Asian Journal of Control, Vol 6, No 4, pp 447-453, December 2004 447 ADAPTIVE SLIDING MODE BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS WITH UNMATCHED UNCERTAINTY Ali J Koshkouei, Alan S I Zinober, and Keith J Burnham ABSTRACT This paper considers an adaptive backstepping algorithm for designing the control for a class of nonlinear continuous uncertain processes with disturbances that can be converted to a parametric semi-strict feedback form Sliding mode control using a combined adaptive backstepping sliding mode control (SMC) algorithm, is also studied The algorithm follows a systematic procedure for the design of adaptive control laws for the output tracking of nonlinear systems with matched and unmatched uncertainty KeyWords: Adaptive systems, backstepping, sliding mode control, nonlinear systems, Lyapunov method I INTRODUCTION The backstepping procedure is a systematic design technique for the globally stable and asymptotically adaptive tracking control of a class of nonlinear systems Adaptive backstepping algorithms have 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 [9] This method has been studied widely in recent years [8,9, 16-19] When plants include uncertainty with lack of information about the bounds of unknown parameters, adaptive control is more convenient; whilst, if some information about the uncertainty, such as bounds, is available, robust control is usually employed If a plant has matched uncertainty, the system may be stabilized via state feedback control [3] Some techniques have been proposed for the case of plants containing unmatched uncertainty [5] The plant may contain unmodelled terms and unmeasurable external disturbances, bounded by known functions The stabilization of dynamical systems with uncertainties has been studied in the recent years [1,3,4,6,14] Manuscript received January 15, 2003; revised April 1, 2003; accepted March 19, 2004 Ali J Koshkouei and Keith J Burnham are with Control Theory and Applications Centre, Coventry University, Coventry CV1 5DB, UK Alan S I Zinober is with Department of Applied Mathematics, The University of Sheffield, Sheffield S10 2TN, UK Most approaches are based upon Lyapunov and linearisation methods to design a control In the Lyapunov approach, it is very difficult to find a Lyapunov function to find a control to stabilise 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 by changing the coordinate [8,9] The robust output tracking of nonlinear systems has been studied by many authors [2,12,16] Sliding mode control (SMC) is a robust control method and backstepping can be considered to be a method of adaptive control The combination of these methods yields benefits from both approaches A systematic design procedure has been proposed to combine adaptive control and SMC for nonlinear systems with relative degree one [21] The sliding mode backstepping approach has been extended to some classes of nonlinear systems which need not be in the PPF or PSF forms [16-19] A symbolic algebra toolbox allows straightforward design of dynamical backstepping control [15] The adaptive sliding backstepping control of semi-strict feedback systems (SSF) [20] has been studied by Koshkouei and Zinober [10] The method ensures that the error state trajectories move on a sliding hyperplane In this paper we develop the backstepping approach for SSF systems with unmatched uncertainty We also design a controller based upon sliding backstepping mode techniques so that the state trajectories approach a Asian Journal of Control, Vol 6, No 4, December 2004 448 specified hyperplane These systematic methods not need any extra conditions on the parameters and also any sufficient conditions for the existence of the sliding mode A backstepping method for designing a sliding mode control for a class of nonlinear system without uncertainties, has been presented by Rios-Bolívar and Zinober [15-17] Their method needs some sufficient conditions for existence of the sliding mode In this paper a method for designing sliding mode control is proposed which assures that the sliding mode occurs without satisfying these conditions and also guarantees the stability of the system We extend the classical backstepping method to the parametric semi-strict feedback form in Section to achieve the output tracking of a dynamical reference signal The sliding mode control design based upon the backstepping approach is presented in Section We consider an example to illustrate the results in Section with some conclusions in Section II ADAPTIVE BACKSTEPPING CONTROL Consider the uncertain system ρ = F (ρ) + G(ρ)θ + G(ρ)u + Q(ρ, w, t ) (1) which is transformable into the semi-strict feedback form (SSF) [10,11,20] xi = xi +1 + ϕTi ( x1 , x2 , , xi )θ + ηi ( x, w, t ), ≤ i ≤ n − xn = f n ( x ) + gn ( x )u + ϕ ( x )θ + ηn ( x, w, t ) T n y = x1 (2) First, a classical backstepping method will be extended to this class of systems to achieve the output tracking of a dynamical reference signal The sliding mode control design based upon backstepping techniques is then presented in Section 2.1 Backstepping algorithm The design method based upon the adaptive backstepping approach is now presented [10,11] The functions that compensate the system disturbances, are continuous This method ensures that the output tracks a desired reference signal Step Define the error variable z1 = x1 − yr then z1 = x2 + ϕ1T ( x1 ) θ + η1 ( x, w, t ) − yr From (4) z1 = x2 + ω1T θˆ + η1 ( x, w, t ) − yr + ω1T θ estimate of the unknown parameter vector θ Consider the stabilization of the subsystem (4) and the Lyapunov function V1 ( z1, θˆ ) = ~T −1 ~ z1 + θ Γ θ 2 ~ V1 ( z1, θˆ ) = z1 ( x2 + ω1T θˆ + η1 ( x, w, t ) − yr ) + θT Γ−1 (Γω1 z1 − θˆ ) (7) Define τ1 = Γω1z1 Let Assumption The functions ηi(x, w, t), i = 1, …, n are bounded by known positive functions hi(x1, … xi) ∈ R, i.e Then i = 1, ,n (3) The output y should track a specified bounded reference signal yr(t) with bounded derivatives up to n-th order The system (1) is transferred into system (2) if there exists a diffeomorphism x = x (ρ) The conditions of the existence of a diffeomorphism x = x (ρ) can be found in [13] and the input-output linearisation results in [7] (6) where Γ is a positive definite matrix The derivative of V1 is with c1, a and variable xi ), (5) ~ with ω1(x1) = ϕ1(x1) and θ = θ − θˆ where θ(t ) is an where x =[x1 x2 … xn]T is the state, y the output, u the scalar control, and ϕi (x1, …, xi) ∈ Rp, i = 1, …, n, are known functions which are assumed to be sufficiently smooth θ ∈ Rp is the vector of constant unknown parameters and ηi(x, w, t), i = 1, …, n, are unknown nonlinear scalar functions including all the disturbances w is an uncertain time-varying parameter | ηi ( x, w, t ) | ≤ hi ( x1, (4) n at α1 ( x1 , θˆ , t ) = −ω1T θˆ − c1 z1 − h1 z1e (8) positive numbers Define the error z2 = x2 − α1 ( x1, θˆ , t ) − yr n at = x2 + ω1T θˆ + c1 z1 − yr + h1 z1e z1 = −c1 z1 + z2 + ω1T θ + η1 ( x, w, t ) − n at h1 z1e (9) (10) and V1 is converted to V1 ( z1 , θˆ ) ≤ −c1 z12 + z1 z2 + e − at + θT Γ −1 ( τ1 − θˆ ) n Step k (1 < k ≤ n − 1) The time derivative of the error variable zk is A Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems zk = xk +1 + ωTk θˆ − k −1 ∑ i =1 where ω n ( x, θˆ ) is defined in (12) for k = n Extend the Lyapunov function to be ∂αk −1 ∂α xi +1 − k −1 θˆ + ξk − yr ( k ) (t ) ∂xi ∂θˆ ∂α +ωTk θ − k −1 ∂t 449 (11) Vn = Vn −1 + ( n + 1) − at zn + e = 2a 2a where i =n ∑z i + θT Γθ + i =1 ( n + 1) − at e 2a (19) ωk = ϕk ( x1 , …, xk ) − k −1 ∑ i =1 The time derivative of Vn is ∂αk −1 ϕi ( x1 , …, xi ) ∂xi (n + 1) − at e n n−2 ⎛ ∂α ⎞ ≤ − ∑ ci zi2 − ∑ ⎜ i zi +1 ⎟ ( θˆ − τn ) + θT Γ −1 ( τn − θˆ ) ˆ ∂θ ⎝ ⎠ i =1 i =1 (20) Vn = Vn −1 + zn zn − n at ⎛ k −1 ⎛ ∂αk −1 ⎞ ⎞ e ⎜ hk + ∑ ⎜ ⎟ hi ⎟ ⎜ ⎟ i =1 ⎝ ∂xi ⎠ ⎝ ⎠ k −1 ∂α k −1 ξk = ηk − ∑ ηi i =1 ∂xi ζk = Define zk+1 = xk+1 − αk − (12) yr(k) where where τn = τn −1 + Γ ωTn zn k −1 ∂α αk ( x1 , x2 , …, xk , θˆ , t ) = − zk −1 − ck zk − ωTk θˆ + ∑ k −1 xi +1 i =1 ∂xi We select the control ⎛ k −2 ∂α ∂α ∂α ⎞ + k −1 − ζ k zk + k −1 τk + ⎜ zi +1 i ⎟ Γωk ˆ ∂t ∂θ ∂θˆ ⎠ ⎝ i =1 u= ∑ (13) with ck > Then the time derivative of the error variable zk is zk = − zk −1 − ck zk + zk +1 + ωTk θ + ξk − ζ k zk − ⎛ ∂αk −1 ˆ ∂α ( θ − τk ) + ⎜ zi +1 i ∂θˆ ∂θˆ ⎝ i =1 k −2 ∑ ⎞ ⎟ Γωk ⎠ (14) Vk = Vk −1 + zk = 2 k ∑z i + θT Γθ n −1 ∂α ⎡ T ˆ n −1 xi +1 ⎢− zn −1 − cn zn − f n ( x) − ωn θ + ∑ g n ( x) ⎣ i =1 ∂xi + ⎤ ⎛ n −2 ∂αn −1 ∂α ∂α ⎞ τn + n −1 − ⎜ ∑ zi +1 i ⎟ Γωn + yr( n ) − ζ n zn ⎥ ˆ ˆ t ∂ ∂θ ∂θ ⎠ ⎥⎦ ⎝ i =1 (22) ~ with cn > Taking θˆ = τn , θ is eliminated from the right-hand side of (20) Then n Vn ≤ −∑ ci zi2 ≤ −c || z ||2 < Consider the extended Lyapunov function (21) (23) i =1 (15) i =1 where c = ci This implies that limt→∞ zi = 0, i = i ≤i ≤ n 1,2, …, n, particularly lim t→∞ (x1 − yr) = The time derivative of Vk is III SLIDING BACKSTEPPING CONTROL k ( k + 1) − at e 2n k Vk ≤ − ∑ ci zi2 + zk zk +1 + i =1 ~ ⎛ k −1 ∂α ⎞ + θ T Γ −1 (τ k − θˆ ) + ⎜ ∑ i z i +1 ⎟ (τ k − θˆ ) ˆ ⎝ i =1 ∂θ ⎠ (16) since k τ k = τ k −1 + Γωk z k = Γ ∑ ωi z i (17) i =1 Step n Define zn = xn − αn −1 − yr( n ) with αn−1 obtained from (13) for k = n Then the time derivative of the error variable zn is n −1 ∂α ∂α z n = f n ( x) + g n ( x)u + ωTn ( x, t ) θˆ − ∑ n −1 xi +1 − n −1 θˆ i =1 ∂xi ∂θˆ ∂α − n −1 + ωTn ( x, t )θ + ξn − yr( n ) (18) ∂t Sliding mode techniques yielding robust control and adaptive control techniques are both popular when there is uncertainty in the plant The combination of these methods has been studied in recent years [16-19] In general, at each step of the backstepping method, the new update tuning function and the defined error variables (and virtual control law) take the system to the equilibrium position At the final step the system is stabilized by suitable selection of the control The adaptive sliding backstepping control of SSF systems has been studied by Koshkouei and Zinober [10,11] The controller is based upon sliding backstepping mode techniques so that the state trajectories approach a specified hyperplane The sufficient condition for the existence of the sliding mode, given by Rios-Bolívar and Zinober [15-17], is no longer needed To provide robustness, the adaptive backstepping Asian Journal of Control, Vol 6, No 4, December 2004 450 algorithm can be modified to yield an adaptive sliding output tracking controller The modification is carried out at the final step of the algorithm by incorporating the following sliding surface defined in terms of the error coordinates σ = k1z1 + … + kn −1 zn −1 + zn = (24) where ki > 0, i = 1, …, n − 1, are real numbers Additionally, the Lyapunov function is modified as follows n −1 Vn = ∑z i i =1 1 (n − 1) − at + σ2 + (θ − θˆ )T Γ −1 (θ − θˆ ) + e 2 2a (25) Let n −1 ⎛ n −1 ⎛ ⎞ ⎛ ⎞⎞ τn = τn −1 + Γσ ⎜ ωn + ki ωi ⎟ = Γ ⎜ zi ωi + σ ⎜ ωn + ki ωi ⎟ ⎟ ⎜ ⎟ i =1 i =1 ⎝ ⎠ ⎝ ⎠⎠ ⎝ i =1 n −1 ∑ ∑ ∑ (26) The time derivative of Vn is Vn ≤ − ∑ ci zi2 − zn −1 (k1 z1 + k2 z2 + … + kn −1 zn −1 ) n −1 ∂α ⎡ ∂α +σ ⎢ zn −1 + f n ( x ) + g n ( x )u + ωTn θˆ − ∑ n −1 xi +1 − n −1 θˆ i =1 ∂xi ∂θˆ ⎣ − ∂αn −1 ˆ ∂αn −1 n θ− + ξn − yr( n ) + k1 ( z2 − c1 z1 − h1 z1 eat + η1 ) ∂t ∂θˆ + ∑k ∂αi −1 ˆ ⎛ ⎜ − zi −1 − ci zi + zi +1 + ξi − ζ i zi − ˆ (θ − τi ) ∂θ ⎝ i +Γwi i −2 ∑z l +1 l =1 n−2 −∑ zi +1 i =1 n −1 ⎞⎤ ∂αl ⎞ ⎛ n − ∂α ⎞ ⎛ − ⎜ ∑ zi +1 i ⎟ Γ ⎜ ωn + ∑ ki ωi ⎟ ⎥ ⎟ ˆ ˆ ∂θ ⎠ ⎝ i =1 ∂θ ⎠ ⎝ i =1 ⎠ ⎥⎦ ∂αi ˆ (θ − τn ) + θT Γ −1 ( τn − θˆ ) ∂θˆ (27) since from (24), zn = σ − k1z1 − k2z2 − … − k n −1 zn −1 ~ Setting θˆ = τn , θ is eliminated from right-hand side of (27) Consider the adaptive sliding mode output tracking control n −1 ∂α ∂α ⎡⎣ − zn −1 − f n ( x ) − ωTn θˆ + n −1 τn + ∑ n −1 xi +1 u= ˆ gn ( x ) ∂θ i =1 ∂xi +y − (n) r n −1 ⎛ i i −1 i =2 − ci zi + zi +1 − ζ i zi − 1≤ i ≤ n (29) (n − 1) − at e ≤ − [ z1 z2 … zn −1 ] Q [ z1 z2 … zn −1 ]T − K | σ | −Wσ2 Vn = Vn −1 + σσ − where ⎡ c1 ⎢ c2 Q=⎢ ⎢ ⎢ ⎣⎢ k1 k2 ⎤ ⎥ ⎥ ⎥ ⎥ kn −1 + cn −1 ⎦⎥ (30) which is a positive definite matrix (31) which yields limt→∞ σ = and lim t→∞ zi = 0, i =1, 2, …, n−1 Particularly, lim t→∞ (x1 − yr) = Since zn = σ − k1z1 − k2z2 − … − kn − zn − 1, lim t→∞ zn = Therefore, the stability of the system along the sliding surface σ = is guaranteed There is a close relationship between W ≥ and K > To reduce the chattering obtained from the discontinuous term, they should be tuned so that the desired performances are achieved If K is very large with respect to W, unwanted chattering is produced If K is sufficiently large, one can select W so that stability with a chattering reduction is established W also affects the reaching time of the sliding mode By increasing the value W, the reaching time is decreased Remark Alternatively, one can apply a different procedure at the n-th step which yields u= [− zn −1 − f n ( x) − ωTn θˆ + ∂α nˆ−1 τn g n ( x) ∂θ n −1 ∂α i =1 n −1 ∂xi xi +1 + yr( n ) + ∂αn −1 ∂t ⎛ ⎞ n − k1 ⎜ −c1 z1 + z2 − h1 z1 eat ⎟ ⎝ ⎠ ∂αi −1 ( τ n − τi ) ∂θˆ ⎞ ⎛ n −2 ∂αi ⎞ ⎟ Γωi ⎟ + ⎜ ∑ zi +1 ˆ ∂θ ⎠ ⎠ ⎝ i =1 n ⎤ ⎛ ⎞ − Wσ − ⎜ K + ∑ ki vi ⎟ sgn(σ)⎥ i =1 ⎝ ⎠ ⎦ ∂α ⎛ i −2 + ⎜ ∑ zl +1 l ∂θˆ ⎝ l =1 ∂α k −1 hj , ∂xi Then substituting (28) in (27) yields +∑ ⎛ ⎞ ∂α n h1 z1 e at ⎟ + n −1 − k1 ⎜ − c1 z1 + z2 − ∂t ⎝ ⎠ ∑ k ⎜⎝ − z j =1 Vn ≤ − Wn < i =1 i =2 i −1 vi = hi + ∑ Let Wn = [ z1 z2 … zn −1 ] Q [ z1 z2 … zn −1 ]T + K | σ | + W σ2 then n −1 n −1 where kn = 1, K > and W ≥ are arbitrary real numbers and n −1 ⎞ ⎛ ⎞ ⎟ Γ ⎜ ωn + ∑ ki ωi ⎟ i =1 ⎠ ⎠ ⎝ (28) n −1 ⎛ − ∑ k1 ⎜ − zi −1 − ci zi + zi +1 − ζi zi i =2 ⎝ − ⎞ ⎛ i −2 ∂αi −1 ∂α ⎞ ( τn − τi ) + ⎜ ∑ zl +1 l ⎟ Γωi ⎟ ⎟ ˆ ˆ ∂θ ∂θ ⎠ ⎝ l =1 ⎠ A Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems n −1 ∂α ⎞ ⎛ ⎛ n −2 ⎞ + ⎜ ∑ zi +1 i ⎟ Γ ⎜ ωn + ∑ ki ωi ⎟ ˆ i =1 ∂θ ⎠ ⎝ ⎠ ⎝ i =1 ∂α ∂α ∂α1 x2 + τ + + y r( 2) ˆ ∂x1 ∂t ∂θ ⎛ ∂α1 ⎞ (36) + h1 z1 eat − W σ − ⎜ K + k1 + ⎟ h1 sgn( σ) ⎜ ∂x1 ⎟⎠ ⎝ u = (c1 k1 − 1) z1 − k1 z − ωT2 θˆ + n ⎞ ⎤ ⎛ − K sgn(σ) − ⎜W + ∑ ki vi ⎟ σ⎥ i =1 ⎠ ⎦ ⎝ (32) with kn = 1, K > and W ≥ arbitrary real numbers and for all i, ≤ i ≤ n vi = i −1 ∂αi −1 ⎞ n at ⎛ e ⎜ hk + + ∑ hi ⎟ ⎜ ⎟ ∂x j j =1 ⎝ ⎠ (33) where τ2 = Γ (z1ω1 + σ(ω2 + k1ω1)) 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 reduced and also the reaching time is shorter than when w = IV EXAMPLE V CONCLUSION Consider the second-order system in SSF form x1 = x2 + x1 θ + η ( x1 , x2 ) x2 = u (34) where |η| ≤ h1 = 2x12 2x12.We , z1 = x1 − yr , 451 Backstepping is a systematic Lyapunov method to design control algorithms which stabilize nonlinear sys- have ω1 = x1 z2 = x2 + x1 θˆ + c1 z1 + α1 = − x1 θˆ − c1 z1 − x14 z1 e at , τ2 = Γ( ω1 z1 + ω2 z2 ) , ω2 = − ζ2 = x14 z1 e at − yr ∂α1 x1 ∂x1 ⎛ ∂α ⎞ e ⎜ x1 ⎟ ⎝ ∂x1 ⎠ at Then the control law (22) becomes ∂α ∂α ∂α u = − z1 − c2 z − ωT2 θˆ + x2 + τ + + y r( 2) − ζ z ˆ ∂x1 ∂t ∂θ (35) Simulation results showing desirable transient responses = 10, are presented in Fig with yr = 0.4, a = 0.1, Γ = 1, c1 = 12, c2 = 0.1 and η(x1, x2) = 2x12 cos (3x1 x2) Alternatively, we can design a sliding mode controller for the system Assume that the sliding surface is σ = k1z1 + z2 = with k1 > The adaptive sliding mode control law (28) is Fig Regulator responses with nonlinear control (35) for PSSF system Fig Tracking responses with sliding control (36) for PSSF system with K = 10 and W = Fig Tracking responses with sliding control (36) for PSSF system with K = 10 and W = 452 Asian Journal of Control, Vol 6, No 4, December 2004 tems Sliding mode control is a robust control design method and adaptive backstepping is an adaptive control design method In this paper the control design has benefited from both design approaches Backstepping control and sliding backstepping control were developed for a class of nonlinear systems which can be converted to the parametric strict feedback form The plant may have unmodelled or external disturbances The discontinuous control may contain a gain parameter for the designer to select the velocity of the convergence of the state trajectories to the sliding hyperplane We have extended the previous work of Rios-Bolívar and Zinober [15-17] and Koshkouei and Zinober [10], and have removed the sufficient existence condition for the sliding mode to guarantee that the state trajectories converge to a given sliding surface REFERENCES Barmish, B.R and G Leitmann, “On Ultimate Boundedness Control of Uncertain Systems in the Absence of Matching Assumption,” IEEE Trans Automat, Contr., Vol 27, pp 153-158 (1982) 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for the Design of Dynamical Adaptive Nonlinear Control,” Appl Math Comp Sci., Vol 8, pp 73-88 (1998) 16 Rios-Bolívar, M., and A.S.I Zinober, “Dynamical Adaptive Sliding Mode Output Tracking Control of a Class of Nonlinear Systems,” Int J Robust Nonlin Contr., Vol 7, pp 387-405 (1997) 17 Rios-Bolívar, M and A.S.I Zinober, “Dynamical Adaptive Backstepping Control Design Via Symbolic Computation,” Proc 3rd Eur Contr Conf., Brussels, paper no 704 (1997) 18 Rios-Bolívar, M., A.S.I Zinober, and H Sira-Ramírez, “Dynamical Sliding Mode Control via Adaptive InputOutput Linearization: A Backstepping Approach,” in Robust Control via Variable Structure and Lyapunov Techniques, F Garofalo and L Glielmo, Eds., Springer-Verlag, U.S.A., pp 15-35 (1996) 19 Rios-Bolívar, M and A.S.I Zinober, “Sliding Mode Control for Uncertain Linearizable Nonlinear Systems: A Backstepping Approach,” Proc IEEE Workshop Robust Contr Variable Struct Lyapunov Techniques, Benevento, Italy, pp 78-85 (1994) 20 Yao, B and M Tomizuka, “Adaptive Robust Control of SISO Nonlinear Systems in A Semi-Strict Feedback Form,” Automatica, Vol 33, pp 893-900 (1997) 21 Yao, B., and M Tomizuka, “Smooth Adaptive Sliding Mode Control of Robot Manipulators with Guaranteed Transient Performance,” ASME J Dyn Syst Man Cybern., Vol SMC-8, pp 101-109 (1994) A Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems Ali Koshkouei received the Ph.D degree in Control Theory from University of Sheffield in 1997 He worked at the Department of Applied mathematics from 1997 until 2003 as a Research Associate Since 2003, he has worked at Control Theory and Applications Centre, Coventry University as a Senior Research Assistant He has published about 50 papers in the international journals, conferences and as chapters of books His research interests are mainly stabilisation of nonlinear systems and sliding mode control/observers Alan S I Zinober After obtaining his Ph.D degree from Cambridge University in 1974, Alan Zinober was appointed Lecturer in the Department of Applied and Computational Mathematics at the University of Sheffield in 1974, promoted to Senior Lecturer in 1990, Reader in Applied Mathematics in 1993 and Professor in 1995 He has been the recipient of a number of EPSRC research grants and has published over 160 journal and conference publications The central theme of his research is in the field of variable structure sliding mode control systems theory Discontinuous and continuous control algorithms have been developed as part of a CAD control package Other nonlinear control areas studied include sliding observers, nonlinear H infinity control, frequency shaped sliding control, adaptive backstepping techniques and the realization of nonlinear systems Another area of research is in deterministic Operations Research, in particular the field of combinatorial optimization In 1990 he edited a research monograph, Deterministic Control of Uncertain Systems, and Variable Structure and Lyapunov Control, was published in 1994 He has been an invited speaker at a number of international conferences He organised the IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical Systems in September 1992, and the Fourth NCN Workshop in 2002, and has organized invited sessions at many IFAC and IEEE Conferences He is a past Chairman of the IEEE (UK and RI) Control Systems Chapter and the IMA (UK) Control Theory Committee 453 Keith J Burnham has been Professor of Industrial Control Systems, School of Mathematical and Information Sciences, Coventry University, and Director of the University’s Control Theory and Applications Centre since 1999 This is a multidisciplinary research centre in which effective collaboration takes place amongst staff from across the University There are currently a number of research programmes with UK based industrial organisations, many of which are involved with the design and implementation of adaptive control systems Keith Burnham obtained his BSc (Mathematics), MSc (Control Engineering) and PhD (Adaptive Control) at Coventry University in 1981, 1984 and 1991, respectively He is regularly consulted by industrial organisations to provide advice in areas of advanced algorithm development for control and condition monitoring Currently, he is a Member of the Editorial Board of the Transactions of the Institute of Measurement and Control He is also a Member of the Institution of Electrical Engineers, a Member of the Institute of Measurement and Control, and a Member of the Institute of Mathematics and its Applications