in this block.This signal is applied to experimental prototype, using the DS1104 board digital I/O signals, at the NTE3087 in order to turn-on and turn-off transistor of the Buck converter.The control signal u 2 is processed in the ''signal rescaling'' block (interprets u 2 = −1 as u 2 = 0) and through a NOT logic gate the control signal ū2 is obtained.This pair of signals (i.e., u 2 and ū2 ) are then applied to circuits TLP250 and drivers IR2113 in order to turn-on and turn-off transistors of the full bridge inverter.On the other hand, i a , υ, i, and θ are processed by the signal conditioning (SC) blocks.Finally, within this block the velocity ω is also computed by numeric differentiation of angular position, θ, which is obtained directly from the E6B2-CWZ6C encoder.Fig. 4 illustrates the hardware configuration of the MIMO ''DC/DC Buck converter-inverter-DC motor'' system when operated in closed-loop control.The system includes the following components: 1) power supply E, 2) personal computer, 3) DS1104 board from dSPACE, 4) Tektronix P5200A voltage probe, 5) Tektronix A622 current probe, 6) DC/DC-Buck converter, 7) full-bridge inverter, 8) encoder, and 9) Engel GNM 5440E-G3.1 DC motor.The controller gains employed for ( 10)-( 14) are k p1 = 29, k i1 = 2, k p2 = 0.8326, k i2 = 9.1590, f = 1, r a = 0.5, and γ = 50.The proposed desired velocity is described as follows: where ϕ 1 t, t i 1 , t f 1 and ϕ 2 t, t i 2 , t f 2 are Bézier polynomials defined as: and . This ensures that ω d smoothly interpolates from the initial velocity ω i 1 to the final velocity ω f 1 in the time interval [t i 1 , t f 1 ].After this, ω d smoothly interpolates from the initial velocity ω i 2 and the final velocity ω f 2 in the time interval [t i 2 , t f 2 ].All initial and final times are defined for each experiment that follows.

A. EXPERIMENT NUMBER 1: NOMINAL VALUES
In the first experimental test the trajectory tracking capabilities of the developed control scheme is shown, when the system nominal values (32) are considered.With this aim, the following parameters were considered t i 1 = 0 s, t f 1 = 1.5 s, t i 2 = 8 s, t f 2 = 13 s, ω i 1 = 0 rad s , ω f 1 = 13 rad s , ω i 2 = 13 rad s , and ω f 2 = −13 rad s in order to define the desired trajectory in (33).These results are plotted in Figs. 5 and 6.Note that the actual velocity ω tracks very well the desired velocity ω d both signals overlap all the time.Note that voltage at motor terminals ϑ is a smooth signal.This is important to highlight because it verifies that the main purpose to construct and control the MIMO ''DC/DC Buck converterinverter-DC motor'' system has been accomplished: a PWM voltage is not applied at motor terminals but a voltage signal whose magnitude smoothly varies to allow motor to reach the time varying desired velocity.Recall that ϑ = υu 2 .Hence, a smooth ϑ is ensured by a smooth voltage at the converter capacitor υ (excepting at υ = 0) and a piece wise continuous signal u 2 .This property on u 2 might be lost if motor velocity is constant at zero without an applied load torque.
It is observed that the SMC enforces the electric current i to achieve, in an average sense, its desired value i * .However, the sliding mode is lost (see plot of u 1 ) at t = 10 s, i.e., when ω = ω d = 0.This happens because the sliding condition in (15) is not satisfied, i.e., υ + L di * dt = 0 at t = 10 s.Moreover, recall that, in paragraph after (30), it is stated that some performance problems might appear if |ω d | > 0 is not large enough, which is particularly true if ω d = 0. Finally, it is also observed that i a → īa .

B. EXPERIMENT NUMBER 2: TORQUE DISTURBANCE
Here, the same parameters as in Experiment number 1 to define trajectory in (33) were used.Also, a torque disturbance T L is considered which is applied by means of a brake system acting on motor which is applied in the time interval t ∈ [5 s, 16 s].These results are shown in Figs.7 and 8.It can be seen that velocity tracking is very good since the actual velocity overlaps the desired velocity all the time.Note that measured velocity becomes a little noisy when torque disturbance is applied.Also, note that all voltages and electric currents are affected when torque disturbance appears.However, voltage ϑ at motor terminals remains a smooth signal instead of a PWM signal.Finally, the sliding condition in ( 15) is met again, except for around t = 10 s.

C. EXPERIMENT NUMBER 3: FASTER CHANGES IN MOTOR MOVEMENT
The intention of the third experiment is to command a faster change in the desired velocity.This is performed when commanding a reversal change in the desired velocity.With this aim, the following parameters are considered t i 1 = 0 s, t f 1 = 1.5 s, t i 2 = 10 s, t f 2 = 11.5 s, ω i 1 = 0 rad s , ω f 1 = 13 rad s , ω i 2 = 13 rad s , and ω f 2 = −13 rad s in order to define the desired trajectory in (33).The obtained results are shown in Figs. 9 and 10.Note that the actual velocity overlaps the desired velocity all the time.Hence, tracking performance is very good again.It is realized that all of the other variables behave similarly as in the previous experiments.The important thing in the third experiment is that the sliding mode is not lost.This can be seen in the plot of u 1 and in the plot of υ + L di * dt which is always greater than 0 and less than 1.This means that the sliding condition in ( 15) is always satisfied.(32) which are the known values for the controller.All of these experiments have employed the same parameters as in Experiment number 1 to define the desired velocity in (33).
One key highlight of the results presented in Figs.11 and 12 is that considering the value of R ∞ in practice implies disconnecting R from the Buck converter.Based on the results in Fig. 11, it can be concluded that including or excluding R from the system under study is inconsequential.This is because, in both cases, the control objective is achieved (i.e., ω → ω d ).
It can be observed in Figs.[11][12][13][14][15][16], that the main effects of all these parametric uncertainties appear on the electric    current through the converter inductance and its corresponding desired profile.This means that all of these parametric uncertainties are compensated by the sliding modes part of the controller.This explains why motor velocity is not affected (aside from some noisy measurements from time to time) by these uncertainties in these experiments.

V. CONCLUSION
In this paper, a controller is proposed for the MIMO ''DC/DC Buck converter-inverter-DC motor'' system.Such a system was previously proposed but abandoned due to the inability to deliver a smooth voltage signal at the DC motor terminals, instead of a pulse width modulation (PWM) signal.The study demonstrates that by appropriately designing the controller, a smooth voltage signal can indeed be achieved at the DC motor terminals.A formal proof of the asymptotic stability related to the closed-loop system is provided.The proposed controller incorporates multiple proportionalintegral (PI) loops for velocity control, armature current control, and converter capacitor voltage control.Additionally, a sliding modes controller is employed for controlling the current through the converter inductor.Through experimental validation on a prototype system, the robustness against plant parameter uncertainty and external disturbances is verified for the proposed control scheme.

and f 1
= p β k m are satisfied.Note that, based on Definition 2.1:

FIGURE 3 .
FIGURE 3. Block diagram of the closed-loop system.

FIGURE 4 .
FIGURE 4. Picture of prototype used in experiments.

FIGURE 6 .
FIGURE 6. Experiment number 1: Switched control signals, sliding condition, and voltage at the DC motor terminals when nominal values (32) are considered.

FIGURE 7 .
FIGURE 7.Experiment number 2: Closed-loop system variables when a torque disturbance is applied.

FIGURE 8 .
FIGURE 8. Experiment number 2: Switched control signals, sliding condition, and voltage at DC motor terminals when a torque disturbance is applied.

FIGURE 9 .
FIGURE 9. Experiment number 3: Closed-loop system variables when a faster reference change ω d is considered.

FIGURE 10 .
FIGURE 10.Experiment number 3: Switched control signals, sliding condition, and voltage at DC motor terminals, when a faster reference change ω d is considered.

FIGURE 11 .
FIGURE 11.Parametric uncertainty in the converter resistance: Closed-loop system variables.

FIGURE 12 .
FIGURE 12. Parametric uncertainty in the converter resistance: Switched control signals, sliding condition, and voltage at DC motor terminals.

FIGURE 13 .
FIGURE 13.Parametric uncertainty in the power supply voltage: Closed-loop system variables.

FIGURE 14 .
FIGURE 14. Parametric uncertainty in the power supply voltage: Switched control signals, sliding condition, and voltage at DC motor terminals.

FIGURE 15 .
FIGURE 15.Parametric uncertainty in the converter capacitance: Closed-loop system variables.

FIGURE 16 .
FIGURE 16.Parametric uncertainty in the converter capacitance: Switched control signals, sliding condition, and voltage at DC motor terminals.
Figs. 11-16, present the experimental results when parametric uncertainties exist in R, E, C, alone, respectively.The parametric uncertainties that appear in each one of these figures are introduced according to the following.Symbols R p , E p , and C p , represent, respectively, the actual values that the converter resistance, the power supply voltage, and the converter capacitance take, whereas R, E, and C are the nominal values presented in D. EXPERIMENTS UNDER PARAMETRIC UNCERTAINTIES IN R, E , AND C