TUTORIALS ON SLIDING MODE CONTROL
September 20, 2010
Advanced State Observers Observers
Observers-A Survey
Classical observers such as the Kalman Filter and Luenberger Observer depend on accurate mathematical representation of the plant. These state observers are useful in system monitoring and regulation as well as detecting and identifying failures in dynamical systems. The presence of disturbances, dynamic uncertainties and non-linearities pose a great challenge in practical application of these observers.
CLO HGO SMO ESO
Advanced State Observers Observers
Observers-A Survey
Classical observers such as the Kalman Filter and Luenberger Observer depend on accurate mathematical representation of the plant. These state observers are useful in system monitoring and regulation as well as detecting and identifying failures in dynamical systems. The presence of disturbances, dynamic uncertainties and non-linearities pose a great challenge in practical application of these observers.
CLO HGO SMO ESO
Advanced State Observers Observers
Observers-A Survey
Classical observers such as the Kalman Filter and Luenberger Observer depend on accurate mathematical representation of the plant. These state observers are useful in system monitoring and regulation as well as detecting and identifying failures in dynamical systems. The presence of disturbances, dynamic uncertainties and non-linearities pose a great challenge in practical application of these observers.
CLO HGO SMO ESO
Advanced State Observers
Observers-A Survey
The design of a robust observer which overcomes the above challenge has been attempted by many researchers and several advanced observer designs have been proposed proposed.. Som Some e are re::High Gain Observer proposed by Khalil [1 [ 1] and Esfandiari [2 [2] for the design of output feedback controllers. Sliding Mode Observer proposed by Slotine [3] [3] and Utkin [?]. A class of non-linear extended state observers (NESO) proposed by J.Han [4] [4].. Sliding mode control with perturbation estimator(SMCPE) based on time delay control by Elmali and Olgac [5 [ 5]
Observers CLO HGO SMO ESO
Advanced State Observers
Observers-A Survey
The design of a robust observer which overcomes the above challenge has been attempted by many researchers and several advanced observer designs have been proposed proposed.. Som Some e are re::High Gain Observer proposed by Khalil [1 [ 1] and Esfandiari [2 [2] for the design of output feedback controllers. Sliding Mode Observer proposed by Slotine [3] [3] and Utkin [?]. A class of non-linear extended state observers (NESO) proposed by J.Han [4] [4].. Sliding mode control with perturbation estimator(SMCPE) based on time delay control by Elmali and Olgac [5 [ 5]
Observers CLO HGO SMO ESO
Advanced State Observers
Observers-A Survey
The design of a robust observer which overcomes the above challenge has been attempted by many researchers and several advanced observer designs have been proposed proposed.. Som Some e are re::High Gain Observer proposed by Khalil [1 [ 1] and Esfandiari [2 [2] for the design of output feedback controllers. Sliding Mode Observer proposed by Slotine [3] [3] and Utkin [?]. A class of non-linear extended state observers (NESO) proposed by J.Han [4] [4].. Sliding mode control with perturbation estimator(SMCPE) based on time delay control by Elmali and Olgac [5 [ 5]
Observers CLO HGO SMO ESO
Advanced State Observers
Observers-A Survey
The design of a robust observer which overcomes the above challenge has been attempted by many researchers and several advanced observer designs have been proposed proposed.. Som Some e are re::High Gain Observer proposed by Khalil [1 [ 1] and Esfandiari [2 [2] for the design of output feedback controllers. Sliding Mode Observer proposed by Slotine [3] [3] and Utkin [?]. A class of non-linear extended state observers (NESO) proposed by J.Han [4] [4].. Sliding mode control with perturbation estimator(SMCPE) based on time delay control by Elmali and Olgac [5 [ 5]
Observers CLO HGO SMO ESO
Advanced State Observers
Observers-A Survey
The design of a robust observer which overcomes the above challenge has been attempted by many researchers and several advanced observer designs have been proposed proposed.. Som Some e are re::High Gain Observer proposed by Khalil [1 [ 1] and Esfandiari [2 [2] for the design of output feedback controllers. Sliding Mode Observer proposed by Slotine [3] [3] and Utkin [?]. A class of non-linear extended state observers (NESO) proposed by J.Han [4] [4].. Sliding mode control with perturbation estimator(SMCPE) based on time delay control by Elmali and Olgac [5 [ 5]
Observers CLO HGO SMO ESO
Advanced State Observers Observers CLO HGO SMO ESO
Observers-A Survey
Sliding mode state and perturbation observer (SMSPO) by Olgac [6 [6]. Sliding mode state and perturbation observer (SMSPO) by Jiang [7 [7]. ]. Wang and Gao [8 [8] carried out a comparison study of first three advanced state observers.
Advanced State Observers Observers CLO HGO SMO ESO
Observers-A Survey
Sliding mode state and perturbation observer (SMSPO) by Olgac [6 [6]. Sliding mode state and perturbation observer (SMSPO) by Jiang [7 [7]. ]. Wang and Gao [8 [8] carried out a comparison study of first three advanced state observers.
Advanced State Observers Observers CLO HGO SMO ESO
Observers-A Survey
Sliding mode state and perturbation observer (SMSPO) by Olgac [6 [6]. Sliding mode state and perturbation observer (SMSPO) by Jiang [7 [7]. ]. Wang and Gao [8 [8] carried out a comparison study of first three advanced state observers.
Observers-A Survey Observers
Classical Luenberger Observer
Consider a linear, time invariant continuous time dynamical system given by x˙ = Ax + Bu y = Cx
CLO HGO SMO ESO
(1)
where the matrices A,B and C are parameters of the state space model. The Luenberger observer for the above plant is given as ˆ x˙ = Aˆ x + Bu + L(y − Cˆ x) where L is the observer gain matrix which can be found using pole placement.
(2)
Observers-A Survey Observers
Classical Luenberger Observer
Consider a linear, time invariant continuous time dynamical system given by x˙ = Ax + Bu y = Cx
CLO HGO SMO ESO
(1)
where the matrices A,B and C are parameters of the state space model. The Luenberger observer for the above plant is given as ˆ x˙ = Aˆ x + Bu + L(y − Cˆ x) where L is the observer gain matrix which can be found using pole placement.
(2)
Observers-A Survey Observers CLO HGO SMO ESO
Classical Luenberger Observer
The estimation error is given as e = x−ˆ x
(3)
Differentiating the above equation, the error dynamics is arrived at as e˙ = (A − LC)e (4)
The estimation error will converge to zero if (A − LC) has all its eigen values in the left half plane.
Observers-A Survey Observers CLO HGO SMO ESO
Classical Luenberger Observer
The estimation error is given as e = x−ˆ x
(3)
Differentiating the above equation, the error dynamics is arrived at as e˙ = (A − LC)e (4)
The estimation error will converge to zero if (A − LC) has all its eigen values in the left half plane.
Observers-A Survey Observers CLO HGO SMO ESO
Classical Luenberger Observer
The estimation error is given as e = x−ˆ x
(3)
Differentiating the above equation, the error dynamics is arrived at as e˙ = (A − LC)e (4)
The estimation error will converge to zero if (A − LC) has all its eigen values in the left half plane.
Observers-A Survey
Classical Luenberger Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = −a1 x1 − a2 x2 + b0 u
Observers
(5)
Assuming that the output variable (or state) is the only state available for measurement i.e.,y = x 1 , the Luenberger observer for the above plant is given as ˆ x˙ 1 = ˆ x2 + l1 (x1 − ˆ x1 ) ˆ x˙ 2 = −a1ˆ x1 − a 2 ˆ x2 + b0 u + l2 (x1 − ˆ x1 ) where L = [l1 l2 ]T is the observer gain matrix which can be found using pole placement.
(6)
Observers-A Survey
Classical Luenberger Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = −a1 x1 − a2 x2 + b0 u
Observers
(5)
Assuming that the output variable (or state) is the only state available for measurement i.e.,y = x 1 , the Luenberger observer for the above plant is given as ˆ x˙ 1 = ˆ x2 + l1 (x1 − ˆ x1 ) ˆ x˙ 2 = −a1ˆ x1 − a 2 ˆ x2 + b0 u + l2 (x1 − ˆ x1 ) where L = [l1 l2 ]T is the observer gain matrix which can be found using pole placement.
(6)
Observers-A Survey
Classical Luenberger Observer
Observers
Defining estimation error ˜ x = x−ˆ x, diff x, differe erenti ntiati ating ng and substituting the above equations gives ˜ x˙ 1 = ˜ x2 − l1 (˜ x1 ) ˜ x˙ 2 = −a1˜ x1 − a 2 ˜ x2 − l2 (˜ x1 )
(7)
Hence the estimation error dynamics can be given as
˜x˙ 1
˜ x˙ 2
=
1 ˜ x
−l1 −a1 − l2 −a2
1
(8)
˜ x2
or
˜x˙ 0 1
˜ x˙ 2
=
1 ˜ x −l 0 ˜ x
−a1 −a2
1
˜ x2
+
1
−l2 0
1
˜ x2
(9)
CLO HGO SMO ESO
Observers-A Survey
Classical Luenberger Observer
Observers
Defining estimation error ˜ x = x−ˆ x, diff x, differe erenti ntiati ating ng and substituting the above equations gives ˜ x˙ 1 = ˜ x2 − l1 (˜ x1 ) ˜ x˙ 2 = −a1˜ x1 − a 2 ˜ x2 − l2 (˜ x1 )
(7)
Hence the estimation error dynamics can be given as
˜x˙ 1
˜ x˙ 2
=
1 ˜ x
−l1 −a1 − l2 −a2
1
(8)
˜ x2
or
˜x˙ 0 1
˜ x˙ 2
=
1 ˜ x −l 0 ˜ x
−a1 −a2
1
˜ x2
+
1
−l2 0
1
˜ x2
(9)
CLO HGO SMO ESO
Observers-A Survey Observers CLO HGO SMO ESO
Classical Luenberger Observer
Now consider that the above second order plant contains uncertainties, non-linearities and external distur dis turban bances ces.. Hen Hence ce it can be b e exp expres ressed sed as x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
(10)
where f(x) = f 0 (x) + d and f 0 (x) = −a1 x1 − a2 x2 . d is a composite term for uncertainties, non-linearities and external disturbances.
Observers-A Survey Observers
High Gain Observer
The high gain observer (HGO) has an error dynamics structure which is the same as the Luenberger Observer. The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are calculated using pole placement. In the case of HGO, the observer gains calculated using pole placement are divided by a quantity such that 0 < < 1.
CLO HGO SMO ESO
Observers-A Survey Observers
High Gain Observer
The high gain observer (HGO) has an error dynamics structure which is the same as the Luenberger Observer. The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are calculated using pole placement. In the case of HGO, the observer gains calculated using pole placement are divided by a quantity such that 0 < < 1.
CLO HGO SMO ESO
Observers-A Survey Observers
High Gain Observer
The high gain observer (HGO) has an error dynamics structure which is the same as the Luenberger Observer. The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are calculated using pole placement. In the case of HGO, the observer gains calculated using pole placement are divided by a quantity such that 0 < < 1.
CLO HGO SMO ESO
Observers-A Survey Observers
High Gain Observer
The high gain observer (HGO) has an error dynamics structure which is the same as the Luenberger Observer. The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are calculated using pole placement. In the case of HGO, the observer gains calculated using pole placement are divided by a quantity such that 0 < < 1.
CLO HGO SMO ESO
Observers-A Survey
High Gain Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
Observers
(11)
The High Gain Observer for the above plant is given as ˆ x˙ 1 = ˆ x2 + h1 (x1 − ˆ x1 ) ˆ x˙ 2 = f 0 (ˆ x) + b0 u + h2 (x1 − ˆ x1 )
(12)
where H = [h1 h2 ]T is the observer gain matrix which can be found by dividing the values calculated using pole placement by the quantity such that 0 < < 1 i.e., h1 = l1 and h2 = l22 .
Observers-A Survey
High Gain Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
Observers
(11)
The High Gain Observer for the above plant is given as ˆ x˙ 1 = ˆ x2 + h1 (x1 − ˆ x1 ) ˆ x˙ 2 = f 0 (ˆ x) + b0 u + h2 (x1 − ˆ x1 )
(12)
where H = [h1 h2 ]T is the observer gain matrix which can be found by dividing the values calculated using pole placement by the quantity such that 0 < < 1 i.e., h1 = l1 and h2 = l22 .
Observers-A Survey Observers
High Gain Observer
Defining estimation error ˜ x = x−ˆ x, diff x, differe erenti ntiati ating ng and substituting the above equations gives ˜ x˙ 1 = ˜ x2 − h1 (˜ x1 ) ˜ x˙ 2 = f (x (x) − f 0 (ˆ x) − h2 (˜ x1 ) ˜ x˙ 2 = δ (x) − h2 (˜ x1 )
(13)
Hence the estimation error dynamics can be given as
˜x˙ −h 1 ˜x 0 1
˜ x˙ 2
=
1
−h2 0
1
˜ x2
+
1
δ (x)
(14)
CLO HGO SMO ESO
Observers-A Survey Observers
High Gain Observer
Defining estimation error ˜ x = x−ˆ x, diff x, differe erenti ntiati ating ng and substituting the above equations gives ˜ x˙ 1 = ˜ x2 − h1 (˜ x1 ) ˜ x˙ 2 = f (x (x) − f 0 (ˆ x) − h2 (˜ x1 ) ˜ x˙ 2 = δ (x) − h2 (˜ x1 )
(13)
Hence the estimation error dynamics can be given as
˜x˙ −h 1 ˜x 0 1
˜ x˙ 2
=
1
−h2 0
1
˜ x2
+
1
δ (x)
(14)
CLO HGO SMO ESO
EXERCISE Observers CLO HGO SMO ESO
High Gain Observer-Exercise
What is the physical significance of dividing the observer gains calculated using pole placement by a quantity such that 0 < < 1 in the case of HGO? What are the disadvantages of HGO?
EXERCISE Observers CLO HGO SMO ESO
High Gain Observer-Exercise
What is the physical significance of dividing the observer gains calculated using pole placement by a quantity such that 0 < < 1 in the case of HGO? What are the disadvantages of HGO?
Observers-A Survey
Sliding Mode Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
Observers
(15)
The Sliding Mode Observer (SMO) for the above plant with y = x1 is given as ˆ x˙ 1 = ˆ x2 + l1 (y − ˆ x1 ) + k1 sgn(y − ˆ x1 ) (16) ˆ x˙ 2 = f 0 (x) + b0 u + l2 (y − ˆ x1 ) + k2 sgn(y − ˆ x1 ) where L = [l1 l2 ]T is the observer gain matrix which can be calculated using pole placement and K = [k1 k2 ]T > 0.
Observers-A Survey
Sliding Mode Observer
CLO HGO SMO ESO
Considering a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
Observers
(15)
The Sliding Mode Observer (SMO) for the above plant with y = x1 is given as ˆ x˙ 1 = ˆ x2 + l1 (y − ˆ x1 ) + k1 sgn(y − ˆ x1 ) (16) ˆ x˙ 2 = f 0 (x) + b0 u + l2 (y − ˆ x1 ) + k2 sgn(y − ˆ x1 ) where L = [l1 l2 ]T is the observer gain matrix which can be calculated using pole placement and K = [k1 k2 ]T > 0.
Observers-A Survey Observers
Extended State Observer
Consider a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
CLO HGO SMO ESO
(17)
The non-linear function f(x) is now considered as an extended state x3 and the above set of equations can be modified as x˙ 1 = x2 x˙ 2 = x3 + b0 u x˙ 3 = h
(18)
Observers-A Survey Observers
Extended State Observer
Consider a second order dynamic system, x˙ 1 = x2 x˙ 2 = f(x f(x) + b0 u
CLO HGO SMO ESO
(17)
The non-linear function f(x) is now considered as an extended state x3 and the above set of equations can be modified as x˙ 1 = x2 x˙ 2 = x3 + b0 u x˙ 3 = h
(18)
Observers-A Survey Observers CLO HGO SMO ESO
Extended State Observer
The Extended State Observer (ESO) for the above plant with y = x1 is given as ˆ x˙ 1 = ˆ x2 + β1 (y − ˆ x1 ) ˆ x˙ 2 = ˆ x3 + b0 u + β2 (y − ˆ x1 ) ˆ x˙ 3 = β3 (y − ˆ x1 )
(19)
where β = [β1 β2 β3 ]T is the observer gain matrix which can be calculated using pole placement in case of Linear ESO. How are the gains selected in case of Non-linear ESO (NESO)?
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode Control Perturbation Estimator
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode Control Perturbation Estimator
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode State and Perturbation Observer - Olgac
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode State and Perturbation Observer - Olgac
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode State and Perturbation Observer - Jiang
Observers-A Survey Observers CLO HGO SMO ESO
Sliding Mode State and Perturbation Observer - Jiang
Khalil, H., “High Gain Observers in Nonlinear Feedback Control:New Directions in Nonlinear Observer Design,” Lecture Notes in Control and Information Sciences, Vol. 24, No. 4, 1999, pp. 249–268. Esfandiari and Khalil, “Output feedback stabilisation of fully linearisable systems,” International Journal of Control, Vol. 56, 1992, pp. 1007–1037. Slotine, J. J. E. and Misawa, E. A., “On Sliding Mode Observers for Nonlinear Systems,” Journal of Dynamic Systems, Measurement and Control, Vol. 109, 1987, pp. 245–252.
Elmali, H. and Olgac, N., “Sliding Mode Control with Perturbation Estimation (SMCPE): A New Approach,” International Journal of Control, Vol. 56, No. 4, 1992, pp. 923–941.
Observers CLO HGO SMO ESO
Moura, J.T., E. H. a. O. N., “Sliding Mode COntrol with Sliding Perturbation Observer,” Transactions of the ASME, Journal of Dynamic Systems,Measurement, and Control, Vol. 119, 1997, pp. 657–665. Jiang, L.and Wu, Q., “Nonlinear Adaptive Control via Sliding-mode State and Perturbation Observer,” IEE Proc.- Control Theory Applications, Vol. 149, No. 4, 2002, pp. 269–277. Wang, W. and Gao, Z., “A Comparison Study of Advanced State Observer Design Techniques,” Proceedings of the American Control Conference, Denver, Colorado, 2003, pp. 4754–4759.
Observers CLO HGO SMO ESO
