Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems

Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems Ali J Koshkouei1 , Keith Burnham1 , and Alan Zinober2 Control Theory and Applications Centre, Coventry University, Coventry CV1 5FB, UK {a.koshkouei, k.burnham}@coventry.ac.uk Department of Applied Mathematics, The University of Sheffield, Sheffield S10 2TN, UK a.zinober@sheffield.ac.uk Introduction Sliding mode control (SMC) is a powerful and robust control method SMC methods have been widely studied in the last three decades from theoretical concepts to industrial applications [1]-[3] Higher-order sliding mode controllers have recently been addressed to improve the system responses [1] However, when designing a control for a plant it is sometimes more beneficial to use combined techniques, using SMC in conjunction with other methods such as backstepping, passivity, flatness and even other traditional control design methods including H∞ , proportional-integral-derivative (PID) and self-tuning Note that PID control design techniques may also be used for designing the sliding surface A drawback of the SMC methods may be unwanted chattering resulting from discontinuous control There are many methods which can be employed to reduce chattering, for example, using a continuous approximation of the discontinuous control, and a combination of continuous and discontinuous sliding mode controllers Chattering may also be reduced using the higher-order SMC [4] and dynamic sliding mode control [4, 5] When plants include uncertainty with a lack of information about the bounds of unknown parameters, adaptive control is more convenient; whilst, if sufficient information about the uncertainty, such as upper bound is available, a robust control is normally designed The stabilisation problem has been studied for different classes of systems with uncertainties in recent years [6]-[10] Most control design approaches are based upon Lyapunov and linearisation methods In the Lyapunov approach, it is very difficult to find a Lyapunov function for designing a control and stabilising the system The linearisation approach yields local stability The backstepping approach presents a systematic method for designing a control to track a reference signal by selecting an appropriate Lyapunov function and changing the coordinates [11, 12] The robust output tracking of nonlinear systems has been studied by many authors [13]-[15] Backstepping technique guarantees global asymptotic stability Adaptive backstepping algorithms have G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 269–290, 2008 c Springer-Verlag Berlin Heidelberg 2008 springerlink.com 270 A.J Koshkouei, K Burnham, and A Zinober been applied to systems which can be transformed into a triangular form, in particular, the parametric pure feedback (PPF) form and the parametric strict feedback (PSF) form [12] This method has been studied widely in recent years [11, 12], [15]-[19] If a plant has matched uncertainty, a state feedback control may stabilise the system [7] Many techniques have been proposed for the case of plants containing unmatched uncertainty [20] The plant may contain unmodelled terms and unmeasurable external disturbances bounded by known functions or their norm is bound to a constant SMC is a robust control method and backstepping can be considered to be a method of adaptive control The combination of these methods, the so-called adaptive backstepping SMC, yields benefits from both approaches This method can be used even if the system does not comprise of an unknown parameter The backstepping sliding mode approach has been extended to some classes of nonlinear systems which need not be in the PPF or PSF forms [15]-[19] A symbolic algebra toolbox allows straightforward design of dynamical backstepping control [16] A backstepping method for designing an SMC for a class of nonlinear system without uncertainties, has been presented by Rios-Bol´ ıvar and Zinober [16, 17] The adaptive sliding backstepping control of semi-strict feedback systems (SSF) [21] has been studied by Koshkouei and Zinober [22] In this chapter, a systematic design procedure is proposed to combine adaptive control and SMC techniques for a class of nonlinear systems In fact, the backstepping approach for SSF systems with unmatched uncertainty is developed A controller based on SMC techniques is designed so that the state trajectories approach a specified hyperplane These systematic methods not need any extra condition on the parameters and also any sufficient conditions for the existence of the sliding mode to guarantee the stability of the system On the other hand, flatness is an important property in control theory which assures that the system can be stabilised by imposing an artificial output [23][25] A linear system is flat if and only if it is controllable A SISO system with an output is not flat if the relative degree of the system with respect to the output (if it is defined and finite) is not the same as the order of the system In general, there is no comprehensive systematic method for classifying flat and non-flat systems, and also for finding a suitable flat output for nonlinear systems However, the controllability matrix yields a flat output for a linear system [23] and flat time-varying linear systems have been studied by Sira-Ram´ ırez and Silva-Navarro [26] In addition, the control of non-flat systems is an important issue which has been studied since the last decade [24, 27, 28] Flat outputs may not be the actual outputs of the system Flatness for the tracking problem of linear systems in differential operator representation has been considered by Deutscher [29] For MIMO nonlinear systems, there are differences between exact feedback linearisablility and differential flatness (for example see [24, 28]) However, most published papers have dealt with flatness or non-flatness of SISO systems Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems 271 Exact feedforward linearisation based upon differential flatness has been studied by Hagenmeyer and Delaleau [30] in which a flat system is linearised via feedforward control using the differential flatness trajectory satisfying a certain condition on the initial conditions In fact there is a relationship between the flatness and linearisability of nonlinear systems by feedback In particular, for single input systems, flatness is equivalent to linearisability by static state feedback and static feedback linearisability is equivalent to dynamic feedback linearisability [31] In other words, linearisation via static (dynamic) state feedback and coordinate transformation is equivalent to the linearisation by the static (dynamics) feedback of some outputs and a finite number of their derivatives The practical and asymptotic tracking problems for nonlinear systems when only the output of the plant and the reference signal are available has been considered in [32] In addition the concept of global flatness has been presented A system is not globally flat if either the relative degree of the associated augmented system is not well-defined everywhere or the change of coordinates using a particular transformation is not a global diffeomorphism [32] SMC and second-order SMC for nonlinear flat systems are also considered in this chapter The method benefits from the advantages of both approaches The important and main property of SMC is its robustness in the presence of matched uncertainties whilst the flatness property guarantees that the control can be obtained as a function of the flat output and its derivatives In this case, the sliding surface is also introduced in terms of the flat output and its derivatives Differential flatness property and the second-order SMC for a hovercraft vessel model has been studied in [33] The technique has been proposed for the specification of a robust dynamic feedback multivariable controller accomplishing prescribed trajectory tracking tasks for the earth coordinate position variables Moreover, in this chapter a gravity-flow tank/pipeline system is stabilised via an SMC obtained from flatness and sliding mode control theory This combined method inherits the robustness property from SMC If sufficient information about the flat output is available then the control is accessible and applicable without requiring further knowledge of the system variables This chapter is organised as follows: The classical backstepping method to control systems in the parametric semi-strict feedback form is extended in Section to achieve the output tracking of a dynamical reference signal The SMC design based upon the backstepping approach is presented in Section An example which illustrates the results of the backstepping method, is presented in Section In Section the definition and properties of flatness for nonlinear systems are considered In Section a control design method for a class of nonlinear systems with unknown parameters using SMC and the flatness techniques is proposed A suitable estimate for unknown parameters is also obtained In Section the SMC flatness results are applied to a gravity-flow tank/pipeline model for controlling the volumetric flow rate of the liquid leaving the tank and the height of the liquid in the tank presented Conclusions are given in Section 272 A.J Koshkouei, K Burnham, and A Zinober Adaptive Backstepping Control In this section the backstepping procedure for a class of nonlinear systems with unmatched disturbances is presented Consider the uncertain system χ = F (χ) + G(χ)θ + Q(χ)u + D(χ, w, t) ˙ (1) where χ ∈ Rn is the state and u the scalar control The functions F (χ) ∈ Rn , G(χ) ∈ Rn×p and Q(χ) ∈ Rn are known D(χ, w, t) ∈ Rn and w are unknown function and an uncertain time-varying parameter, respectively Also θ ∈ Rp is the vector of constant unknown parameters Assume that the system (1) is transformable into the semi-strict feedback form (SSF) [21, 22, 34] x1 = x2 + ϕT (x1 )θ + η (x, w, t) ˙ x2 = x3 + ϕT (x1 , x2 )θ + η (x, w, t) ˙ xn = fn (x) + gn (x)u + ϕT (x)θ + η n (x, w, t) ˙ n (2) y = x1 where x = [x1 x2 xn ]T is the state, y the output, fn (x), gn (x) ∈ R and ϕi (x1 , , xi ) ∈ Rp , i = 1, , n, are known functions which are assumed to be sufficiently smooth η i (x, w, t), i = 1, , n, are unknown nonlinear scalar functions including all the disturbances Assumption The functions η i (x, w, t), i = 1, , n are bounded by known positive functions hi (x1 , xi ) ∈ R, i.e |η i (x, w, t)| ≤ hi (x1 , xi ), i = 1, , n (3) The output y should track a specified bounded reference signal yr (t) with bounded derivatives up to the n-th order The system (1) is transformed into system (2) if there exists an appropriate diffeomorphism x = x(χ) The conditions of the existence of a diffeomorphism x = x(χ) can be found in [35] and the input-output linearisation results in [36] First, a classical backstepping method will be extended to this class of systems to achieve the output tracking of a dynamical reference signal The SMC design based upon backstepping techniques is then presented in Section 2.1 Backstepping Algorithm The design method based upon the adaptive backstepping approach has been presented in [22, 34] and is recalled afterwards This method ensures that the output tracks a desired reference signal Step Define the error variable z1 = x1 − yr then ˙ z1 = x2 + ϕT (x1 )θ + η (x, w, t) − yr ˙ (4) Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems 273 From (4) ˙ z1 = x2 + ωT ˆ + η (x, w, t) − yr + ωT ˜ ˙ 1θ 1θ (5) with ω (x1 ) = ϕ1 (x1 ) and ˜ = θ − ˆ where ˆ is an estimate of the unknown θ θ θ(t) parameter vector θ Consider the stabilisation of the subsystem (4) and the Lyapunov function θ) V1 (z1 , ˆ = ˜T −1 ˜ z + θ Γ θ 2 (6) where Γ is a positive definite matrix The derivative of V1 is T ˙ ˙ V1 (z1 , ˆ = z1 x2 + ω T ˆ + η (x, w, t) − yr + ˜ Γ −1 Γ ω z1 − ˆ θ θ) ˙ θ 1θ (7) Define τ = Γ ω1 z1 Let n α1 (x1 , ˆ t) = −ωT ˆ − c1 z1 − h2 z1 eat θ, 1θ with c1 , a and (8) positive numbers Define the error variable θ, ˙ z2 = x2 − α1 (x1 , ˆ t) − yr n = x2 + ω T ˆ + c1 z1 − yr + h2 z1 eat ˙ 1θ Then (9) n z1 = −c1 z1 + z2 + ω T ˜ + η (x, w, t) − h2 z1 eat ˙ 1θ (10) ˙ and V1 is converted to ˙ V1 (z1 , ˆ ≤ −c1 z1 + z1 z2 + θ) Step k (1 < k ≤ n − 1) n T ˙ θ θ e−at + ˜ Γ −1 τ − ˆ (k−1) Define zk = xk − αk−1 − yr where k−2 αk−1 (x1 , , xk−1 , ˆ t) = −zk−2 − ck−1 zk−1 − ωT ˆ + θ, k−1 θ i=1 + ∂αk−2 ∂αk−2 − ζ k−1 zk−1 + τ k−1 + ∂t ∂ˆ θ k−3 zi+1 i=1 ∂αk−2 xi+1 ∂xi ∂αi ∂ˆ θ Γ wk (11) with ck−1 > Then the time derivative of the error variable zk is k−1 zk = xk+1 + ˙ ωT ˆ kθ − i=1 + where ωT ˜ kθ ∂αk−1 ˆ ∂αk−1 ˙ (k) xi+1 − θ + ξ k − yr (t) ˆ ∂xi ∂θ − ∂αk−1 ∂t (12) 274 A.J Koshkouei, K Burnham, and A Zinober k−1 ωk = ϕk (x1 , , xk ) − i=1 ζk = k−1 n at e i=1 ξ k = ηk − i=1 ∂αk−1 ∂xi h2 + k k−1 ∂αk−1 ϕ (x1 , , xi ) ∂xi i h2 i (13) ∂αk−1 η ∂xi i (k) Define zk+1 = xk+1 − αk − yr where k−1 αk (x1 , x2 , , xk , ˆ t) = −zk−1 − ck zk − ωT ˆ + θ, kθ i=1 −ζ k zk + ∂αk−1 ∂αk−1 xi+1 + ∂xi ∂t k−2 ∂αk−1 τk + ∂ˆ θ zi+1 i=1 ∂αi ∂ˆ θ Γ wk (14) with ck > Then the time derivative of the error variable zk is ∂αk−1 ˆ ˙ zk = −zk−1 − ck zk + zk+1 + ω T ˜ + ξ k − ζ k zk − ˙ θ − τk kθ ˆ ∂θ k−2 zi+1 + i=1 ∂αi ∂ˆ θ Γ wk (15) Consider the extended Lyapunov function Vk = Vk−1 + zk = 2 i=k T zi + ˜ Γ ˜ θ θ (16) i=1 The time derivative of Vk is k ˙ Vk ≤ − ci zi + zk zk+1 + i=1 k−1 + i=1 k(k + 1) −at ˜T −1 ˙ τk − ˆ e +θ Γ θ 2n ∂αi zi+1 ∂ˆ θ ˙ θ τk − ˆ since (17) k τ k = τ k−1 + Γ ωk zk = Γ ω i zi (18) i=1 Step n Define (n) zn = xn − αn−1 − yr with αn−1 obtained from (11) for k = n Then the time derivative of the error variable zn is Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems n−1 zn = fn (x) + gn (x)u + ω T (x, t)ˆ − ˙ θ n i=1 275 ∂αn−1 ∂αn−1 ˆ ˙ θ xi+1 − ∂xi ∂ˆ θ ∂αn−1 (n) − + ω T (x, t)˜ + ξ n − yr θ n ∂t (19) where ω n (x, t) is defined in (13) for k = n Extend the Lyapunov function to be (n + 1) −at Vn = Vn−1 + zn + e = 2a 2a i=n T (n + 1) −at e zi + ˜ Γ ˜ + θ θ 2a i=1 (20) The time derivative of Vn is (n + 1) −at ˙ ˙ Vn = Vn−1 + zn zn − e ˙ n ≤− n−2 ∂αi zi+1 ∂ˆ θ ci z i − i=1 i=1 ˙ ˙ ˆ − τ n + ˜T Γ −1 τ n − ˆ θ θ θ (21) where τ n = τ n−1 + Γ ω T zn n (22) Select the control u= gn (x) + [ −z n−1 Tˆ n−1 − cn zn − fn (x) − ω n θ + ∂αn−1 − ∂t i=1 n−2 zi+1 i=1 ∂αi ∂ˆ θ ∂αn−1 ∂αn−1 τn xi+1 + ∂xi ∂ˆ θ (n) Γ wn +yr − ζ n zn (23) ˙ with cn > Taking ˆ = τ n , ˜ is eliminated from the right-hand side of (21) θ θ Then n ˙ Vn ≤ − ci zi ≤ −c z 0, i = 1, , n − 1, are real numbers In addition, the Lyapunov function (20) is modified as follows Vn = n−1 1 (n − 1) −at θ) e zi + σ + (θ − ˆ T Γ −1 (θ − ˆ + θ) 2 2a i=1 (26) Let n−1 τ n = τ n−1 + Γ σ ω n + ki ω i i=1 n−1 =Γ n−1 zi ω i + σ ω n + i=1 ki ω i (27) i=1 The time derivative of Vn is n−1 ˙ Vn ≤ − ci zi − zn−1 (k1 z1 + k2 z2 + + kn−1 zn−1 ) i=1 n−1 [ +σ zn−1 + fn (x) + gn (x)u + ωT ˆ − nθ i=1 ∂αn−1 ∂αn−1 ˆ ˙ θ xi+1 − ∂xi ∂ˆ θ ∂αn−1 n (n) − + ξ n − yr + k1 z2 − c1 z1 − h2 z1 eat + η ∂t n−1 ki + ( −z i−1 − ci zi + zi+1 + ξ i − ζ i zi − i=2 i−2 +Γ wi zl+1 l=1 n−2 − zi+1 i=1 ∂αl ∂ˆ θ n−2 − zi+1 i=1 ∂αi ∂ˆ θ ∂αi−1 ˆ ˙ θ − τi ∂ˆ θ n−1 Γ ∂αi ˆ T ˙ ˙ θ θ θ − τ n + ˜ Γ −1 τ n − ˆ ˆ ∂θ ωn + ki ω i i=1 (28) Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems 277 ˙ since from (25), zn = σ − k1 z1 − k2 z2 − − kn−1 zn−1 Setting ˆ = τ n , ˜ is θ θ eliminated from the right-hand side of (28) Consider the adaptive sliding mode output tracking control u= gn (x) [ −z Tˆ n−1 − fn (x) − ω n θ + ∂αn−1 τn + ∂ˆ θ n−1 i=1 ∂αn−1 xi+1 ∂xi ∂αn−1 n (n) − k1 −c1 z1 + z2 − h2 z1 eat +yr + ∂t n−1 − ki ( −z i−1 − ci zi + zi+1 − ζ i zi − i=2 i−2 zl+1 + l=1 n−2 ∂αl ∂ˆ θ zi+1 + Γ wi i=1 ∂αi−1 (τ n − τ i ) ∂ˆ θ ∂αi ∂ˆ θ n−1 ωn + Γ ki ω i i=1 n −W σ − ki ν i K+ sgn(σ) (29) i=1 where kn = 1, K > and W ≥ are arbitrary real numbers and i−1 ν i = hi + j=1 ∂αi−1 hj , ∂xj ≤ i ≤ n (30) Then substituting (29) in (28) yields (n − 1) −at ˙ ˙ Vn = Vn−1 + σ σ − e ˙ T ≤ − [z1 z2 zn−1 ] P [z1 z2 zn−1 ] − K |σ| − W σ where ⎡ c1 ⎢0 ⎢ P =⎢ ⎣ c2 k1 k2 0 ⎤ ⎥ ⎥ ⎥ ⎦ (31) kn−1 + cn−1 which is a positive definite matrix T ˜ Let Wn = [z1 z2 zn−1 ] P [z1 z2 zn−1 ] + K |σ| + W σ Then ˙ ˜ Vn ≤ −Wn < (32) lim which yields t→∞ σ = and t→∞ zi = 0, i = 1, 2, , n − In particular, lim lim (x1 − yr ) = Since zn = σ − k1 z1 − k2 z2 − − kn−1 zn−1 , t→∞ zn = lim t→∞ Therefore, the stability of the system along the sliding surface σ = is guaranteed There is a close relationship between W ≥ and K > A trade-off between two sliding mode gains W and K which may reduce the chattering obtained from discontinuous term and the desired performances may be achieved 278 A.J Koshkouei, K Burnham, and A Zinober If K is very large with respect o W , unwanted chattering is produced If K is sufficiently large, one can select W so that stability with a significant chattering reduction is established W also affects the reaching time of the sliding mode In fact by increasing the value W , the reaching time is decreased Remark Alternatively, at the n-th step, one can select the following control in preference to (29) u= gn (x) [ −z Tˆ n−1 − fn (x) − ω n θ + ∂αn−1 τn + ∂ˆ θ n−1 i=1 ∂αn−1 xi+1 ∂xi ∂αn−1 n (n) − k1 −c1 z1 + z2 − h2 z1 eat +yr + ∂t n−1 − ki ( −z i−1 − ci zi + zi+1 − ζ i zi − i=2 i−2 zl+1 + l=1 ∂αl ∂ˆ θ n−2 Γ wi zi+1 + i=1 ∂αi−1 (τ n − τ i ) ∂ˆ θ ∂αi ∂ˆ θ n−1 Γ ωn + ki ω i i=1 n −Ksgn(σ) − W+ ki ν i σ (33) i=1 with kn = 1, K > and W ≥ arbitrary real numbers and for all i, ≤ i ≤ n ⎛ ⎞ i−1 ∂αi−1 ⎠ n at ⎝ hi νi = e hk + + (34) ∂xj j=1 The sliding mode gains in (29) and (27) are different which may effect the chattering phenomenon Example To illustrate the results the following second-order system which is in the SSF form is considered: x1 = x2 + x1 θ + η(x1 , x2 ) ˙ x2 = u ˙ (35) where η is the disturbance signal and |η| ≤ 2x2 Then from (13) h1 = 2x2 z1 = x1 − yr α1 = −x1 ˆ − c1 z1 − x4 z1 eat θ ω1 = x1 z2 = x2 + x1 ˆ + c1 z1 + x4 z1 eat − yr ˙ θ ∂α1 ω2 = − ∂x1 x1 τ = Γ (ω z1 + ω z2 ) ζ = eat x1 ∂α1 ∂x Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems State behaviour, x (t) 279 State behaviour, x (t) 0.7 0.6 0.5 0.5 0.4 −0.5 0.3 −1 0.2 −1.5 0.1 −2 −2.5 t t Parameter estimate, θ Control action −5 −10 −15 −1 −20 −2 t −25 t Fig Regulator responses with nonlinear control (36) for PSF system Then the control law (23) becomes u = −z1 − c2 z2 − ω T ˆ + 2θ ∂α1 ∂α1 ∂α1 (2) + yr − ζ z2 τ2 + x2 + ∂x1 ∂t ∂ˆ θ (36) Simulation results showing desirable transient responses are presented in Fig with yr = 0.4, a = 0.1, = 10, Γ = 1, c1 = 12, c2 = 0.1 and η(x1 , x2 ) = 2x2 cos(3x1 x2 ) Alternatively, one can design an appropriate SMC for the system Assume that the sliding surface is σ = k1 z1 + z2 = with k1 > The adaptive SMC law (29) is ∂α1 ∂α1 ∂α1 (2) + yr τ + x2 + u = (c1 k1 − 1) z1 − k1 z2 − ω T ˆ + 2θ ˆ ∂x1 ∂t ∂θ ∂α1 + h2 z1 eat − W σ − K + k1 + | | h1 sgn(σ) ∂x1 (37) where τ = Γ (z1 ω + σ(ω + k1 ω )) Simulation results showing desirable transient responses are shown in Fig with the same values as the case without sliding mode and k1 = 1, K = 10, W = The simulation results with K = 10, W = 5, are shown in Fig If W > the chattering of the sliding motion is 280 A.J Koshkouei, K Burnham, and A Zinober State behaviour, x1(t) State behaviour, x2(t) 0.8 0.6 0.4 −2 0.2 −4 0 Parameter estimate, θ −6 10 6 0 Control action 5 −5 −5 Sliding function −10 t −5 t Fig Tracking responses with sliding control (37) for PSF system with K = 10 and W =0 reduced and also the reaching time is shorter than when w = So trade off for a suitable selection of the gain pair K and W is an important issue which may affect the chattering Flatness As stated, there is a link between the differential flatness and the feedback linearisation problem If the derivative of the state can be expressed in terms of the system state and the derivatives of input variables then the state is called the generalised state and the preceding equations are referred to as a generalised state representation of the system [37] If the generalised state representations are used for designing a feedback control, the time derivatives of the input variables may appear in the feedback laws This feedback is known as a quasi-static state feedback (see [38] and references therein) A flat nonlinear system is linearisable via a generalised quasi-static state feedback For SISO systems, the linearisability and flatness properties are equivalent Therefore the control obtained stabilises the systems without including any extra dynamics If the system Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems State behaviour, x1(t) 281 State behaviour, x2(t) 0.8 0.6 0.4 −2 0.2 0 Parameter estimate, θ −4 4 Control action 6 20 10 −10 −1 Sliding function −20 t −2 −4 t Fig Tracking responses with sliding control (37) for PSF system with K = 10 and W =5 includes uncertainties, particularly matched uncertainties, sliding mode control is an appropriate approach to achieve the system tracking stability Backstepping method is applicable to minimum-phase nonlinear systems [15] with unknown parameters and disturbances In particular, systems in the form of SFF can benefit from this technique Flatness is a geometric system property which does not change the coordinates and indicates that the system is transformable to an associated linear system Therefore, a flat system has a well-structured system which enables one to design a controller and solve the stabilisation problem One can also use the dynamic feedback linearisation method for control of flat systems However, backstepping method is applicable for a wide class of nonlinear systems Note that there is no systematic method for constructing a flat output To study the performance of flatness, the definition of flatness is first considered Definition [30] Consider the nonlinear system x(t) = f (x(t), u(t)) ˙ (38) where x ∈ R is the state, t ∈ R, f (x, u) ∈ R is a smooth vector field and u ∈ Rm is the control The system (38) is (differentially) flat if there exists a n n 282 A.J Koshkouei, K Burnham, and A Zinober T set of m independent variables y = [y1 y2 ym ] , the so-called flat output, such that y = η(x, u, u, , u(i) ) ˙ x = φ(y, y, , y (j) ) ˙ u = ϕ(y, y, , y (k) ) ˙ (39) where η, φ and ϕ are smooth functions in open sets of Rm×(i+1) , Rn×(j+1) and Rm×(k+1) , respectively A necessary condition for flatness of a single input system is that the relative degree is the system order n Since the relative degree is invariant under coordinate transformation and feedback, the flatness property is independent of the selection of w SMC Design for Flat Nonlinear Systems with Unknown Parameters In this section a class of flat nonlinear systems with unknown parameters are considered For simplicity, it is assumed that the unknown parameters appear in the same equation as the control A suitable estimate is obtained so that SMC can stabilise the flat system and the output tracks a desired value Consider the system x1 = a2 x2 + f1 (x1 ) ˙ x2 = a3 x3 + f1 (x1 , x2 ) ˙ xn−1 = an xn + fn (x1 , x2 , , xn−1 ) ˙ xn = fn (x) + gn (x)u + ϕT (x)θ ˙ n y = x1 (40) where fi , gn ∈ R and ϕ ∈ R1×p are smooth functions and = 0, i = 2, , n are known The vector θ ∈ Rp×1 consists of constant unknown parameters The states can be expressed in terms of the output and a finite number of its derivatives x2 = (y − f1 (y)) ˙ a2 = y − α1 (y) ˙ a2 1 dα1 x3 = y − f2 y, y (y) y ă a3 a2 dy a2 = y (y, y) ă ˙ a2 a3 Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems xn = y (n−1) − an a2 an−1 n−1 i=1 283 dαn−1 (i−1) y − fn dy (i−2) = ˙ y (n−2) − αn−1 y, y, , y (n−2) a an u= ˙ ˙ y (n) − Fn (y, y, , y (n−1) ) − φT (y, y, , y (n−1) )θ n Gn (y, y, , y (n−1) ) ˙ (41) where φ y, y, , y ˙ (n−1) = ϕ(x), Fn y, y, , y ˙ (n−1) = fn (x) and ˙ Gn y, y, , y (n−1) = gn (x) So the system is flat Consider the control (41) in which θ is replaced with ˆ θ Define the sliding function ˙ s = k1 y + k2 y + + y (n−1) n−1 where p(λ) = k1 + k2 λ + + λ (42) is a Hurwitz polynomial Then s = k1 y + k2 y + + y (n) ˙ ă and to obtain s = Ws sgn(s) it is required that ă y (n) = k1 y + k2 y + + y (n−1) + Ws sgn(s) (43) Select the control u= Gn (y, y, , y (n−1) ) ˙ n−1 ki y (i) − Fn (y, y, , y (n−1) − Ws sgn(s)) ˙ − i=1 −φT (y, y, , y (n−1) )ˆ ˙ θ n (44) where ˆ is an estimate of θ and kn−1 = Consider the Lyapunov function θ V = s + (θ − ˆ T Γ −1 (θ − ˆ θ) θ) (45) with γ > Then ˙ ˙ V = ss + (θ − ˆ −1 (−ˆ ˙ θ)Γ θ) ˙ = s k1 y + k2 y + + y (n) + (θ − ˆ −1 ( ă ) ) ă ) ) = s k1 y + k2 y + + uGn + Fn + φT θ + (θ − ˆ −1 (−ˆ n ˙ ˆ θ)Γ θ) = s −Ws sgn(s) + φT (θ − θ) + (θ − ˆ −1 (−ˆ n ˙ = −Ws |s| + (θ − ˆ −1 (Γ sφT − ˆ θ)Γ θ) n (46) 284 A.J Koshkouei, K Burnham, and A Zinober Consider the following estimate function ˙ ˆ = Γ sφT θ Then (46) implies ˙ V = −Ws |s| < (47) Integrating from (47) yields t V (t) − V (0) = − Ws (μ)|s(μ)|dμ o t t So V (t) + o Ws (μ)|s(μ)|dμ = V (0) In particular, o Ws (μ)|s(μ)|dμ ≤ V (0) t Therefore, t→∞ o Ws (μ)|s(μ)|dμ exists According to Barbalat’s lemma lim lim Ws (t)|(s(t)| = which guarantees the sliding mode stability Since s and s ˙ t→∞ tend to zero, (42) implies that y = x1 , y, y y (n) also tend to zero Then, from ˙ ¨ (41), one can conclude the trajectories approach an equilibrium point along the sliding surface s = For greater accuracy of the SMC design, one can design a second-order sliding mode The second-order sliding mode occurs if s = s = and the sufficient ˙ condition s = −β|s|p − α sgn(s)dt ˙ (48) where α, β > and < p ≤ 0.5, is satisfied [4] The following control law satisfies the condition (48) u= Gn (y, y, , y (n−1) ) ˙ −α n−1 ki y (i) − Fn (y, y, , y (n−1) − β|s|p ˙ − i=1 sgn(s)dt) − φT (y, y, , y (n−1) )ˆ ˙ θ n (49) Example: Gravity-Flow/Pipeline System A gravity-flow/pipeline System is a liquid system in which the water supply is higher than all points in the pipeline and no pump is normally required (see Fig 4) It is assumed that the flux cannot be reversed Consider the following gravity-flow/pipeline system including an elementary static model for an ‘equal percentage valve’ [39] Ap g Kf x2 − x L ρA2 p FCmax α−(1−u) − x1 + θf (x1 , x2 ) x2 = ˙ At x1 = ˙ (50) Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems 285 with x1 : volumetric flow rate of liquid leaving the tank : height of the liquid in the tank x2 FCmax : maximum value of the volumetric rate of fluid entering the tank g : gravitational acceleration constant L : the pipe length : friction of the liquid Kf ρ : density of the liquid : cross sectional area of the pipe Ap : cross sectional area of the tank At α : rangeability parameter of the value u : control input, taking values in the closed interval [0, 1] θ : an unknown parameter f (x1 , x2 ) : a known perturbation function depending on the waves produced by entering the liquid Fig A gravity-flow tank/pipeline system The equilibrium point of the system (50) is X1 = FCmax α−(1−U) ; X2 = LKf X gρA3 p corresponding to a constant value U ∈ [0, 1] The operating region of the system is R2 Using the auxiliary control w = FCmax α−(1−u) , the system (50) becomes + Ap g Kf x2 − x L ρA2 p x2 = ˙ (w − x1 ) + θf (x1 , x2 ) At x1 = ˙ (51) 286 A.J Koshkouei, K Burnham, and A Zinober Assume ˆ is an estimate of θ It is desired that the state x1 tracks the constant θ value X1 Select y = x1 − X1 as the output x1 = y L Kf y θ y+ ˙ Ap g ρA2 p LAt 2LAt Kf w= y+ ă y y + At f (y, y) ˙ ˙ gAp ρgA3 p x2 = (52) So the system is flat with the output y Consider the sliding function s = ky + y ˙ (53) where k > real number To obtain s = −Ws sgn(s) ˙ (54) y = − (k y + Ws sgn(s)) ă (55) it is required that From (52) y= ă gAp gAp Kf gAp f (y, y) ˙ w− y − 2 yy + ˙ LAt LAt ρAp L Select the control w=y+ LAt LAt Kf LAt y+2 ă y y At (y, y) − ˙ Ws sgn(s) θf ˙ gAp ρgA3 gAp p (56) where ˆ is an estimate of θ Select the Lyapunov function θ V = θ) s + γ(θ − ˆ 2 (57) where γ > Then from (52)-(56), the time-derivative of the Lyapunov function is obtained ˙ ˙ V = ss + (θ − ˆ ˙ θ)(−ˆ θ)γ ˙ = s (k y + y ) + ( ă )( ) = s (−Ws sgn(s) − θAt f (y, y) + θAt f (x1 , x2 )) ˙ ˙ θ) ˙ θ = −Ws |s| + γ(θ − ˆ γ −1 At sf (y, y) − ˆ (58) The adaptation mechanism is obtained from (58) ˙ ˆ = γ −1 At sf (y, y) ˙ θ (59) Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems Volumetric flow rate of liquid leaving the tank 287 Height of liquid 14 12 10 x 1.6 x 1.8 1.4 1.2 20 40 60 Time(sec) 80 100 20 Estimate of parameter 40 60 Time(sec) 80 100 80 100 Control input 0.25 2.5 0.2 θ u 0.3 0.15 1.5 0.1 0.05 20 40 60 Time(sec) 80 100 0.5 20 40 60 Time(sec) Fig The responses of the gravity-flow tank/pipeline system using the continuous approximation of the SMC (56) The control can be obtained in terms of the original states using (52) and (56) w = −kAt x2 + − 2At LKf kLAt Kf 2At Kf x1 − x1 + x1 x2 − x ρgA3 ρA2 gρ2 A5 p p p At L Ws sign(s) − At ˆ (x1 , x2 ) θf gAp (60) This method with the flat output y which is the volumetric flow rate of the liquid leaving the tank yields the appropriate control with a suitable estimate of the unknown parameter The simulation results are shown in Fig for g = 9.81, L = 900, k = 1, f (x1 , x2 ) = sin(0.1πx1 ), ρ = 998, At = 10.5, Ap = 0.653, α = 9.3, FCmax = 2.5, K = 4.1, γ = 0.06, Ws = 0.05 and θ = 4.4739 The desired equilibrium for U = 0.89 is X1 = and X2 = 6.66 Note that the simulation results have been carried out using the continuous approximation of the SMC control Conclusions In this chapter, backstepping, flatness and SMC for nonlinear systems have been studied Backstepping is a systematic Lyapunov method for designing control 288 A.J Koshkouei, K Burnham, and A Zinober algorithms which stabilise nonlinear systems SMC and adaptive backstepping are a robust control and an adaptive control design methods, respectively A combination of these two control design methods may benefit from the advantages of the both methods In this chapter backstepping control and sliding mode backstepping control were developed for a class of nonlinear systems which can be converted to the parametric strict feedback form The systems may have unmodelled or external disturbances The discontinuous control obtained may contain a gain parameter for the designer to select the velocity of the convergence of the state trajectories to the sliding hyperplane The method does not require any existence of a sufficient condition for the sliding mode to guarantee that the state trajectories converge to a given sliding surface On the other hand, flatness is an important property which one can use for designing a control, since a flat system can be considered as a controllable system In fact for linear systems controllability and flatness are equivalent The system is flat if there exists an artificial output such that the states and the control can be expressed as functions of the output and a 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Variable structure systems, sliding mode and nonlinear control Lecture Notes in Control and Information Sciences Springer, Berlin (1999) ... method for classifying flat and non-flat systems, and also for finding a suitable flat output for nonlinear systems However, the controllability matrix yields a flat output for a linear system [23] and. .. and differential flatness (for example see [24, 28]) However, most published papers have dealt with flatness or non-flatness of SISO systems Flatness, Backstepping and Sliding Mode Controllers for. .. t) − yr ˙ (4) Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems 273 From (4) ˙ z1 = x2 + ωT ˆ + η (x, w, t) − yr + ωT ˜ ˙ 1θ 1θ (5) with ω (x1 ) = ϕ1 (x1 ) and ˜ = θ − ˆ