Two-Valued Control for a Second-Order Plant with Additive External Disturbance

In this work a two-valued state feedback control for a plant of second order with known constant coefficients and an additive bounded disturbance is designed. In this controller the control signal can take only two possible values. The controller design is based on Lyapunov-like function method, achieving the convergence of the tracking error to a user-defined residual set. A boundedness condition for the user-defined reference signal is defined, which is necessary to allow out-put tracking. The developed scheme avoids large commutation rate of the control input. The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, i conditions to guarantee the expected convergence of the tracking error are established. These conditions are imposed on the reference signal and the extreme values of the control input. The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat’s Lemma and includes ii the bounded nature of the Lyapunov function, iii the monotonic convergence of the Lyapunov function to a residual set, and iv the asymptotic convergence of the tracking error to a residual set of userdefined size.


Introduction
The design of two-valued feedback controllers has attracted important research 1-8 .It is intended for two-valued actuators, which lead to a control input that takes only two possible preassigned finite values.Thus, the control input is discontinuous with respect to time 1, 9 .The two-valued actuators have the following advantages with respect to proportional ones 1, 10, 11 : they are simple, relatively cheaper, lead to fast output response, and overcomes the issue of actuator static gain.The electrical kiln 5 , hydrogel valves 12 , the compressor 6, 13 , the satellite antenna 2, page 25 are some systems with two-value actuators.
A basic method for two-valued control is the relay feedback with fixed hysteresis band.The aim of the hysteresis is to avoid high commutation rate of the control input, what is known as input chattering 10, 13 and may lead to high power consumption and wear of mechanical components cf. 5, 6 .The width of the hysteresis band determines the commutation rate of the control signal and the size of the residual set to which the output error converges.The width has to be chosen to obtain a trade-off between commutation rate and size of the residual set, because one fact improves at the expense of the other 6 .An improper designed relay feedback may lead to overshoot, large amplitude oscillation of the output, or large settling time cf.10, 13 .These can be overcomed by two-valued control based on the direct Lyapunov method, as shown in 1 .In that paper the authors consider a single input single output SISO linear plant in controllable form, with time varying coefficients and additive disturbances.They ensure the convergence of the tracking error to a residual set whose size is user defined.They show that large commutation rate is avoided if the initial values of the tracking error and its time derivative are such that the initial value of the Lyapunov function is inside a target region of user defined size.
In the case of input output stable plants with fixed preset values of the control input extremes, a two-valued input implies the following: i the output remains inside some bounded region, regardless the controller, ii a user defined trajectory with periodic behavior and large frequency or large magnitude cannot be reached by the system output.The suitability of the frequency and magnitude of the user defined trajectory depends on the extreme values of the control input, and the plant model coefficients, as it will show in this work.Nevertheless, as far as we know, there is not a condition that indicates such suitability, in the literature on finite-valued control based on the direct Lyapunov method.In this work such condition is established.The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, i conditions to guarantee the expected convergence of the tracking error are established.These conditions are imposed on the reference signal and the extreme values of the control input.The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat's Lemma and includes ii the bounded nature of the Lyapunov function, iii the monotonic convergence of the Lyapunov function to a residual set, and iv the asymptotic convergence of the tracking error to a residual set of user defined size.
The work is only valid for a second-order plant with constant coefficients and an unknown external disturbance with known upper bounds.Upper bounds can be known from previous modelling tasks.A controller for the plant is developed, the controller is based on the direct Lyapunov method, ensuring the convergence of the tracking error to a residual set whose size are user defined.The work is organized as follows.In Section 2 the plant model and the goal of the control design is detailed.In Section 3 the condition that indicates if the desired output is suitable to achieve tracking is established.In addition, a preliminary rough control law, which achieves the convergence of the tracking error to a residual set whose size is user defined is formulated.Nevertheless, it may lead to large commutation rate of the control input.This will remedy in subsequent sections.In Section 4 the control law to avoid large commutation rate of the control input is formulated.In Sections 5 and 6 the boundedness of the closed loop signals and the convergence of the tracking error are proven.In Section 7 numerical simulation is presented.Finally, in Sections 8 and 9 discussion and conclusions are presented.

Problem Statement
The plant, the reference model, and the state goal of the control design are detailed in this section.Consider the following second-order plant systems analyzed with this model can be found in 14 where a m1 , a mo are positive constant of the user choice, the command signal r is bounded and user defined, y d is the desired output or reference signal, C be is a positive constant, user defined.It is important that the initial conditions y d t o , ẏd t o be chosen such that the initial value of the Lyapunov function be inside the target set, to avoid large commutation rate see 1 .This requirement is included in the control scheme.The aim of the reference model 2.5 is to provide an adequate nature of y d and ẏd , such that the control input can induce tracking while large input commutation rate is avoided.
The objective of the control design is to formulate a control law for the control input u, provided the plant model 2.1 , subject to assumptions Ai to Aiv , such that Gi the tracking error e t converges asymptotically to the residual set Ω e , Gii large commutation rate of the control input is avoided.Other goal of the control design is to develop a condition that indicates if a given desired output is suitable to achieve tracking.

A Preliminary Rough Control Law
In this section, a preliminary control law for the control input u, provided by the plant 2.1 , subject to assumptions Ai to Aiv is developed.The main contribution is that the tracking error asymptotically converges to Ω e , being Ω e defined in 2.6 .The Lyapunov-like function method, which is commonly used to design robust controllers for plants with continuous control inputs is used.A preliminary control law is developed.This control does not prevent large commutation rate but will be improved in subsequent sections.
Subtracting ÿd from both sides of 2.1

3.3
The term v is introduced with two objectives.The first one is for notational simplicity, and the second one is to simplify the design of a control law that overcomes the effect of y d and its time derivatives.Equation 3.2 can be rewritten as ẋ1 x 2 , ẋ2 −a 1 x 2 − a o x 1 bv,

3.5
Consider the following Lyapunov function: The time derivative of V along trajectory 3.
where v is defined in 3.3 , whereas c 1 is constant, user defined, and satisfies c 1 ∈ 0, 1 .Constant c 1 is introduced in 3.9 with the objective to change the commutation rate.A complete expression of the improved controller is given in 4.9 and it shows explicitly the usefulness of this constant.The above equation suggests that if the control law for u is properly defined, then

3.10
The condition V ≤ −c 1 a 1 x 2 1 implies that the tracking error converges asymptotically to a small value.To find a rough control law that achieves this, the term Sv can be written as

3.11
From assumption Aiv it follows that d/b S ≤ μ o /b |S|.Substituting this into 3.11 , it is obtained:

3.12
If the term in parenthesis is positive, then Sv ≤ 0. It can be achieved with the following rule: where the following property is needed.Substituting 3.17 in 3.15 , it follows Sv ≤ 0 for S / 0, and from 3.9 it follows that 1 if rule 3.13 is used.These results are summarized in the following Theorem.
Theorem 3.4.Consider the plant model 2.1 subject to assumptions (Ai) to (Aiv), the Lyapunov function V x t defined in 3.6 and the signal S defined in 3.8 .If condition 3.16 is fulfilled, and rule 3.13 is used, then

3.18
Therefore, the control law 3.13 implies the convergence of the tracking error x 1 to a small value.
Remark 3.5.The control rule 3.13 operates as follows.For t t o , if S x t o 0 the control input may take any on the values u mn , u mx .If S x t o / 0, the control input is defined by u u − δ sgn S x t o for t t o .The control input u retains its initial value until the signal sgn S x t changes its value with respect to sgn S t o .Then, u changes according to the rule u u − δ sgn S .The input u retains such value until the value of sgn S changes again, so that u changes according to u u − δ sgn S .This is repeated successively.If S 0 in some instant time, the input u does not change its value.Remark 3.6.According to 2, 7 , the use of discontinuous control law may lead to i loss of trajectory unicity, ii sliding motion of trajectories along the discontinuity surface, what may imply chattering see 2, pages 282-283 , and iii input chattering, which is an undesired large commutation rate component in the control input see 2, page 292 .Large commutation rate may lead to high power consumption and wear of mechanical components cf. 5, 6 .A rigorous design of a direct Lyapunov method should include the following tasks cf.7 : i ensure that trajectory unicity is preserved, ii develop the Filippov's construction for the case that sliding motion occurs, in order to avoid chattering.In the case under study, there may be sliding motion of the states x 1 , x 2 along the surface S 0. In 1 sliding motion for a closely related control scheme is illustrated.Thus, the control scheme 3.13 may lead to undesired large commutation rate in the input u when S takes on small values, so that goal Gii is not fulfilled.If large commutation rate is not a problem, sliding mode control could be used as an alternative approach 17-19 .
In next section, the control law given by 3.13 is improved.

The Final Control Law
In previous section it was formulated a control law that achieves adequate convergence of the tracking error but leads to large commutation rate.In this section, the convergence of the quadratic function V to a small residual set and adequate initial values of the Lyapunov function V are considered and the large commutation rate is overcome.In 1, 20 authors show that the convergence of Lyapunov function to a target manifold leads to the convergence of tracking error to a residual set of user defined size.From 3.6 it follows that if V converges to some small residual set of adequate size, then the tracking error e converges to the residual set Ω e defined in 2.6 .Proposition 4.1.Consider the function V defined in 3.6 , the tracking error e defined in 2.4 and the set Ω e defined in 2.6 .Let The convergence of V and e are related as follows: If V converges to Ω v , then e converges to Ω e , where C be is a positive constant defined by the user.The proof is presented in Appendix B.
Therefore, it is necessary to formulate a control law for u that ensures: so as to achieve the expected convergence of the tracking error.Indeed, if the above condition is ensured, then: i V converges asymptotically to Ω v , where ii if in addition V reaches Ω v for some instant, V remains inside thereafter, iii the tracking error e converges asymptotically to Ω e .Control scheme 3.13 can achieve condition 4.3 as mentioned in Theorem 3.4.does not involve the case V < C bv , then the control input u can take arbitrary values for V < C bv without disrupting the convergence of e to Ω e .As in 1 , it Mathematical Problems in Engineering is advisable to stop the commutation of u when V < C bv , in order to avoid large commutation rate: stops commutation otherwise.

4.4
This control rule implies that large commutation rate is prevented for V ≤ C bv , but not for V > C bv .Indeed, there may be sliding motion of the trajectories along S 0 if V > C bv , as occurs in 1 .A possible remedy is to impose a boundary layer around S 0 for V > C bv .From 4.3 it follows that commutation can be stopped when the condition V ≤ −c 1 a 1 x 2  1 is satisfied for V ≥ C bv under arbitrary values of the control input u.The restrictions over x 1 and x 2 ensuring the above condition will be determined at the following.Expression 3.9 can be rewritten as The requirement 3.16 implies that |v| is bounded by a constant, as the following proposition shows.

4.6
The proof is presented in Appendix C.
Substituting 4.6 into 4.5 yields Therefore, where b/a o |S||u mx − u mn | ≤ 0 defines a boundary layer around S 0 in the x 1 − x 2 state space, for V > C bv .The control law can take advantage of the above expression.Equation 4.8 indicates that it is possible to turn off commutation of the control input u while obtaining V ≤ −c 1 a 1 x 2 1 for V ≥ C bv if x 1 , x 2 satisfy the condition therein.The control law for u is then formulated as follows: i u follows the rule u u − δ sgn S for the case that V ≥ C bv and condition in 4.8 is not fulfilled, ii u follows the rule u u − δ sgn S for the case that V C bv and S / 0, and iii u stops commutation stops cm otherwise ow .Equivalently: Remark 4.3.The control law 4.9 operates as follows.For t t o , u follows 4.10 .The control input u retains its initial value until some of the conditions in 4.9 is fulfilled.At that instant time, the control input u follows the rule u u − δ sgn S , inducing the decrease of V .The input u retains such value until some of the conditions in 4.9 are fulfilled again.This procedure is repeated in the same way.Condition 3.16 should be fulfilled.Remark 4.4.Equation 4.9 indicates that the control signal u commutes as less as possible.When the commutation stops, the control signal keeps the value acquired during previous commutation mode.The commutation rate of the control input u does not reach excessive values, because if S becomes zero, then u stops commutation.
Remark 4.5.The constant C be cannot be zero, as we explain at the following.From 4.9 it follows that the control input u commutes when V C bv and S / 0. A value C be 0 would imply C bv 0, as it follows from the definition 4.2 .Therefore, the control input u would commute when V 0 and S / 0. Such condition is not possible, according to definition 3.6 .If C be is overly small, then C bv is also small, as it follows from the definition 4.2 .Therefore, the time that V takes to reach V C bv is small, implying a larger commutation rate, according to 4.9 .12 .The control input u retains its initial value until V C bv and S / 0. At that instant time, the control input u follows the rule u u − δ sgn S , inducing the decrease of V .The input u retains such value until V C bv and S / 0 is fulfilled again.This procedure is repeated in the same way.If V C bv and S 0, then u does not change.Condition 3.16 should be fulfilled.Notice that the constant c 1 is not necessary to formulate the control law 4.11 .

The discussion and simulation examples shown in 1 indicate that if condition
Remark 4.7.Equation 4.11 indicates that the control signal u commutes as less as possible.When the commutation stops, the control signal keeps the value acquired during previous commutation mode.The commutation rate of the control input u does not reach excessive values, because if S becomes zero, then u stops commutation.

Implementation Issues
In experimental implementation could be difficult to detect the exact moment when V C bv , so that it could be difficult to use the control scheme 4.11 .Then, with the aim to apply the control strategy to one system, it is possibility to use the control scheme given by 4.9 instead of 4.11 , because 4.9 includes the case V > C bv .Other possibility is to use a threshold δ V in 4.11 as follows: where δ V is a positive constant that satisfies δ V < C bv , and y d t o and ẏd t o are chosen such that

Boundedness Analysis
In From 4.9 it follows that

5.2
Therefore, it follows from Theorem 3.4 that and controller 4.9 is applied.From 4.7 it follows that or V C bv , S 0.

5.4
From 5.3 , 5.4 it follows that Since the above expression is not valid for V < C bv , it does not lead to a straightforward proof of the boundedness and convergence of V .In order to show that the Lyapunov function V is bounded, a Lyapunov-like function f a f a V that satisfies iii ḟa ≤ 0 ∀t ≥ t o , 5.6 will be used, being c a , c b , c c positive constants.If f a satisfies the above three conditions, then f a ∈ L ∞ and consequently V ∈ L ∞ .One example of such function is

5.7
The reader is referenced to 2, page 309 , 22, 23 for closely related functions.Its time derivative is: since V − C bv is positive or zero for V ≥ C bv , it can be multiplied by 5.12 without changing the order of the inequality: substituting it into 5.9 , it is obtained:

5.11
Then, f a satisfies properties 5.6 , as discussed at the following.From the definition 5.7 it follows that f a ≥ 0, V ≤ C bv 2f a , so that properties 5.6 i and 5.6 ii are satisfied.From 5.11 it follows that property 5.6 iii is true, so that f a ∈ L ∞ .Since 5.6 ii is true, then V ∈ L ∞ .From 3.6 it follows that x 1 ∈ L ∞ , S ∈ L ∞ , and from 3.8 it follows that x 2 ∈ L ∞ .This completes the proof.
If the control law 4.11 is applied to the plant 2.1 , and the initial value of the function V is located inside the target region, then V remains inside the target region thereafter, as is proven at the following.Theorem 5.2 boundedness of the Lyapunov function for V ≤ V x t o .Consider the plant model 2.1 , subject to assumptions (Ai) to (Aiv), the tracking error e, the desired output y d and the function S provided by 2.4 , 2.5 , and 3.8 , respectively.If condition 3.16 is fulfilled, the controller 4.11 is applied and Proof.The nature of V for V C bv must be examined considering each of the cases S 0 and S / 0 separately.From Theorem 3.4 it follows that if the control law 3.13 is used, S / 0 and V C bv , then V ≤ −a 1 x 2 1 .To show that this expression is also valid for the case when S 0 and V C bv , 3.9 is used.From 3.9 it follows that V ≤ −a 1 x 2 1 if S 0, regardless the value of V .Consequently, if the control law 4.11 is used, S 0 and V C bv , then V ≤ −a 1 x 2 1 .So far, it has been shown that if the control law 4.11 is used and V C bv , then 12 regardless the value of S. This implies that if and from 3.8 it follows that x 2 ∈ L ∞ .This completes the proof.

Convergence of the Tracking Error
In this section it is proven that if the controller 4.11 is applied to the plant model 2.1 , the tracking error e t converges to a residual set Ω e {e ∈ R : |e| ≤ C be }.
Equation 5.11 will be arranged into a single expression.Using

6.1
Equation 5.11 can be rewritten as A difficulty is that df b /dt is not continuous, as ∂f b /∂V is discontinuous at V C bv .Consequently, the Barbalat's Lemma can not be applied on f b .One remedy is to express 6.2 in terms of a new function whose first derivative with respect to V is continuous.One instance of that function is

Simulation Example
The aim of the following example is to show that the controller 4.11 achieves the benefits mentioned in Theorem 6.3.To that end, assumptions Ai to Aiv of Section 2 and condition 3. 16 1.The chosen values of λ r , y d t o , ẏd t o imply that y d ≈ r for all t ≥ t o .Since a 1 1, the signal S defined in 3.8 is then S a 1 e ė e ė.The closed loop behavior of y, e, u is shown in Figure 2, whereas the state plane is shown in

Discussion
The discussion presented here is only valid for the plant model 2.1 , which is a model second order with known constant coefficients and an additive disturbance.The time derivative of the function V defined in 3.6 results in a negative semidefinite term, so that it is necessary to use Lyapunov-like function method to prove the convergence of the tracking error.Hereafter, some conclusions based on both stability proofs and numerical simulations are stated.In the classical relay feedback method 10, 13 the commutation of the control input is function of the tracking error, but it does not take into account the time derivative of the tracking error.
The developed condition 3.16 indicates wether a given desired output y d is suitable to achieve the convergence of the tracking error e to a residual set of user defined size, that is, Ω e , for the plant 2.1 .For any instance of the plant 2.1 , one may modify the coefficients of the reference model 2.5 to satisfy condition 3.16 , by means of trial and error.This procedure does not involve the output y, nor the input u, and is previous to the implementation of the controller.
It is important to set the initial values of the desired output and its time derivative such that V x t o ≤ C bv .This implies that the Lyapunov function V is always located inside the target region Ω v {V ∈ R : V ≤ C bv }, so that it avoids convergence towards this region, and consequently large commutation rate of the control input are avoided.
The contribution of the scheme with respect to classical relay feedback control based on hysteresis is to ensure the convergence of the tracking error e t to a residual set whose size is user defined.The main contribution with respect to closely related control based on the direct Lyapunov method is that the condition that the desired output y d has to fulfill in order to achieve tracking is defined.

Conclusions
The controller achieves the convergence of the tracking error to a residual set of user defined size if the desired output satisfies the formulated condition.This condition uses the upper bound of the additive disturbance, as in a basic nonadaptive robust controller.It allows us to develop a rigorous proof of the tracking error convergence to a residual set that is user defined, by means of the Lyapunov-like function.If the initial values of the desired output and its time derivative are properly defined, the Lyapunov function is located inside a target region at initial time and thereafter.In this case, the control input only commutes when the Lyapunov function reaches the boundary.

B.1
The value of C * b will be fond out by means of two different ways.On the one hand, it follows from 3. C.8 This completes the proof.

3 . 1 since
the first-and second-time derivatives of the tracking error are ė ẏ − ẏd , ë ÿ − ÿd , it is obtained: ë −a 1 ė − a o e bu − ÿd − a 1 ẏd − a o y d d −a 1 ė − a o e bv, 3.2 v u − ÿd a 1 ẏd a o y d b d b .

Figure 2 :Figure 3 :
Figure 2: Example 1, a output y continuous line and desired output y d dashed line ; b tracking error e t ; c control input u.

Aÿd a 1
ẏd a o y d b − u ≤ δ − μ o b .Proposition 4.1 will be proven by finding the value of a positive constant C * b such that:

B. 2 FromSee 2 , 4 From
6 and the definition of Ω v in B.1 that 1 2a o S 2 ≤ V, ⇒ |S| ≤ 2a o V ⇒ if V converges to Ω v , then S converges to Ω s , Ω s S : |S| ≤ 2a o C * b .the definitions 3.8 , 3.5 , S can be expressed as a linear function of e, ė: S a 1 e ė.Thus, e in terms of S is given by this equation and according to 2, page 279-280 , and 29 , it follows: If S converges asymptotically to Ω s , then e converges asymptotically to Ω e , B.4 where Ω s S : |S| ≤ 2a o C * b , Ω e e : |e| ≤ 1 a 1 2a o C * b .B.5 This and B.2 imply: if V converges to Ω v , then e converges to Ω e , Ω e {e : |e| ≤ c a }, page 279-280 and 29 for closely related results.The value of C * b that leads to c a C be has been found, thus B.1 is satisfied.This value is C * b a 2 1 C 2 be / 2a o , which is a first value of C * b .On the other hand, it follows from 3.6 that 1 2 e 2 ≤ V ⇒ |e| ≤ √ 2V .B.7This and the definition ofΩ v in B.1 imply if V converges to Ω v ,then e converges to Ω e , Ω e {e : |e| ≤ c c }, value of C * b that leads to c c C be has been found out so that B.1 is satisfied.This value is C * b 1/2 C 2 be , which is a second value of C * b .Since both values of C * b are valid, then C bv can be defined as the maximum: a 1 ẏd a o y d b − μ o b ≥ u mn .C.1 with C.5 and C.2 with C.6 , and using definition 3.3 , yields −v ÿ a 1 ẏd a o y d b − d b − u ≤ u mx − u mn , −v ÿ a 1 ẏd a o y d b − d b − u ≥ u mn − u mx , C.7 combining the above equations yields: − u mx − u mn ≤ −v ≤ u mx − u mn , ⇒ |−v| ≤ u mx − u mn , |v| ≤ u mx − u mn .
R is the system output, u t ∈ R is the input, a o , a 1 , b are plant coefficients, being b the control gain, and d is an uncertain term that may result from modelling error or an external disturbance.Let us consider the following assumptions.Ai The coefficients a o , a 1 , b are constant, known, and positive.Aii The signals y, ẏ are available for measurement.Aiii The values of u mn , u mx are constant, user defined, satisfy u mx > u mn , and are not restricted to positive values.Aiv The uncertainty d is time varying and satisfies either i |d| ≤ μ o , μ o > 0, where μ o is a known positive constant, or ii d μ o , μ o 0.
2.3 Now, the control goal can be established.Let e t y t − y d t , 2.4 ÿd −a m1 ẏd − a mo y d a mo r, 2.5 Ω e {e ∈ R : |e| ≤ C be }, 2.6 can take any of the values u mn or u mx , ii if V x t o < C bv , u can take any on the values u mn , u mx , iii if S x t o / 0 and V x t o C bv , u takes on the value u u − δ sgn S x t o ,iv if V x t o > C bv and − 1 − c 1 a 1 x 2 1 b a o |S| u mx − u mn > 0,u takes on the value u u − δ sgn S x t o 4.10 the signals necessary for the computation of u are: u and δ 3.14 , C bv 4.2 , S 3.8 , e 3.5 , y d provided by 2.1 , c 1 is a user defined positive constant.
≤ C bv for all t ≥ t o and large commutation rate is avoided.Indeed, from definition 3.6 it follows that the condition V x t o ≤ C bv is fulfilled if e t o , ė t o have adequate magnitude, or equivalently, if the distance between y d t o and y t o , and distance between ẏd t o and ẏ t o are adequate.Thus, the following control strategy is chosen: , ẏd t o are chosen such that V x t o ∈ Ω v , Ω v {V x t ∈ R : V x t ≤ C bv }.For t t o , the signal control is computed as: i if V x t o < C bv , u can take any on the values u mn , u mx , ii if S x t o / 0, V x t o C bv , u takes on the value u u − δ sgn S x t o , iii if S x t o 0, V x t o C bv , u can take any on the values u mn , u mx, 4.12 Remark 4.6.The control law 4.11 operates as follows.For t t o , u follows 4.
this section we analyze the boundedness properties of the closed loop signals.As in 21 , the notation • ∈ L ∞ is used.This notation indicates that • is bounded.boundedness of the closed loop signals .Consider the plant model 2.1 , subject to assumptions (Ai) to (Aiv), the tracking error e, the desired output y d and the function S provided by 2.4 , 2.5 , 3.8 , respectively.If condition 3.16 is fulfilled and the controller 4.9 is applied, then the signals x 1 , x 2 , S remain bounded.
have to fulfill.Considerations of Section 4 are taken into account.Consider the plant: example of the plant 2.1 with a 1 1, a o 1, b 1, u mn −1.2, u mx 1.2, μ o 0.11.Therefore, assumptions Ai to Aiv stated in Section 2 are fulfilled.Since a o > 1/4 a 2 1 , the linear part of the plant is underdamped.The aim is that y converges towards the value r cos 0.5t with an accuracy of 0.05.Thus, set C be 0.05 and y d is defined by means of the second order system: Figure3.The state plane confirms that V x t o ≤ C bv , so that V x t ≤ C bv for all t ≥ t o and u only commutes when V C bv .The figure of the control input u and the state plane indicate that large commutation rate is avoided.This confirms the importance of choosing y d t o , ẏd t o such that V x t o ≤ C bv .