* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Una propuesta de control por modo deslizante para convertidores
Electrical substation wikipedia , lookup
Stray voltage wikipedia , lookup
Power inverter wikipedia , lookup
Alternating current wikipedia , lookup
Voltage optimisation wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Amtrak's 25 Hz traction power system wikipedia , lookup
Voltage regulator wikipedia , lookup
Distributed control system wikipedia , lookup
Mains electricity wikipedia , lookup
Wassim Michael Haddad wikipedia , lookup
Integrating ADC wikipedia , lookup
Resilient control systems wikipedia , lookup
Two-port network wikipedia , lookup
Opto-isolator wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Variable-frequency drive wikipedia , lookup
PID controller wikipedia , lookup
Pulse-width modulation wikipedia , lookup
Control theory wikipedia , lookup
REVISTA INGENIERÍA UC. Vol. 11, No 2, 26-38, 2004 Una propuesta de control por modo deslizante para convertidores electrónicos de potencia Rubén Rojas, Alfredo R. Castellano, Ramón O. Cáceres, Oscar Camacho Postgrado en Automatización e Instrumentación. Av. Tulio Febres Cordero. Facultad de Ingeniería. Escuela de Ingeniería Eléctrica, ULA. Mérida – Venezuela Email: [email protected] Resumen Este artículo describe la síntesis de un controlador por modo deslizante basado en un modelo lineal reducido de segundo orden usando una superficie integro-diferencial de error, con salida aproximadamente continua. A diferencia de estrategias similares, los parámetros de sintonización guardan estrecha relación con la dinámica del sistema en términos de especificaciones convencionales de respuesta transitoria. El controlador propuesto requiere solamente la realimentación de la variable de salida del sistema y podría ser empleado satisfactoriamente en control de sistemas eléctricos no lineales de una entrada y una salida, tal como el caso de los convertidores electrónicos de potencia a PMW (modulación por ancho de pulsos). Palabras clave: Modo deslizante, control no lineal, convertidor de potencia, PWM. A sliding mode control proposal for electronic power converters Abstract This article describes the synthesis of a sliding mode controller SMCr based on a reduced second order linear model using an integral-differential surface of the tracking-error. Different from similar strategies, the tuning parameters keep a close relationship with the system dynamics in terms of conventional specifications of transient response. The proposed controller only needs the output feedback of system and could be satisfactorily used in control of single input-single output nonlinear electric systems such as electronic power converters with pulse-width modulation. Keywords: Sliding mode, nonlinear control, power converter, Pulse-width modulation 1. INTRODUCTION Numerous electrical systems exhibit a behavior whose dynamics can be represented by a single inputsingle output (SISO), second order model. However, such approach discards process inherent nonlinearities that sometimes would cause degradation of control system, such as when the conventional PID strategy is used. This becomes more evident when it is working in extreme conditions or in zones where the parametric uncertainty of the model becomes more accented. dynamics of the system in the sliding mode have stimulated development of multiple procedures for the synthesis of controllers in a wide spectrum of applications. Such as are the cases of processes with multiinput / multi-output configuration [1], with strong non linearities [2], with variable dead time [3] or non minimum phase [4]. In addition, excellent results have been reported in control of electric motors [5] and electronic power converters [6, 7] where discontinuous action in Variable Structure Control (VSC) is compliant with the nature of their elements. In this framework, robustness proper of the sliding mode control strategy (SMC) should provide better performance than conventional control strategies. Properties such as order reduction and invariant The main drawback in VSC is the chattering generally associated with a high control activity that sometimes could not be tolerated by the system. 26 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 Rojas, Castellano, Cáceres y Camacho It could excite high frequency unmodeled dynamics or decrease their efficiency [5, 8]. This last aspect highlights the convenience of synthesizing the SMCr under an approximately continuous control law [9]. Based on the SMC robustness, this article shows the synthesis of a controller based on an integral-differential surface of the error and a continuous approximation of nonlinear part of the controller. The design is pointed to obtain a simple structure that only needs the output feedback making it attractive for control of nonlinear electric systems such as power converters with schemes of pulse width modulation (PWM). In a sliding mode control strategy, the system dynamics is forced by the controller to stay confined in a subset of the state space denominated sliding surface, σ(t). The system is directed toward and reaches the sliding surface at a finite time due to the control action. Finally, once the system dynamics reach the user-chosen sliding surface, it will behave according to that surface, which is of a lower order than that of the system and independent of the model parametric uncertainties [10]. Let M be the second order model of system (lower order approximation of the real system Q), with an output variable θ1(t) that tracks the reference input θr(t) with a tracking error e1(t) under control law U(t) of the SMCr . The dynamics of the controlled system M can be described by the following state space equation: . θ 1 (t ) = θ 2 (t ) . (2) y (t ) = e1 (t ) here, f(t) contains the external disturbances m(t), the reference signal qr(t) and its derivatives. Whenever the external disturbance acts in the same state space of the control U, a control Uf will exist such that: bU f (t ) = f (t ) (3) and the sliding mode existence will be ensured if a known upper bound, the supreme F(t), for f(t) exists such that for any instant t: 2. SYSTEM MODELING θ 2 (t ) = −a1θ1 (t ) − a2θ 2 (t ) + m(t ) + bU (t ) y (t ) = θ1 (t ) . e1 ( t ) = e 2 ( t ) . e 2 (t ) = − a1e1 (t ) − a 2 e2 (t ) + f (t ) + bU (t ) (1) where a1, a2, b are the parameters of the second order model approximation of the system and m(t) represents the external disturbances. This equation can be rewritten in terms of the tracking error as follows: f (t ) < F (t ) (4) 3. CONTROLLER SYNTHESIS The desired performance of the system is usually described in terms of the transient response specifications. This performance must be satisfied in presence of disturbances and set point changes. The transient response specifications are thoroughly used in conjunction with PID control to regulate systems with dominant second order dynamics. This approach is used in this article to evaluate the system response. In general, the most used set of specifications for the transient response can be summarized as follows: S1. S2. S3. S4. Steady State Error, ess. Percent Overshoot, Mp. Peak Time, Tp. Settling Time, Ts. The design problem consists first on choosing the sliding surface so that the system exhibits the desired dynamics defined by the given specifications; and second to guarantee the conditions so that the system reaches that surface at a finite time. There are many options to choose s(t); the Sliding Surface selected in this work is an integral-differential equation acting on the tracking-error [11], which is represented by the following expression: Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 27 Una propuesta de control por modo deslizante ⎞ ⎛d + λ⎟ ⎠ ⎝ dt σ (t ) = ⎜ ∫ t (5) e(t )dt de (t ) + 2λ1λ0 e(t ) + λ0 dt (9) ∀ t ≥ t0 0 The advantage of this surface is that it contains an integral term, which ensures the cancellation of the steady state error. Then s(t) is a function of the error between the reference, qr, and the output, q1, values as was described in Eq. (2), such that: σ (t ) = σ& (t ) = e& 2 (t ) + 2 λ 1 λ 0 e 2 (t ) + λ 0 2 e1 (t ) = 0 2 t ∫ e (t )dt = 0 (6) 0 λ0 , λ1 > 0 represents the system dynamics in the sliding mode. The satisfaction of expression s(t) = 0 means that in steady state the tracking-error of the system should decrease to zero with a dynamics that depends on the selection of parameters l0 and l1. For all SISO systems [10], the condition for existence of a sliding mode is satisfied if: σ (t ) σ& (t ) < 0 (7) Geometrically, this inequality means that the time derivatives of the state error vector always point toward the sliding surface when system is in reaching mode, and therefore, the system dynamics will approach to the surface dynamics in a finite time. To satisfy the given specifications S1–S4 and Eq. (6), an augmented equivalent control law [1] was used: U ( t ) = U eq ( t ) + U N ( t ) ( ) As can be appreciated in Eq. (10), Ueq(t) evidences a proportional-derivative nature, attenuated by the static gain of the model. Finally, the chattering problem could be solved satisfactorily if the control UN(t) is designed according to Zinober [12]: U N (t ) = K d σ (t ) The continuous part Ueq(t) could be determined supposing that in the instant t0, the state trajectory of the system intercepts the surface and enters the sliding mode. The existence of sliding mode implies that Eq. (6) is satisfied, so: (11) ; Kd ,δ > 0 σ (t ) + δ where Kd is a tuning parameter to assure the reaching of the sliding surface and d is set to obtain suppression of chattering. An estimate of the Kd parameter could be obtained from Eq. (3) and Eq. (4). The overall control law structure is shown in Figure 1. d/dt + 2λ0 λ1 e ∫ dt + σ (t) Σ + Kd × ( ) abs( ) + δ UN + λ02 Σ U + (8) Ueq(t) is denominated the Equivalent Control, and represents the continuous part of the controller that maintains the output of the system restricted to the sliding surface. UN(t) is the nonlinear part of the controller that ensure to reach the sliding surface and therefore, it should satisfy the inequality given in Eq. (7). 28 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 Then the equivalent control Ueq(t) is obtained from Eq. (9), after substituting the error derivatives by the state equations (2), when the system is not disturbed (f (t) = 0): 1 2 Ueq ( t ) = ⎡ λ0 − a1 e1 ( t ) + ( 2 λ0 λ1 − a2 ) e2(t ) ⎤ (10) ⎣ ⎦ b d/dt 2λ0 λ1 – a2 + Σ 1/ b + Ueq λ02 – a1 Figure 1. Structure of the proposed SMCr. 4. CONTROLLER TUNING In sliding mode, system dynamics depends on the sliding surface σ(t), and therefore the desired transient response will depend on (10) and (11) parameters selection. Both parameters could be determined solving Eq. (6). Depending on values range in (11), it is possible to obtain three different solutions for the set of specifications: Rojas, Castellano, Cáceres y Camacho If 0 < λ1 < 1; λ0 > 0 4.1 In this case, the dynamics of the surface will present a behavior strongly underdamped and steadystate error will approach to zero after an oscillation around the equilibrium point (see Figure 2). The response of the system can be evaluated through the following equations: e ss = 0 Ts ≈ Mp = 13.53% (18) Tp = exp α ⎛ 2λ β ⎞ −⎜ 1 ⎟ ⎝ α ⎠ sin β 2 (19) λ0 5.4 Ts = (20) λ0 (12) 100 Tp = 2 (17) 1 (13) λ0 = 15 λ1 = 1 0.8 β λ0α Tracking error e(t) Mp = e ss = 0 (14) 4 (15) λ0 λ1 0.6 0.4 Mp = 13. 53 % Tp = 0. 13 s 0.2 Ts = 0. 36 s 0 Mp -0.2 α = 1 − λ12 ⎛α ⎞ ; β = tan −1 ⎜ ⎟ ⎝ λ1 ⎠ (16) -0.4 Ts Tp 0 0.1 0.2 0.3 0.4 0.5 0.6 Time t , (s) Figure 3. Dynamics of the sliding surface when λ1 = 1. Notice that the percent overshoot Mp only depends on λ1. 1 Tracking error e(t) In this case, the system response can be evaluated through the following equations: λ0 = 15 λ1 = 0.7 0.8 0.6 Mp = 21. 03 % 0.4 Tp = 0. 15 s 0 Mp -0.2 Ts Tp -0.4 0 0.1 0.2 0.3 0.4 0.5 e ss = 0 (21) −⎜ −1⎟ Ln ⎜ ⎟ 50 ⎡ ⎢( λ1 − α ) exp ⎝ α ⎠ ⎝ λ1 −α ⎠ Mp = α ⎢ ⎣ ⎛ λ ⎞ ⎛ λ +α ⎞ ⎤ − ⎜ 1 +1⎟ Ln ⎜ 1 ⎟ α λ −α − ( λ1 + α ) exp ⎝ ⎠ ⎝ 1 ⎠ ⎥ ⎥⎦ ⎛ λ1 Ts = 0. 33 s 0.2 If λ1> 1; λ0 > 0 4.3 0.6 Time t , (s) Figure 2. Dynamics of the sliding surface when 0 < λ1 < 1. ⎛ λ1 +α ⎞ (22) Tp = ⎛ λ +α ⎞ Ln ⎜ 1 ⎟ λ0α ⎝ λ1 − α ⎠ (23) Ts ≈ ⎛ 50 ( λ1 − α ) ⎞ Ln ⎜ ⎟ λ0 ( λ1 − α ) ⎝ α em % ⎠ (24) 4.2 If λ1= 1; λ0 > 0 In this case, the system response will present a moderate underdamped behavior (see Figure 3). Because Mp only depends on λ 1 , it is constant (13.53%), and the response of the system can be evaluated through the following equations: ⎞ 1 1 α = λ12 − 1 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 (25) 29 Una propuesta de control por modo deslizante where em% is the specified error band inside which Ts is specified. The response, Figure 4, exhibits a reduced overshoot but a larger settling time. response in terms of peak time or settling time. This tuning procedure is straightforward and it is as simpler as a PID controller tuning. Due to overdamping, the settling time will be severely influenced by the specified error band. Damping ratio associated with λ1 parameter is shown in Figure 5. Note that overshoot only could be reduced if λ1 >> 1. Another interesting aspect is the strong relationship between the controller parameters and the system natural response. In Eq. (1), the system approximated by the second order model should contain the dominant poles of Q, so that an alternative representation of M could be obtained in terms of natural frequency ωn and damping radio ξ from the original system at any operation point, such that: 1 λ0 = 15 λ1 = 1. 5 Tracking error e(t) 0.8 a1 = ω n 0.6 Mp = 7. 56 % Tp = 0. 12 s Ts = 0. 37 s 0.4 0.2 -0.2 -0.4 a 2 = 2ξω n Mp 0 Ts Tp 0 0.1 0.2 0.3 0.4 0.5 0.6 Time t , (s) Figure 4. Dynamics of the sliding surface when λ1> 1. Percent Overshoot Mp, (%) 14 12 10 8 Mp = 7. 56% 6 4 2 0 1 1.5 2 2.5 3 3.5 4 4.5 2 5 λ1 parameter Figure 5. Overshoot -λ1 relationship. (26) (27) From Eq. (10) it is evident that l0 is related with ωn and λ1 with ξ. As usual, the controlled system must respond faster than the open-loop system and therefore, it is desirable that λ0 > ω n. According to Eqs. (13), (18) and (22), the percent overshoot only depends on λ1, which provides a straight way to achieve the desired percent overshoot in system response. For instance, choosing λ1 > ξ is convenient if ξ < 0.707. Then, next step is to adjust λ0 to provide the desired response in terms of settling time or peak time, which are totally determined according to the λ1 range. A reasonable value for Kd could be determined assuming that in the meantime of an external disturbance or when step change is introduced, the sliding surface value s(t) >> d. Therefore, the nonlinear part of the controller UN approximates Kdsign(s(t)) and its magnitude will be Kd. In order to guarantee the reaching condition in any circumstance, from Eq. (3) and Eq. (4), it is enough that: Kd > 1 F (t ) b (28) As shown in the above results for the three cases, the relationship between the controller parameters and the transient response of the closeloop system is evident. This facilitates the controller tuning considerably. For instance, if a desired overshoot should be lower than 8 % , λ1 >1.5 would be a suitable election (see Figure 5). For systems with saturated single input, the superior estimate F(t), in Eq. (4), is given by the control input bounds: Once defined the percent overshoot Mp, the parameter λ0 is selected to satisfy the desired time because is impossible to allow, without loss of stability, a disturbance in the system that requires a 30 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 f ( t ) < b (U max − U min ) (29) Rojas, Castellano, Cáceres y Camacho 5. EXAMPLES In order to show the proposed controller performance and its parameter tuning simplicity, a couple of examples of SISO systems, increasing in complexity, are presented in this section, beginning with the original system state variables description and their inherent characteristic dynamics. When the order of the original system is higher (greater than two) a second order model approximation around the operation point is determined for tuning the controller. Finally, the proposed control strategy is evaluated by simulation using the original state variable description in MATLAB and the controller with the parameters set at the previous established tuning; in each case the resulting response is compared an contrasted with the predicted response given by the controller equations. 5.1 A boost converter In general, if energy conversion is performed in continuous conduction mode, the boost converter can be described in the average state space according to: diL ( t ) Vs 1 = − v0 ( t ) (1 − u ( t ) ) dt L L dv0 ( t ) 1 1 =− v0 ( t ) + iL ( t ) (1 − u ( t ) ) dt RC C d (s) =− R 2⎞ ⎛ ⎜ s − (1 − D ) ⎟ L ⎝ ⎠ 2 ⎛ 2 1− D) ( 1 s+ ⎜s + ⎜ RC LC ⎝ Vs RC (1 − D ) 2 Switch PWM L D’T 2 3 Vs 1 + DT where iL(t) is the inductor average current, v0(t) is the output voltage, R is the load in ohms, vs is the external dc source voltage and u(t) is the control input. This control input is assumed as saturated continuous function in the open interval (0, 1). In order to derive an approximate model of the converter, the electronic switch is replaced by the PWM switch [13], resulting in the circuit shown in Figure 6. Substituting the PWM switch by its fundamental frequency model, around a stationary point of operation D, the dynamics of system in Eq. (30) is described approximately as follows: + R C v0 – Figure 6. Equivalent circuit of the boost converter. The behavior of the open-loop system corresponding to reference step changes and load disturbances is shown in Figure 7. 21.5 25 21 20.5 (30) ⎞ (31) ⎟ ⎟ ⎠ where d(s) represents the incremental duty cycle. Equation (31) characterizes a non-minimum phase system with respect to the output voltage, and therefore, it is difficult to control. Output voltage v0 , (V) Finally, the parameter d (always positive) is adjusted to suppress the chattering. v0 ( s ) Output voltage v0 , (V) control magnitude larger than the range of values of U(t) for an indefinite period of time. 20 0.5 0.6 0.7 0.8 Time t , (s) 0.9 1 20 15 10 0 (a) 100 to 50Ω in t = 0. 5 s 0.5 50 to 100Ω in t = 1 s 1 Time t , (s) 1.5 (b) Figure 7. Open loop response of the converter against a: (a) Reference step change response (20 - 21V) (b) Step change in the load resistor (±50%R). To obtain the tuning parameters of the controller, the system in Eq. (31) is described in terms of the natural response of the approximate model M: ωn = (1 − D ) ξ LC = LC 2 RC (1 − D ) (32) where D is the steady state duty cycle. A parameter summary is shown in Table 1, according to the following circuit elements: inductor, 170 mH Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 31 Una propuesta de control por modo deslizante capacitor, 1000 µF; resistor, 100 Ω; and power supply voltage, 10 Vdc. The duty cycle D is computed for an output voltage of 20 V, (D = 0.5). Table 1. Model parameters M(s). Parameter Value b 5.882x104 a1 1.471x103 a2 10 ωn , natural frequency 38.35 ξ , damping ratio 0.130 Lower limits for l0 and l1 are immediately obtained from the natural response: λ0 > ω n = 38.35 λ1 > ξ = 0.130 (33) Considering the closed-loop gain restriction, that imposes an unstable zero dynamics of the system, is advisable to choose moderate gains, for instance, λ0 = 1.05 ωn or 1.1 ωn and then increase λ1 from a value equal to ξ until the desired response is achieved. The gain Kd is obtained from the control input bounds (0 < u < 1). Hence, to reach the sliding surface is enough to choose Kd = 1. In systems with high natural frequency, λ0 will be larger than λ1 and therefore, the instantaneous value of the surface s(t) could reach an unusual magnitude. Because d y s(t) have the same order of magnitude, for practical implementation purposes, the sliding surface s(t) can be normalized with respect to the natural frequency without degrading the UN behavior. 5.2 A Buck-boost converter The buck-boost DC-AC converter is an inverter of commuted mode that works in for quadrants. It is characterized by its capacity to generate sinusoidal voltages that can be greater or lower than the DC supply voltage, depending on the duty cycle. Its main advantage is that it does not need a second power conversion stage [8]. The voltage control on this inverter topology is 32 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 a troublesome task due to the inherent nonlinear behavior of the system and the sensitivity to abrupt load changes that eventually could drive it to instability, particularly when classical PID control methods are used [13]. Control is even more difficult because of the non minimum phase characteristic of the system. The design and experimentation of a boost type similar converter show very satisfactory experimental results following a sliding mode control strategy with a frequency modulation scheme [8]. To reach the control goal, nevertheless, two controllers (one for each branch of the inverter) and four state variables feedback (two voltages and two currents for each branch), were needed. In the proposed control strategy, the control input is a continuous signal to be used beside a PWM auxiliary circuit with a fix commutation frequency. The DC-AC conversion is obtained using two DC-DC bidirectional buck-boost converters, modulating the output voltage in a sinusoidal way in counter-phase as it is shown by the basic electrical scheme of Figure 8. Each converter generates a unipolar sinusoidal voltage with a DC component. This unipolar component of the voltage DC is suppressed by the differential connection of the load. S1 Vs + S3 L1 u PWM S2 L2 C1 – C2 v0 + S4 R Figure 8. The Buck-boost DC-AC converter. The average state variable representation of the converter is shown in Figure 9 according with the following relationships [14]: diL1 1 ⎛ Vs − vC1 ⎞ = vC1 + ⎜ ⎟u dt L ⎝ L ⎠ dvC1 1 ⎛ vC 2 − vC1 ⎞ i = ⎜ − i L1 ⎟ + L1 u dt C⎝ R ⎠ C (34) Rojas, Castellano, Cáceres y Camacho diL 2 1 ⎛ Vs − vC 2 ⎞ = Vs + ⎜ ⎟u dt L L ⎠ ⎝ (35) dvC 2 1 ⎛ vC1 − vC 2 ⎞ iL 2 = ⎜ ⎟− u dt C⎝ R ⎠ C v0 = vC 2 − vC1 + Va where iL1 and iL2 are the average currents in L1 and L2 inductors, respectively, vC1 are vC2 the average voltages through C1 and C2, respectively. Vs is the DC external voltage supply, R is the load connected to the converter, v0 is the average voltage on the load, and u is the average control input. – X u 0<u<1 X • 1 L1 Σ + Σ – + 1 C1 vC1 + X ( ) ∫ • 1 L2 Σ iL2 iL2 ∫ v0 + X X + • 1 C2 Σ – vC2 vC2 ∫ After replacing each pair of commuters S1-S4, S2-S3 by the PWM switch to the equivalent diagram of the inverter is obtained as it is shown in Figure 10. During the time interval, DT, energy is stored on the L1, inductor, simultaneously the storage energy L2 is transferred to the RC net. Similarly, during the following interval, D’T, energy is transferred from the L1 inductor to the RC net while energy is stored on L2 inductor. a 2 1 D’T 3 Vs – C1 L1 v0 b + vC1 – D’T 1 2 DT R + + + Vs 3 Ib 1 Vs 1− D (36) 1 Vs D (37) Vb = − 2D−1 Vs 2 D(1− D) R Ib = − Figure 9. Average state variable model of the converter. DT – + DVb Ia Ia = – DVa 3 Va = vC1 Σ 1 Vs 2 DIb The following equations are obtained from the steady state model: iL1 ∫ – + R iL1 • – X – + + – 1 R DIa + Vb + Figure 11. Buck-boost converter steady state model. + + X 1 V0 Σ + – – – 2 3 + vC2 – + Vs C2 L2 (38) 2D −1 Vs D2 (1− D) R (39) V0 2D − 1 = Vs D (1 − D ) (40) Equation (40) defines the nonlinear static gain of the converter shown in Figure 12. It can be observed that the duty cycle is constrained to the open interval (0,1) and the output voltage is zero when D = 0.5. 4 Voltage Static Gain, v0 / Vs Vs Assuming converter operates on the DC conduction mode during all the commutation period T and this commutation occurs on a sufficiently high frequency, the average model of the PWM switch is used to obtain the steady state model (see, Figure 11). 2 0 -2 -4 0 0.2 0.4 0.6 0.8 1 Duty Cycle, D Figure 10. Buck-boost converter equivalent diagram. Figure 12. Static gain characteristic. Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 33 Una propuesta de control por modo deslizante Converter fundamental frequency model is obtained replacing the PWM switch by its equivalent around steady state (Figure 13). The converter control input is obtained by adding the incremental variation d and the steady state duty cycle D. + va – 2 – v0 D ia + 1 D ib R – + D va d Ia 1 – + d Va ia 3 + vb – d Ib + + vC1 vC2 – – 2 – D vb + d Vb – + ib L 3 L ( z1 = −L D2 Ia + (1 − D) Ib 2 ) P2 = RLC(1−2D+2D2 ) z 2 = LC ( (1 − D ) Va − DVb ) p3 = 2 L2 C z 3 = − L2 C ( I a + I b ) p 4 = R ( LC ) 2 Figure 14 show the root locus of Q(s) around an operation point defined in V0 = 200 V (D=0.707) with L = 1 mH, C = 10 µF, R = 60 ohm, Vs = 100 Vdc. From Eq. (44) it can be observed that the system presents right half-plane zeros that causes a nonminimum phase response respect to the output voltage. Figure 13. Buck-boost converter fundamental frequency model. 1.5 x j 104 Analyzing the above model in the frequency domain the following equations are obtained: v0 ( s ) = v a ( s ) + vb ( s ) (41) ⎛ (1 − D ) Va − I a L s ⎞ va ( s ) = d ( s ) ⎜ ⎟ ⎜ (1 − D )2 + LC s 2 ⎟ ⎝ ⎠ Imaginary Axis 1 0.5 0 -0.5 -1 (42) -1.5 -0.5 0 0.5 1 1.5 2 ⎛ ⎞ Ls −v0 ( s ) ⎜ ⎟ 2 ⎜ R (1 − D ) + LC s 2 ⎟ ⎝ ⎠ (43) ⎛ ⎞ Ls ⎟ −v0 ( s ) ⎜ ⎜ R ( D 2 + LC s 2 ) ⎟ ⎝ ⎠ The non linear characteristic of the static gain, Eq. (40), causes the open loop output voltage distortion during a sinusoidal reference tracking as it is shown in Figure 15. 300 Vref: 250 Vpk R: 60 ohm Equations from (41) to (43) represent the open loop converter dynamics around the operation point D. The transfer function Q(s) = vo(s)/d(s), derived from Figure 13 model [12], is given by: z3 s 3 + z 2 s 2 + z1 s + z 0 p 4 s 4 + p3 s 3 + p 2 s 2 + p1 s + p0 (44) Output Voltage v0, (V) 200 where: 3 x 10 4 Figure 14. Q(s) root locus. ⎛ DV + I L s ⎞ vb ( s ) = − d ( s ) ⎜ 2 b b 2 ⎟ ⎝ D + LC s ⎠ Q (s) = k 2.5 Real Axis 100 0 -100 -200 -300 0 10 20 30 40 50 Simulation time t , (ms) k=R p 0 = R (1 − D ) D 2 Figure 15. Static gain effect on the average output voltage. z0 = D (1 − D) ( DVa − (1− D)Vb ) p1 = L (1 − 2 D + 2 D 2 ) Figures, 16 and 17, show the converter response against abrupt changes in load. Due to energy 2 34 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 Rojas, Castellano, Cáceres y Camacho storage on the inductor and filters, the converter experiments severe variations in the loop gain when abrupt decrements in load occur. On the other hand, it is less sensitive to overloads. 400 Output Voltage v0, (V) Vref: 200 Vpk 200 -200 0 10 20 30 40 50 Simulation time t , (ms) b=k z0 2 ωn p0 (47) Vref: 200 Vpk 200 Table 2 Converter and system Q(s) parameters. Converter Design Parameters Figure 16. Step load change converter response (from 60 to 6000 ohm). Output Voltage v0, (V) (46) ξ where Re(p) and Im(p) are the dominant pole, p, real and imaginary part magnitudes, respectively, k, z0, p0 are the original system Q(s) coefficients. The converter design parameters, Q(s) coefficients and the parameters given by the root locus, are summarized in Table 2. 0 -400 Re ( p ) ωn = D L1, L2 C1, C2 R Vs 0.5 1 mH 10 µF 60 W 100 Vdc System Q(s) coefficients for Vo = 200Vdc (D=0.707) 100 k 0 z3 x10-11 z2x10-6 z1 x 10-3 z0 -6. 67 2.00 -5.29 58.58 p1x10 p0 60 -100 p4x10 -200 -15 p3x10 6.00 0 10 20 30 40 3.51 -4 5.86 2.57 Table 3 Model M(s) parameters. Figure 17. Step load change converter response (from 60 to 42 ohm). M(s) model parameters obtain by root locus 5.2.1 Controller tuning The control aim is a 60 Hz sinusoidal reference tracking to generate output voltages from 0 to 200 volts approximately, using a supply source of 100 Vdc. As it was described in the controller synthesis procedure, first of all the system is approximated by a second order model that contains the dominant poles of Q(s). The model parameters static gain, b, natural frequency, ωn, and damping coefficient, ξ , can be estimated using linear systems simplification techniques such as the root locus as follows: ⎞ ⎟ ⎟⎟ ⎠ 2.00 p2x10 -7 50 Simulation Time t , (ms) 2 ⎛ ⎛ Im ( p ) ⎞ ⎜ ξ = 1 + ⎜⎜ ⎟ ⎜⎜ Re ( p ) ⎟⎠ ⎝ ⎝ -11 −1 (45) Re (p) 895 Im (p) 2 898 ξ 0. 295 ωn 3 033 K 1. 256x1010 a1 9. 199x106 a2 1. 789x103 Figure 18 shows the frequency response of the original system Q(s) and the second order approximation M(s). Using the second order model parameters, the tuning parameters range for l0 and l1 are defined as: λ0 > ω n = 3.033 (48) λ1 > ξ = 0.295 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 35 Una propuesta de control por modo deslizante even better than the predicted by the tuning equations. The ITAE performance index, Figure 8 (b), becomes constant confirming the annulment of steady state error. Q(s) -50 100 M(s) 101 102 103 104 21.5 105 106 0 M(s) Mp = 12.93 % Tp = 0.092 s Ts = 0.186 s ess = 0 20 Q(s) 30 Ts 20.5 -360 ess = 0 ess 21 0.5 -720 0.6 0.7 0.8 Time t , (s) 0.9 20 10 0.5 1 0.6 (a) 10 2 10 3 10 4 10 5 10 6 Frequency (rad/s) Figure 18. Open loop systems frequency responses. Considering the non-minimum phase characteristic (that constrains the system direct loop gain) and the calculated natural frequency, the procedure to follow is to choose initial moderated gains as l0 = 1.1 ω n (or 1.2 ω n ) . Then, increase progressively the l1 parameter from a base value, equal to ξ , up to reach another value that guaranties a satisfactory reference tracking. The gain of the controller nonlinear part, Kd, is obtained from the control input signal bounds (in this case: 0 < u < 1). The sufficiency condition to reach the sliding surface is satisfied if: As in the case of the buck converter, the controller must include a DC component Ubias = 0.5 to cancel the output voltage. 6. RESULTS 6.1 Boost converter Figure 19, shows the system step response when a reference step change is introduce at t = 0.5 s. It shows a small overshoot and a short settling time 36 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 1 A good system response behavior to load disturbances is shown in Fig. 20 (a), when the output voltage is set to 20 V. The load disturbances are simulated by step changes of ±50% of the nominal load. Figure 20 (b) shows that the steady state error is annulled in similar times. 25 0.3 ess= 0 20 50 to 100Ω in t = 1 s 15 0 0.5 1 Time t , (s) 0.2 ess= 0 0.1 100 to 50Ω in t = 0. 5 s ess= 0 10 0 1.5 0 (a) 0.5 1 Time t , (s) 1.5 (b) Figure 20. Attenuation of load disturbances, step changes ± 50% nominal load. ITAE performance index. 1 0.8 0.6 0.4 0.2 0 Using the average description according to Eq. (30), the system was simulated using MATLAB with step time equal to 1x10–4 s. The controller synthesis was performed with the following parameters: l0 = 42.18 (1.1 ωn); l1 = 0.417 (3.2x); Kd = 1; d = 1.5; the error surface was normalized with respect to a factor of 10 ωn. 0.9 Figure 19. System step response, from 20 up to 21 V, in t = 0.5 s. ITAE performance index. (49) Kd ≥ 1 0.7 0.8 Time t , (s) (b) ITAE 10 1 Inductor current iL , (A) 10 0 Output voltage v0, (V) Phase φ , (degrees) Frequency (rad/s) -3 40 x 10 Mp ; Tp ITAE 0 Output voltage v0, (V) 50 Control u (normalized) Gain |G(jω)|, (dB) 100 0 0.5 1 Time t , (s) (a) 1.5 1 0.8 0.6 0.4 0.2 0 0 0.5 1 Time t , (s) 1.5 (b) Figure 21. Load step change response, control input magnitude (normalized), average inductor current. Figure 21 shows the control signal and the average inductor current when the load change was simulated. The results are very similar to those obtained using a SMC adaptive strategy (Escobar et al., 1997) evaluated on experimental prototype of the boost converter. Rojas, Castellano, Cáceres y Camacho 6.2 Buck-boost converter 250 vref 200 0. 109 × 10–6 d/dt + 0. 321 × 10–3 + e σN Σ + + Ubias + 0. 5 Σ + d/dt 0. 154 × Output Voltage v0, (V) 0 -50 -100 -150 -250 0 10 20 30 40 50 Simulation Time t , (ms) 40 Vref: 200 Vpk 30 20 10 0 -10 -20 -40 u Σ Ueq + 50 -30 + 0. 093 × 10–6 100 -200 UN 1. 0 × ( ) abs ( ) + 0. 35 ∫ 1. 210 v0 150 Tracking Error e (t), (V) Using the average model, Figure 9, the system was simulated using MATLAB with a sampling period of 5 ms, following 60 Hz sinusoidal references of amplitude 150, 200 and 250 V. The controller realization is shown in Figure 22 with the tuning parameters set as follows: l0 = 3.336 (1.1ωn); l1 = 2 0.443 (1.5 ξ ); Kd = 1; d = 0.35; sN = s/ ωn (sliding 2 surface normalized respect to ωn ). Figure 23, 24 and 25 shows the system tracking response for a 200 V amplitude, 60 Hz sinusoidal reference signal against abrupt load and DC voltage changes at t =21 ms, when the system is more sensitive to disturbances (on the top of the sinusoidal wave approximately). 0 10 20 30 40 50 Simulation time t , (ms) Figure 24. System tracking response for a 60 Hz sinusoidal, reference signal of amplitude 200 V, when abrupt load was applied. (a) Step load change from 60 to 42 ohm. (b) Instantaneous tracking error. 10–3 Figure 22. Controller realization for the buck-boost converter. 300 vref v0 Output Voltage v0, (V) 200 100 Output Voltage v0, (V) 250 0 200 vref 150 v0 100 50 0 -50 -100 -150 -200 -250 -100 0 10 20 30 40 50 Simulation time t , (ms) -200 0 10 20 30 40 50 40 Simulation time t , (ms) Vref: 200 Vpk 30 40 Vref: 200 Vpk 20 Tracking Error e (t), (V) 30 Tracking Error e (t), (V) 20 10 0 -10 -20 -30 -40 0 -10 -20 -30 -40 -50 -60 10 0 10 20 30 40 50 0 10 20 30 40 50 Simulation time t , (ms) Simulation time t , (ms) Figure 23. System tracking response for a 60 Hz sinusoidal, reference signal of amplitude 200 V, when abrupt load was applied. (a) Step load change from 60 to 6000 ohm. (b) Instantaneous tracking error. Figure 25. System tracking response for a 60 Hz sinusoidal, reference signal of amplitude 200 V, when abrupt DC voltage changes were applied. (a) Step DC voltage change from 100 to 90 V. (b) Instantaneous tracking error. Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 37 Una propuesta de control por modo deslizante 7. CONCLUSIONS The designed SMC controller, based on an integral-differential surface of the tracking-error, performed very well when reference and load step changes were introduced, with zero steady state error and without chattering. Its main attribute is the close relationship between the controller tuning parameters and the desired closed-loop transient response. The possibility to obtain the desired percent overshoot unilaterally (adjusting l1) allows achieving demanding agreements between percent overshoot and settling time. Since the control law has been synthesized from a second order model of the system, a simple structure controller structure is obtained, and it requires only an output feedback loop. Just one parameter (d) in the controller structure is subject to the designer selection. Once a satisfactory Kd/d relationship has been determined, the sliding surface s(t) can be normalized with respect to the natural frequency, without degrading the UN behavior because it only depends on the magnitude and sign of the sliding surface s (t) with respect to the numeric value of d. This aspect is very important for practical implementation of the SMCr in analog schemes. REFERENCES [1] DeCarlo R., Zak S. and Matthews G. (1988). “Variable structure control of nonlinear multivariable systems”. A tutorial. In: Proceedings of the IEEE, Vol. 76, Nº 3, pp. 212232. [2] Camacho O. (1996). “A new approach to design and tune sliding mode controllers for chemical processes”. Doctoral Dissertation, University of South Florida, Tampa, Florida. [3] Camacho O. and Smith C. (1997). “Application of sliding mode control to a nonlinear chemical processes with variable dead time”. Proceedings of II Congreso Colombiano de Automática (ACA), Bucaramanga, Colombia, pp. 122-128. [4] Camacho O., Rojas R. and García W. (1999). “Variable structure control applied to chemical processes with inverse response”. ISA Transactions (38), pp. 55-72. 38 Rev. INGENIERÍA UC. Vol. 11, No 2, Agosto 2004 [5] Utkin, V. J. (1993). “Sliding mode control design principles and applications to electric drives”. IEEE Transactions on Industrial Electronics, Vol. 40, Nº 1, pp. 23-36. [6] Rojas R., Camacho O., Cáceres O., Castellano A. (2002) “A Sliding Mode Control Proposal for Nonlinear Electrical Systems” 15th IFAC World Congress, IFAC 2002, Barcelona, España. [7] Cáceres R. and Barbi I. (1999). “A boost DC-AC converter: Analysis, design and experimentation”. IEEE Transactions on Power Electronics. Vol. 14. No. 1, pp.134 - 141. [8] Escobar G., Ortega R., Sira-Ramirez H., Vilain J.P. and Zein I. (1997). “An experimental comparison of several non linear controllers for power converters”. IEEE International Symposium on Industrial Electronics, Portugal, pp. SS89-SS94. [9] Rojas R., Camacho O., Cáceres O., Castellano A. (2004) “On Sliding Mode Control for Nonlinear Electrical Systems” 3rd WSEAS Int. Conf. on APPLICATIONS OF ELECTRICAL ENGINEERING (AEE '04), Cancún, México. [10] Utkin V. J. (1977). “Variable structure systems with sliding modes”. IEEE Transactions on Automatic Control, Vol. AC-22, pp. 212-222. [11] Slotine J. J. and Li W. (1991). “Applied Nonlinear Control”. Prentice Hall, New Jersey. [12] Zinober, A.S. (1994). “Variable Structure and Liapunov Control”. Spring-Verlag. London. [13] Tymerski R., Vorpérian V., Lee F. and Baumann W. (1988). “Nonlinear modelling of the PWM switch”. IEEE Power Electronics Specialist Conference, Apr., pp. 968-976. [14] Cáceres R., Rojas R. and Camacho Q. (2000) “Robust PID Control of a Buck-Boost DC – AC Converter”. IEEE International Telecommunications Energy Conference. INTELEC-2000, pp. 180 – 185.