Analysis of Closed-Loop Performance and Frequency-Domain Design of Compensating Filters for Sliding Mode Control Systems Igor Boiko Honeywell and University of Calgary, 2500 University Dr NW, Calgary, Alberta, T2N 1N4, Canada i.boiko@ieee.org Introduction It is known that the presence of parasitic dynamics in a sliding mode (SM) system causes high-frequency vibrations (oscillations) or chattering [1]-[3] There are a number of papers devoted to chattering analysis and reduction [4]-[10] Chattering has been viewed as the only manifestation of the parasitic dynamics presence in a SM system The averaged motions in the SM system have always been considered the same as the motions in the so-called reduced-order model [11] The reduced-order model is obtained from the original equations of the system under the assumption of the ideal SM in the system Under this assumption, the averaged control in the reduced-order model becomes the equivalent control [11] This approach is well known and a few techniques are developed in details However, the practice of the SM control systems design shows that the real SM system cannot ensure ideal disturbance rejection Therefore, if the difference between the real SM and the ideal SM is attributed to the presence of parasitic dynamics in the former then the parasitic dynamics must affect the averaged motions Yet, the effect of the parasitic dynamics on the closed-loop performance can be discovered only if a non-reduced-order model of averaged motions is used Besides improving the accuracy, the non-reduced-order model would provide the capability of accounting for the effects of non-ideal disturbance rejection and nonideal input tracking The development of the non-reduced order model becomes feasible owing to the locus of a perturbed relay system (LPRS) method [12] that involves the concept of the so-called equivalent gain of the relay, which describes the propagation of the averaged motions through the system with self-excited oscillations Further, the problems of designing a predetermined frequency of chattering and of the closed-loop performance enhancement may be posed The solution of those two problems may have a significant practical impact, as it is chattering that prevents the SM principle from a wider practical use, and it is performance deterioration not accounted for during the design that creates the situation of “higher expectations” from the SM principle In the present chapter, analysis of chattering and of the closed-loop performance is carried out in the frequency domain via the LPRS method Further, G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 51–70, 2008 c Springer-Verlag Berlin Heidelberg 2008 springerlink.com 52 I Boiko Fig Relay feedback system it is proposed that the effects of the non-ideal closed-loop performance can be mitigated via the introduction of a linear compensator The methodology of the compensating filter design is given The chapter is organized as follows At first, the approach to the analysis of averaged motions in a SM control system is outlined, and the concept of the equivalent gain of the relay with respect to the averaged signals is introduced After that the basics of the LPRS method are given In the following section, the non-reduced-order model of the averaged motions is presented Then, a motivating example illustrating the problem is considered In the following section, the compensation mechanism is analyzed Finally, an illustrative example of the compensating filter design is given Averaged Motions in a Sliding Mode System It is known that the SM control is essentially a relay control with respect to the sliding variable Therefore, the SM system can be analyzed as a relay system If no parasitic dynamics and switching imperfections were present in the system the ideal SM would occur It would feature infinite frequency of switching of the relay and infinitely small amplitude of the oscillations at the system output However, the inevitable presence of parasitic dynamics (in the form of the actuator and sensor dynamics) in series with the plant and switching imperfections of the relay result in the finite frequency of switching and finite amplitude of the oscillations, which is usually referred to as chattering Chattering was the subject of analysis of a number of publications [1]-[4], [11] and was analyzed as a periodic motion caused by the presence of fast actuator dynamics To obtain the model of averaged motions in a SM control system that has parasitic dynamics and switching imperfections, let us analyze the relay feedback system under a constant load (disturbance) or a constant input applied (Fig.1) At first obtain the model that relates the averaged values of the variables when a constant input is applied to the system Later we can extend the results obtained for the constant input (and constant averaged values) to the case of slow inputs Let the SM system be described by the following equations that would comprise both: the principal dynamics and the parasitic dynamics, which are assumed to be type servo system (not having integrators): x = Ax + Bu ˙ y = Cx (1) Analysis of Closed-Loop Performance for Sliding Mode Control Systems 53 c if σ(t) = f0 − y(t) ≥ b or σ(t) > −b, u(t−) = c −c if σ(t) = f0 − y(t) ≤ −b or σ(t) < b, u(t−) = −c (2) u(t) = where A ∈ Rn×n , B ∈ Rn×1 , C ∈ R1×n are matrices, A is nonsingular, c and b are the amplitude and the hysteresis value of the relay nonlinearity respectively, f0 is a constant input, u ∈ R1 is the control, x ∈ Rn is the state vector, y ∈ R1 is the system output, σ ∈ R1 is the sliding variable (error signal), u(t− ) is the control at time instant immediately preceding time t Let us call the part of the system described by equations (1) the linear part Alternatively the linear part can be given by the transfer function Wl (s) = C(Is − A)−1 B The parasitic dynamics can be present in both: the linear part (1) or/and in the nonlinearity (2) as a non-zero hysteresis value Assume that: (A) In the autonomous mode (f (t) ≡ 0) the system exhibits a symmetric periodic motion (B) In the case of a constant input f (t) ≡ f0 , the switches of the relay become unequally-spaced and the oscillations of the output y(t ) and of the error signal σ(t) become asymmetric (Fig 2) The degree of asymmetry depends on f0 (a pulse-width modulation effect); (C) All external signals f (t) applied to the system are slow in comparison with the self-excited oscillations (chattering) We shall consider as comparatively slow signals the signals that meet the following condition: the external input can be considered constant on the period of the oscillation without significant loss of accuracy of the oscillation estimation In spite of this being not a rigorous definition, it outlines the framework of the subsequent analysis First limit our analysis to the case of the input being a constant value f (t) ≡ f0 Under that assumption each signal has a periodic and a constant term: u(t) = u0 + up (t), y(t) = y0 + yp (t), σ(t) = σ + σ p (t), where subscript ”0” refers to the constant term (the mean value on the period), and subscript ”p” refers to the periodic term (having zero mean) If we quasi-statically vary the input (slowly enough, so that the input value can be considered constant on the period of the oscillation – as per assumption C) from a certain negative value to a positive value and measure the values of the constant term of the control u0 (mean control) and the constant term of the error signal σ (mean error) we can determine the constant term of the control signal as a function of the constant term of the error signal: u0 = u0 (σ ) Two examples of this function are given in Fig (for the first-order plus dead time plant; and are those functions for two different values of dead time) Let us call this function the bias function It is worth noting that despite the fact that the original nonlinearity is a discontinuous function the bias function is smooth and even close to the linear function in a relatively large range The described effect is known as the chatter smoothing phenomenon [13] The derivative of this function (mean control) with respect to the mean error taken in the point σ = (corresponding to zero constant input) provides the so-called equivalent gain of the relay kn [13] The equivalent gain concept can be used for building the model of the SM system for averaged values of the variables kn = du0 /dσ |σ =0 = lim (u0 /σ ) f0 →0 (3) 54 I Boiko Fig Asymmetric oscillations at unequally spaced switches Fig Bias functions (negative part is symmetric) Once the equivalent gain kn is found via analysis of the relay system having constant input f (t) ≡ f0 , we can extend the equivalent gain concept to the case of slow inputs (as per assumption C) Therefore, the main point of the inputoutput analysis is finding the equivalent gain value, and all subsequent analysis of the forced motions can be carried out exactly like for a linear system with the relay replaced by the equivalent gain The concept of the equivalent gain is used within the describing function method Yet, only approximate value of the equivalent gain can be obtained due to the approximate nature of the method The LPRS method [12] can provide an exact value of this characteristic The basics of the LPRS method are presented below The Locus of a Perturbed Relay System Let us consider at first the solution of this problem via the describing function (DF) analysis [14] The DF of the relay function and the equivalent gain can be given as follows: Analysis of Closed-Loop Performance for Sliding Mode Control Systems N (a) = kn(DF ) = 4c πa ∂u0 ∂σ b a 1− = σ0 =0 2c πa −j 4cb , πa2 1− b a 55 (4) , (5) where a is the amplitude of the symmetric oscillations The oscillations in the relay feedback system can be found from the harmonic balance equation: Wl (jΩ) = −1/N (a), which can be transformed into the following form via the replacement of N (a) with expression (4), accounting for (5) and using the switching condition y(DF ) (0) = −b, where t = is the time of the relay switch from “−c” to “+c”: 1 π Wl (jΩ) = − + j y(DF ) (t) (6) kn(DF ) 4c t=0 It follows from (4)-(6) that the frequency of the oscillations and the equivalent gain in the system (1), (2) can be varied by changing the hysteresis value 2b of the relay Therefore, the following two mappings can be considered: M1 : b → Ω, M2 : b → kn Assume that M has an inverse mapping (it follows from (4)-(6) for the DF analysis and is proved below via deriving an analytical formula of −1 that mapping) M1 : Ω → b Applying the chain rule consider the mapping −1 : Ω → b → kn Now let us define a certain function J exactly as the M2 M1 expression in the right-hand side of formula (6) but require from this function that the values of the equivalent gain and the output at zero time should be exact −1 : ω → b → kn , ω ∈ [0; ∞), in which we values Applying mapping M2 M1 treat frequency ω as an independent parameter, write the following definition of this function: 1 π y(t)|t=0 J(ω) = − +j (7) kn 4c −1 −1 where kn = M2 M1 (ω) , y(t)|t=0 = M1 (ω) Thus, J(ω) comprises the two mappings and is defined as a characteristic of the response of the linear part to the unequally spaced pulse input u(t) subject to f0 → as the frequency ω is varied The real part of J(ω) contains information about gain kn , and the imaginary part of J(ω) comprises the condition of the switching of the relay and, consequently, contains information about the frequency of the oscillations If we derive the function that satisfies the above requirements we will be able to obtain the exact values of the frequency of the oscillations and of the equivalent gain Let us call function J(ω) defined above as well as its plot on the complex plane (with the frequency ω varied) the locus of a perturbed relay system (LPRS) The word “perturbed” refers to an infinitesimally small perturbation f0 applied to the system Assuming that the LPRS of a given system is available we can determine the frequency of the oscillations and the equivalent gain kn as illustrated in Fig.4 The point of intersection of the LPRS and of the straight line, which lies at the distance πb/(4c) below (if b>0) or above (if b < 0) the horizontal axis and is 56 I Boiko Fig The LPRS and oscillations analysis parallel to it (line “−πb/4c“) offers computing the frequency of the oscillations and the equivalent gain kn of the relay According to (7), the frequency Ω of the oscillations can be computed via solving equation: ImJ(Ω) = −πb/(4c) (8) (i.e y(0) = −b is the condition of the relay switch) and the gain kn can be computed as: kn = −1/(2ReJ(Ω)) (9) that is a result of the definition of the LPRS The LPRS, therefore, can be seen as a characteristic similar to the frequency response W (jω) (Nyquist plot) of the plant with the difference that the Nyquist plot is a response to the harmonic signal but the LPRS is a response to the square wave Clearly, the methodology of analysis of the periodic motions with the use of the LPRS is similar to the one with the use of the DF In such an analysis, the LPRS serves the same purpose as the Nyquist plot, and the line “−πb/4c“ serves the same purpose as the negative reciprocal of the DF (−N −1 (a)) Obviously, formula (7) cannot be used for the outlined analysis It is a definition In [12], a formula of the LPRS that involves only parameters of the linear part was derived as follows: J(ω) = −0.5C[A−1 + 2π (I − e ω A )−1 e ω A ]B+ ωπ π +j π C(I + e ω A )−1 (I − e ω A )A−1 B 2π π (10) Let us derive another formula of J(ω) for the case of the linear part given by a transfer function, which will have the format of infinite series and be instrumental at analyzing some effects in the SM systems Suppose, the system is a type Analysis of Closed-Loop Performance for Sliding Mode Control Systems 57 servo system (the transfer function of the linear part does not have integrators) Write the Fourier series expansion of the signal u(t ) depicted in Fig 2: u(t) = u0 + 4c π ∞ sin(πkθ1 /(θ1 + θ2 ))/k× k=1 ×{cos(kωθ1 /2) cos(kωt) + sin(kωθ1 /2) sin(kωt)} where u0 = c(θ1 − θ2 )/(θ1 + θ2 ), and ω = 2π/(θ1 + θ2 ) Therefore, y(t) as a response of the linear part with the transfer function W l (s) can be written as: y(t) = y0 + 4c π ∞ sin(πkθ1 /(θ1 + θ2 ))/k × {cos(kωθ1 /2) cos[kωt + ϕ1 (kω)]+ k=1 + sin(kωθ1 /2) sin[kωt + ϕ1 (kω)]} Al (ωk) (11) where ϕl (kω) = argWl (jkω), Al (kω) = |Wl (jkω)|, y0 = u0 |Wl (j0)| The conditions of the switches of the relay can be written as: f0 − y(0) = b f0 − y(θ1 ) = −b (12) where y(0) and y(θ1 ) can be obtained from (11) if we set t = and t = θ respectively: y(0) = y0 + ∞ 4c π [0.5 sin(2πkθ1 /(θ1 + θ2 ))Re Wl (jkω) k=1 (13) +sin2 (πkθ1 /(θ1 + θ2 ))Im Wl (jkω)]/k 4c π y(θ1 ) = y0 + ∞ [0.5 sin(2πkθ1 /(θ1 + θ ))Re Wl (jkω) k=1 (14) −sin (πkθ1 /(θ1 + θ2 ))Im Wl (jkω)]/k Differentiating (12) with respect to f0 (and taking into account (13) and (14)) we obtain the formulas containing the derivatives in the point θ = θ2 = θ = π/ω: c|Wl (0)| 2θ dθ1 df0 2c − θ2 c|Wl (0)| 2θ − dθ1 df0 dθ1 df0 2c + θ2 dθ2 df0 − dθ1 df0 + + dθ2 df0 dθ2 df0 + + dθ2 df0 dθ1 df0 c θ ∞ k=1 c θ − dθ2 df0 ∞ cos(πk)ReWl (ω k )− k=1 (ω sin (πk/2) dImWlk k ) dω dθ1 df0 − dθ2 df0 −1=0 (15) ∞ cos(πk)ReWl (ω k ) k=1 ∞ (ω sin2 (πk/2) dImWlk k ) dω k=1 −1=0 (16) where ω k = πk/θ Having solved the set of equations (15), (16) for d(θ1 − θ2 )/df0 and d(θ + θ2 )/df0 we obtain: 58 I Boiko d(θ1 +θ ) df0 f0 =0 =0 which corresponds to the derivative of the period of the oscillations, and: d(θ − θ ) df0 2θ = f0 =0 c |Wl (0)| + ∞ (17) cos(πk) ReWl (ω k ) k=1 Considering the formula of the closed-loop system transfer function we can write: 2θkn d(θ − θ2 ) (18) = df0 c (1 + kn |Wl (0)|) f0 =0 Solving together equations (17) and (18) for kn we obtain the following expression: (19) kn = ∞ (−1)k Re Wl (kπ/θ) k=1 Taking into account formula (19) and the definition of the LPRS (7) we obtain the final form of the expression for Re J(ω) Similarly, having solved the set of equations (12) where θ1 = θ2 = θ and y(0) and y(θ1 ) have the form (13) and (14) respectively, we obtain the final formula of Im J(ω) Having put the real and the imaginary parts together, we can obtain the final formula of the LPRS J(ω) for type servo systems: ∞ ∞ (−1)k+1 Re Wl (kω) + j J(ω) = k=1 k=1 Im Wl [(2k − 1)ω] 2k − (20) It is easy to show that series (20) converges for every strictly proper transfer function (see Theorem below) The LPRS J(ω) offers solutions of the periodic and input-output problems for a relay feedback system – as illustrated by Fig 4, and does not require involvement of the filtering hypothesis Let us formulate that as the following theorem, the proof of which is given as the formulas above Theorem For the existence of a periodic motion of frequency Ω with infinitesimally small asymmetry of switching in u(t) caused by infinitesimally small constant input f0 in the relay feedback system (1), (2), it is necessary that equation (8) should hold The ratio of infinitesimally small averaged control u0 to infinitesimally small averaged error σ (the equivalent gain of the relay) will be determined by formula (9) The LPRS in (8) and (9) can be computed per (20) Analysis of Chattering Usually two main types of chattering are considered in continuous-time SM control The first one is the high-frequency oscillations due to the existence of the discontinuity in the system loop along with the presence of parasitic dynamics Analysis of Closed-Loop Performance for Sliding Mode Control Systems 59 or switching imperfections, and the other one is due to the effect of an external noise In the latter case chattering will not be a periodic motion, as the source of chattering is an external signal, which is not periodic However, the first type of chattering is usually considered the main one as being an inherent feature of SM control systems It is perceived as a motion, which oscillates about the sliding manifold [1] In the present chapter only this type of chattering is considered Let us obtain the conditions for the existence of chattering considered as a self-excited periodic motion in a SM system Note that the SM system is a relay feedback system with respect to the so-called sliding variable, which is a function of the system states As a result, the dynamics of a SM system not perturbed by parasitic dynamics is always of relative degree one Also, the hysteresis value of the relay of a SM system is always zero However, if parasitic dynamics are introduced into the system model the relative degree becomes higher than one Let us consider the configuration of the LPRS and in particular the location of its high-frequency segment in dependence on the relative degree of the dynamics Let the transfer function Wl (s) be given as a quotient of two polynomials of degrees m and n respectively: Wl (s) = Bm (s) An (s) = bm sm +bm−1 sm−1 + +b1 s+b0 an sn +an−1 sn−1 + +a1 s+a0 (21) The relative degree of the transfer function Wl (s) is (n − m) Then the following statements hold Lemma If Wl (s) is strictly proper (n > m) then there exists ω ∗ corresponding to any given > such that for every ω ≥ ω ∗ the following inequalities hold: |ReWl (jω) − Re(bm /(an (jω)n−m ))| ≤ (ω ∗ /ω)n−m , (22) |ImWl (jω) − Im(bm /(an (jω)n−m ))| ≤ (ω∗ /ω)n−m (23) ∗ This lemma simply means that at any frequency ω ≥ ω : Wl (s) ≈ bm an sn−m Lemma (monotonicity of high-frequency segment of the LPRS) If ReWl (jω) and ImWl (jω) are monotone functions of frequency ω and |ReWl (jω)| and |ImWl (jω)| are decreasing functions of frequency ω for every ω ≥ ω ∗∗ then within the range ω ≥ ω ∗∗ the real and imaginary parts of the LPRS J(ω) corresponding to that transfer function are monotone functions of frequency ω Proof Since |ReWl (jω)| and |ImWl (jω)| are said to be monotone decreasing functions of ω within the range ω ∈ [ω ∗∗ ; ∞) their derivatives are negative Therefore, functions |ReWl (jkω)| and |ImWl (jkω)|, where k = 1, 2, , ∞, will also have negative derivatives As a result, the derivatives of the following series are negative (being sums of negative addends): ∞ d k=1 |Im W1 [(2k−1)ω]| 2k−1 dω