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Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems Ivan Salgado a, Isaac Chairez b,n, Oscar Camacho a, Cornelio Yañez a a
Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico city, Mexico Unidad Profesional Interdisciplinaria de Biotecnología, Instituto Politécnico Nacional, Av. Acueducto de Guadalupe s/n. Col. La Laguna Ticoman, Mexico city, Mexico
b
art ic l e i nf o
a b s t r a c t
Article history: Received 28 December 2013 Received in revised form 14 March 2014 Accepted 3 April 2014 This paper was recommended for publication by Jeff Pieper
Designing a proportional derivative (PD) controller has as main problem, to obtain the derivative of the output error signal when it is contaminated with high frequency noises. To overcome this disadvantage, the supertwisting algorithm (STA) is applied in closed-loop with a PD structure for multi-input multioutput (MIMO) second order nonlinear systems. The stability conditions were analyzed in terms of a strict non-smooth Lyapunov function and the solution of Riccati equations. A set of numerical test was designed to show the advantages of implementing PD controllers that used STA as a robust exact differentiator. The first numerical example showed the stabilization of an inverted pendulum. The second example was designed to solve the tracking problem of a two-link robot manipulator. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Signal differentiation Sliding modes PD controller
1. Introduction 1.1. Motivation The classical proportional-derivative (PD) and proportionalintegral-derivative (PID) controllers are the most successful techniques implemented in real applications [27,31]. Unfortunately, for unknown plants or systems with high frequency noises at the output, the PD controllers present an evident disadvantage regarding the need for calculating the error derivative [1]. Several approaches have been designed for improving the robustness of classical PD and PID control schemes. These researches have been oriented in two directions: (a) modifying the complete structure of the PD controller and (b) proposing robust techniques for numerical differentiation or state estimation [30]. For the first case, in [2], the classical PD controller structure was tuned with reinforcement learning based on intelligent techniques like fuzzy logic and neural networks. Robust techniques based on sliding modes have also been proposed [26]. The twisting algorithm is a discontinuous second order sliding mode controller that forces finite time convergence for nonlinear systems with a sliding
n
Corresponding author. E-mail addresses: [email protected] (I. Salgado), [email protected] (I. Chairez), [email protected] (O. Camacho), [email protected] (C. Yañez).
surface that has relative degree two with respect to the input signal. The second approach requires the use of state estimators to reconstruct the unmeasurement states (velocity for this case). State estimators based on the Luenberger structure are the most popular solution. However, the exact description of the model associated with the signal is demanded and the observer parameters cannot be tuned easily for reducing sensitivity under measurement noises or perturbations [17,6]. Observers based on fuzzy logic techniques provide robust estimation for uncertain nonlinear systems, see for example [15,28] where small gain observers were designed. On the other hand, high gain observers and extended observers were compared in [16]. When the observer uses position measurements for estimating velocity, the estimator turns to signal differentiators. The most useful methods to approximate the numerical derivative are based on linear filters [32,6]. The main approach for designing a linear differentiator is to approximate a transfer function within a finite frequency band [12]. For nonlinear signal differentiators, a complete review can be found in [18]. For uncertain, perturbed and even unknown systems, sliding modes based differentiators are a suitable option for overcoming the main drawbacks exhibited by other differentiators [29,12]. 1.2. Sliding mode theory for numerical differentiation The main characteristics offered by sliding modes are robustness against parametric uncertainties, external perturbations and finite
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time convergence [29,23]. The first order signal differentiator based on second order sliding modes is the well known super-twisting algorithm (STA) [13]. A complete analysis on second order sliding mode theory was described in [11,14]. The STA has been successfully applied as a controller [8], state estimator [5] and robust exact differentiator (RED) [12]. The stability and finite-time convergence analysis for the STA has been jointly addressed from different points of view, most of them based on the use of scaling symmetries (homogeneity), nonsmooth Lyapunov functions based on the Zubov theorem [25,3,20] and proposing a change of coordinates to analyze the STA with conventional techniques [22]. Moreover, by means of a nonsmooth Lyapunov function the gain selection can be easily obtained for ensuring the STA finite-time convergence [19]. In this way, the emerging of these Lyapunov functions brings out new opportunities to develop several novel applications of the STA that can include observers [19] and adaptive gain versions [8]. In particular, these Lyapunov functions have also been successfully applied to obtain uniform convergence for the STA [4].
1.3. Contribution of the paper In this paper, a novel scheme for controlling partially unknown second order non-linear systems was designed using a basic PD structure in the closed-loop with the STA applied as a RED [12]. The stability of this controller was analyzed using the ideas proposed in [20,19]. This paper analyzed a different contribution from the one presented in [20,19], where only the stability of the differentiator was considered. In this paper, the STA was implemented in the closed-loop with a classical PD scheme. The use of non-smooth Lyapunov functions allowed a constructive method to select an adequate combination of gains for the modified PD controller. The Lyapunov function used to complete the stability analysis considered the full extended system formed by the STA and the PD controller. The contribution proposed in this paper did not use the concept of separation principle when the RED converged. The approach considered the transient period that provides a more accurate solution for the close-loop problem [1]. The behavior of a PD controller can provide asymptotic stability in steady state only if the controlled system is not exhibiting any perturbation in the output signal. Otherwise, for a bounded perturbation, the tracking error will exhibit only ultimately boundedness that coincides with the result obtained in this study. In [9] a first order sliding mode term was added to a PD structure for controlling robot manipulators. In this paper, a second control structure was designed in order to enforce finite time convergence. The classical PD controller structure was slightly modified to include a first order discontinuous term [9]. The Lyapunov analysis for this structure was also studied and the corresponding gains for the controller and the differentiator were obtained. The proposed controllers were evaluated in a simple pendulum and in a two link robot manipulator in order to illustrate the benefits and drawbacks offered by them. Several comparisons were made with a classical PD controller with a linear filter derivative and second order sliding mode controllers [7]. The rest of the paper is organized as follows: in Section 2 the class of nonlinear systems to deal with is introduced as well as the STA working as a RED. In the same section, the extended system that incorporates the STA to estimate the derivative of the error signal in the closed loop with the PD controller is given. In Section 3 the main result is introduced. Numerical results are presented in Section 4 and finally in Section 5, some conclusions are given.
1.4. Notation In this paper, the following notation is used: Rn represents the vector space with n-components. As usual, > is used to define the transpose operation. J z J is used to define the Euclidean norm of z A Rn . ‖z‖2H ≔z > Hz is the weighted norm of the real-valued vector
z A Rn with weight matrix H 4 0, H ¼ H > , H A Rnn . If two matrices N A Rnn and M A Rnn fulfill M 4 Nð Z Þ that means that M N is a positive definite (semidefinite) matrix. The symbol R þ represents the set of positive real scalars. The symbols I nn and 0nn are used to represent the identity matrix and the matrix formed with zeroes of dimension n n.
2. Super-twisting PD controller 2.1. Class of nonlinear systems Consider the nonlinear system described by the following second order nonlinear differential equation: z€ ðtÞ ¼ f ðzðtÞ; z_ ðtÞÞ þ gðzðtÞÞuðtÞ þ ηðz_ ðtÞ; zðtÞ; uðtÞ; tÞ yðtÞ ¼ zðtÞ zð0Þ ¼ z0 and z_ ð0Þ ¼ zd0 given z0 ; zd0 A Rn
ð1Þ
Here z A R and z_ A R , zð0Þ and z_ ð0Þ, are the initial conditions for the differential equation. The drift term f : R2n -Rn is a Lipschitz function and the input associated term g : Rn -Rnn is bounded as it will be explained later. The nonlinear function η : R2n þ 1 -Rn represents some uncertainties affecting the nonlinear system satisfying 2 z þ ‖η‖2 r η0 þ η1 ð2Þ z_ ; η0 ; η1 A R n
n
The signal y A Rn is the available output vector. The control action is represented by u A Rn . The class of systems considered in (1) is a rough generalization of many mechanic, electromechanical, electric, thermodynamic and hydrodynamic systems. The system presented in (1) can be represented as (with the selection of xa ¼ z and xb ¼ z_ ) x_ a ðtÞ ¼ xb ðtÞ x_ b ðtÞ ¼ f ðxðtÞÞ þ gðxa ðtÞÞuðtÞ þ ηðxðtÞ; uðtÞ; tÞ yðtÞ ¼ CxðtÞ x ¼ ½xa>
ð3Þ xb> >
2n
A R and C ¼ ½I nn ; 0nn . where Throughout the paper, the following assumptions are assumed to be fulfilled. Proposition 1. The nonlinear function f ðÞ is unknown but satisfies the Lipschitz condition J f ðxÞ f ðx0 Þ J r L1 J x x0 J ;
8 x; x0 A R2n ; L1 A R þ
ð4Þ
Proposition 2. The nonlinear system (3) is controllable, therefore the function gðxÞ is known and it satisfies 0 o g r J gðxa Þ J r g þ o1;
8 xa A Rn ;
g ; g þ ARþ
ð5Þ
By this assumption, the matrix gðxa Þ is invertible 8 t Z 0. Proposition 3. The control input belongs to the set Uadm defined as U adm ≔fu : ‖uðtÞ‖2 r u0 þ u1 ‖x‖2 g
ð6Þ
þ
with u0 ; u1 A R . The previous condition includes several control techniques such as classical PD controllers and even discontinuous controllers such as sliding modes. 2.2. Nonlinear reference system The problem considered in this paper was to complete the trajectory tracking between the states of (1) and the stable
Please cite this article as: Salgado I, et al. Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.04.003i
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The previous differential equation is a state representation of the signal rðtÞ. The scalar STA to obtain the derivative of rðtÞ looks like
reference model given by n z€ ðtÞ ¼ hðz_ n ðtÞ; zn ðtÞÞ; zn ð0Þ; z_ n ð0Þ are given
yn ðtÞ ¼ zn ðtÞ
ð7Þ
where hðzn ; z_ n Þðzn A Rn Þ is a Lipschitz function. Again, system (7) can be transformed using the state space method with the change of variables xna ¼ zn and xnb ¼ z_ n . The reference system (7) has a stable equilibrium point and by the converse Lyapunov theorem [10], one can ensure that the system in the new coordinates xn ¼ ½ðxna Þ > ðxnb Þ > > satisfies ‖xn ðtÞ‖2 r X nþ ;
xn ¼ ½ðzn Þ > ; ðz_ n Þ > >
ð8Þ
then, the next inequalities are assumed to be valid þ
n
Λ ¼ Λ > 40;
Λ A Rnn
ð9Þ
2n
solution of (7), the last inequality is valid because 8x AR functions f ðxn Þ and hðxn Þ are Lipschitz (continuous) functions and (7) has a stable equilibrium point.
2.3. PD controller design In general, a PD controller is designed using the following structure: _ uðtÞ ¼ k1 eðtÞ k2 eðtÞ
d w 1 ðtÞ ¼ w 2 ðtÞ λ1 jw 1 ðtÞ w1 ðtÞj1=2 signðw 1 ðtÞ w1 ðtÞÞ dt d w 2 ðtÞ ¼ λ2 signðw 1 ðtÞ w1 ðtÞÞ dt dðtÞ ¼
d w 1 ðtÞ dt
ð10Þ
ð13Þ
where λ1 ; λ2 40 are the STA gains. Here d(t) is the output of the differentiator [12]. In this equation 8 if z 4 0 > <1 signðzÞ≔ A ½ 1; 1 > : 1
X nþ A R þ ; 8 t Z 0
‖f ðxn Þ hðxn Þ‖2Λ r h ;
3
if z ¼ 0
To apply the STA as a differentiator, let us represent the uncertain system (3) as the composition of the following n second order systems: x_ a;i ðtÞ ¼ xb;i ðtÞ x_ b;i ðtÞ ¼ f i ðxðtÞÞ þ g i ðxa ðtÞÞui ðtÞ þ η i ðxðtÞ; uðtÞ; tÞ ð15Þ
i ¼ 1; n
where xa;i and xb;i are the i-th and ðn þ iÞth states of (3), respectively. The nonlinear functions f i ðÞ and g i ðÞ are the functions associated with the states xa;i and xb;i . Similarly, ηi ð; Þ is the corresponding uncertainty that affects the same subsystem. In this paper, x_ b;i ðtÞ could be affected by every component of the input vector uðtÞ. The following proposition explaining how the product gðÞuðtÞ is handled. Proposition 4. For the system given in (15), the product of elements g i;j uj for i,j ¼1:n are included in η i ðxðtÞ; uðtÞ; tÞ, that is
η i ðxðtÞ; uðtÞ; tÞ ¼ ηi ðxðtÞ; tÞ þ
n
where k1 ; k2 A R are the controller gains that must be adjusted and e A Rn is the output error given by eðtÞ≔CxðtÞ Cxn ðtÞ
ð11Þ
However, this controller is hardly to be implemented considering _ that eðtÞ and eðtÞ are rarely measured simultaneously without an important resources investment. Therefore, in the classical literature, one can find two important solutions: to construct an observer or using a first order filter to approximate the error derivative. The first one requires the system structure (that is in this paper is assumed to be unknown) and in the second case, the derivative approximation is usually poor specially if the output information is contaminated with noises. One additional option is considering a class of RED that can provide a suitable and accurate approximation of the error derivative despite the presence of noises. STA has demonstrated to be one of the best RED several times.
2.3.1. Super-twisting algorithm The STA application as a RED is described as follows. If w1 ðtÞ ¼ rðtÞ where rðtÞ A R is the signal to be differentiated, w2 ðtÞ ¼ r_ ðtÞ represents its derivative and under the assumption of jr€ ðtÞj r r þ , the following auxiliary differential equation is got: _ 1 ðtÞ ¼ w2 ðtÞ w _ 2 ðtÞ ¼ r€ ðtÞ w
ð14Þ
if z o 0
n
∑
j ¼ 1;j a i
g i;j ðxa ðtÞÞuj
with i a j and j ¼ 1 : n. The term ηi ðxðtÞ; uðtÞ; tÞ is the i-th component of ηðxðtÞ; uðtÞ; tÞ. By Propositions 2 and 3, each element g i;j ðxa ðtÞÞuj is bounded as follows: jg i;j ðxa ðtÞÞuj j2 rg þ ðu0 þ u1 ‖x‖2 Þ
8t Z0
Considering that the kind of perturbations that can affect the nonlinear system (15) satisfies jηi ðxðtÞ; uðtÞÞj r η0 þ η1 ‖x‖2 Then, the joint perturbation η i ðxðtÞ; uðtÞ; tÞ is bounded by jη i ðxðtÞ; uðtÞ; tÞj r η 0;i þ η 1;i ‖x‖2
η 0 ¼ η0 þ g þ u0 η 1 ¼ η1 þ g þ u1
ð16Þ
Before the STA can be applied, the dynamics of the tracking error also must be treated as in (15). The tracking error is defined as e ¼ x xn . The definition for the individual elements of the vector e is given by e_ i ¼ x_ a;i x_ na;i
ð17Þ
and the complete dynamics for the tracking error are composed by the following differential equations: e_ i ðtÞ ¼ ei þ n ðtÞ e_ i þ n ðtÞ ¼ f i ðxðtÞÞ þ g i ðxa ðtÞÞui ðtÞ þ η i ðxðtÞ; uðtÞ; tÞ hi ðxn ðtÞÞ
ð12Þ
ð18Þ
n
where the function hi ðx ðtÞÞ is the i-th component of the vector field hðxn ðtÞÞ.
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2.3.2. PD controller enhancement with the super-twisting algorithm Using the STA, the i-element of the PD controller was proposed as ui ðtÞ ¼ k1;i ðtÞei ðtÞ k2;i ðtÞ di ðtÞ
ð19Þ
where di is the output of the corresponding RED applied over ei. The gains for the i-element of the PD controller were selected as k1;i ðtÞ ¼ k2;i ðtÞ ¼
g i 1 ðxa ðtÞÞk 1;i g i 1 ðxa ðtÞÞk 2;i
ð20Þ
with k 1;i and k 2;i positive scalars. The following extended system describes the complete dynamics of the error signal in the close-loop with the implementation of (13) e_ i ðtÞ ¼ ei þ n ðtÞ e_ i þ n ðtÞ ¼ f i ðxðtÞÞ hi ðxn ðtÞÞ g i ðxÞðk1;i ðtÞei ðtÞþ k2;i ðtÞdi ðtÞÞ þ η i ðxðtÞ; uðtÞ; tÞ
δ_ 1;i ðtÞ ¼ δ2;i ðtÞ λ1;i jδ1;i ðtÞj1=2 signðδ1;i ðtÞÞ δ_ 2;i ðtÞ ¼ λ2;i signðδ1;i ðtÞÞ e€ i ðtÞ
ð21Þ
Theorem 1. Consider the nonlinear system given in (1), supplied with the control law (19) that is adjusted with the gains given in (20) and the derivative of the error signal obtained by (13). If there exist a positive scalar αi and if the gains λ1;i , λ2;i are positive, such that the next Lyapunov inequalities have positive definite solutions P1,i > A1;i P 1;i þ P 1;i A1;i r Q 1;i
2λ2;i
P 2;1
6 6 022 P2 ¼ 6 6 ⋮ 4 022
022
⋯
P 2;2
⋯
⋮
⋱
022
⋯
022
3
7 022 7 7 ⋮ 7 5 P 2;n
ð25Þ
and 1 1 1 þ γ i ¼ 2λmax fΛb;i ghi þ 2λmax fΛb;i gη 0;i þ4λmax fΛb;i gX nþ ;i
ð26Þ
The last theorem was useful to prove practical stability for the tracking error despite the exact convergence provided by the STA of the tracking error derivative. Interestingly, a simple modification of the control action yields to a even more robust result with better convergence properties. The following lemma gives the result associated with this modification. Lemma. If each controller ui ðtÞ is modified to ui ðtÞ ¼ k1;i ðtÞei ðtÞ k2;i ðtÞ di ðtÞ k3;i ðtÞ
si ðtÞ jsi ðtÞj
ð27Þ
where si ¼ C i> Ei with C i A R2 , C i> ¼ ½c1 ; c2 with the control gains ks;i , s¼1, 2, 3 selected as k2;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 2;i
The main result in this paper can be considered as the form of coupling the STA with a classical PD controller with a non-smooth Lyapunov function. In this way, the problem statement to deal with, is To select, an adequate combinations of gains λ1;i ; λ2;i for the STA and k 1;i , k 2;i for the PD controller described in (19) and (20), such that, the error derivatives δ1;i and the output tracking errors ei converge into a boundary layer characterized by the power of uncertainties and noises. The main result of this paper is summarized in the following theorem:
A1;i ¼
where 2
ð24Þ
k1;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 1;i
3. Main result
λ1;i
γi α i¼1 i n
lim E > ðtÞP 2 EðtÞ r ∑
t-1
Proof. The proof is given in the Appendix.
where δ1;i ¼ w 1;i ei and δ2;i ðtÞ ¼ δ_ 1;i ðtÞ. The term di refers to the approximation of ei þ n which is obtained by means of the second order STA presented in (13) and the variable w 1;i is the first state of the corresponding STA implemented as the differentiator. The constants λ1;i , λ2;i were selected according the main theorem that will be introduced below. A set of n differentiators were used to reconstruct the information of ei þ n . The following section shows the main result of this paper. The theorem introduced here gives a constructive way to select the gains of the STA and the PD controller.
"
bound
1 0
#
;
> 4 0; Q 1;i ¼ Q 1;i
Q 1;i A R22
ð22Þ
k3;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 3;i
ð28Þ
þ
k 3;i ¼ μi þ k 3;i
μi ≔4ðhiþ Þ1=2 þ 4η01=2 þ 4ðη1 X þþ Þ1=2 ;
þ
k 3;i 4 0
ð29Þ
and if there exists a positive semidefinite solution for the following matrix inequalities: 1
> P 3;i A3;i þ A3;i P 3;i þ P 3;i Ξ a;i P 3;i þ Ξ a;i r 0
ð30Þ
> a;i 4 0,
with Ξ a;i ¼ Li Ξ a;i and Ξ a;i ¼ Ξ Ξ a;i A R then the trajectories of si ðtÞ converge in finite time Tci to the origin with a convergence time of pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2;i ð0Þ rT ci ð31Þ þ 2k 3;i J C i J 2n2n
>
where V 2;i ð0Þ ¼ s2i ð0Þ and the extended vector z ¼ ½ξ ; s > > converges in finite time to the origin with a convergence time T given by n
n
i¼1
i¼1
∑ T ci þ ∑ T ai rT
ð32Þ
and if for every positive value of L1 satisfying Eq. (4) and a positive þ value h defined in (9), there exist positive gains k 1;i , k 2;i such that the Riccati equations given by
where Tai was defined in the main theorem of this study.
P 2;i ðA2;i þ αi IÞ þ ðA2;i þ αi IÞ > P 2;i þ P 2;i R2;i P 2;i þ Q 2;i r0 " # 0 1 A2;i ¼ ; R2;i ¼ Λa;i þ Λb;i k 1;i k 2;i
Vðξ; EÞ ¼ ∑ V i ðξi ; si Þ
Proof. Consider the Lyapunov candidate function given by n
i¼1
V i ðξi ; Ei Þ ¼ V 1;i ðξi Þ þ V 2;i ðsi Þ
1
Q 2;i ¼ 4λmax fΛb;i gI 22 þ Λ a;i ;
Λ a;i ¼ Li Λa;i ; Λa;i ; Λb;i ; Λc;i 40 and symmetric; Λa;i ; Λb;i ; Λc;i A R22 ; αi A R þ ð23Þ have positive definite solutions P 2;i . Thus, the trajectories of E > ¼ ½e1 ; …; en ; en þ 1 ; …; e2n are globally ultimately bounded with
ð33Þ
The first section of the with V 1;i ðξi Þ ¼ ðξi Þ P 1;i ξi and V 2;i ðsi Þ ¼ previous function V 1;i ðξi Þ is analyzed in the same way as the previous theorem. Now, continuing the analysis for the second section of the candidate Lyapunov function V 2;i ðsi Þ, its full time derivative is >
V_ 2;i ðtÞ ¼ 2si ðtÞs_ i ðtÞ
s2i .
ð34Þ
Please cite this article as: Salgado I, et al. Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.04.003i
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where s_ i ðtÞ ¼ C i> E_ i . The substitution of E_ i in the full time derivative of V 2;i ðsi Þ yields to 0 1 V_ 2;i ðtÞ ¼ 2si ðtÞC i> Ei ðtÞ þ 2si ðtÞC i> N 1 ðxðtÞ; xn ðtÞÞ 0 0 þ 2si ðtÞC i> N 2 ðxðtÞ; xn ðtÞÞ þ 2si ðtÞC i> N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ ð35Þ n
n
Considering the definitions of N 1 ðxðtÞ; x ðtÞÞ, N 2 ðxðtÞ; x ðtÞÞ and N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ used in the proof of the main theorem of this paper, one gets ! " # 0 1 1 V_ 2;i ðtÞ r 2si ðtÞC i> Ei ðtÞ þ si ðtÞC i> Ξ a C i si þ Ei> ðtÞΞ a Ei ðtÞ k 1;i k 2;i þ μi J C i J jsi ðtÞj þ 4η1 J Ei ðtÞ J J C i si ðtÞ J 2si> ðtÞC i> k 3;i 1=2
si ðtÞ jsi ðtÞj
ð36Þ
with μi defined as in (29). This last inclusion can be arranged as follows: 1 > V_ 2;i ðtÞ r Ei> ðtÞ½P 3;i A3;i þ A3;i P 3;i þ P 3;i Ξ a P 3;i þ Ξ a Ei ðtÞ þ ðμi k 3;i ÞJ C i J jsi ðtÞj
A3;i ¼ A2;i þ π I 2n2n
ð37Þ
with P 3;i ¼ C i C i> and π Z 2ðη1 X nþ Þ1=2 . By the assumption given in the corollary statement (30) and considering the gain selection given in (29), then þ V_ 2;i ðtÞ r k 3;i J C i J jsi ðtÞj
ð38Þ
Now, taking into consideration the structure of V 2;i , one gets qffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð39Þ V_ 2;i ðtÞ r k 3;i J C i J V 2;i ðtÞ Taking just the case when the previous inclusion became into an ordinary differential equation and redefining V 2;i ðtÞ as V eq 2;i ðtÞ, one can prove that qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ V eq ð40Þ V eq 2;i ðtÞ ¼ 2;i ð0Þ 2k 3;i J C i J t By the comparison lemma [19] and due to the function V 2;i ðtÞ Z 0, one can prove that 0 r V 2;i ðtÞ r V eq 2;i ðtÞ and V 2;i ðtÞ ¼ 0;
8 t Z T ci
This result finished the proof of this lemma.
ð41Þ □
5
control strategies were selected as Controller
Controller law
PDSTA
u ¼ Jð60e1 ðtÞ þ 49d1 ðtÞÞ
PD
u ¼ Jð110e1 ðtÞ þ 85d1 ðtÞÞ
TWT
u ¼ Jð90 signðe1 ðtÞÞ þ 75 signðd1 ðtÞÞÞ
SPDSTA
u ¼ Jð60e1 ðtÞ þ 20 signðe1 ðtÞÞ þ 49d1 ðtÞÞ
ð43Þ
and the gains for the STA were chosen as λ1 ¼ 10 and λ2 ¼ 9. The solutions for the Lyapunov and Riccati inequalities given in (22) and (23) were obtained according to the assumptions described by Eqs. (4)–(6). Eq. (22) with the values showed in (43) and the matrix Q1 defined as 10 1 A1 ¼ 18 0 " # 590 50 λ1 2λ2 þ λ21 λ1 ¼ ð44Þ Q1 ¼ 2 50 5 λ1 1 The positive definite solution for the matrix Lyapunov inequality with the previous parameters was 29:5139 0:1389 P1 ¼ ð45Þ 0:1389 582:639 For the case of the Riccati equation, the reference signal for the inverted pendulum was selected as a constant, that is xn ðtÞ ¼ π =2. Then, hðxn Þ ¼ 0. Taking into account the worst case when the pendulum position is in 0 rad, and if a DC motor with maximum speed at a free load of 57 rpm was used. The variable L defined in (4) was obtained as follows: 2 3 π 0 4 5 jf ðxÞ f ðxn Þj rL ð46Þ 1:9π 2 ; L Z 0:6822 0 Using this value, selecting Λa ¼ Λb ¼ I 22 and α ¼1, the parameters of the Riccati inequality (23) became 0 1 A2;i ¼ ; R2;i ¼ 2I 22 ; 60 49 Q 2;i ¼ 4:6828I 22
ð47Þ
4. Numerical results 4.1. Stabilization of an inverted pendulum As an illustration of the results presented in this paper, the PD control was applied to control a nonlinear pendulum system. The performance of the regular PD controller and the modified PD controller (supplied by the STA) was presented. Consider a pendulum represented by [20] x_ 1 ðtÞ ¼ x2 ðtÞ 1 MgL Vs sin ðx1 ðtÞÞ x2 ðtÞ þ ηðtÞ x_ 2 ðtÞ ¼ uðtÞ J 2J J y ¼ x1 ðtÞ
ð42Þ
where x1 ¼ θ is the angle of oscillation (rad), x2 ¼ θ_ is the angular velocity (rad/s), M is the pendulum mass, g is the gravitational force, L is the pendulum length, J is the inertia arm, Vs is the pendulum viscous friction coefficient and ηðtÞ is a bounded perturbation. The initial conditions were chosen as x1;0 ¼ 1 and x2;0 ¼ 3. The following numeric values were applied to simulate the pendulum model: M ¼1.1 kg, L¼1 m, g ¼ 9:81 ðm=s2 Þ and V s ¼ 0:18kg m=s2 . For simulation proposes the bounded perturbation was expressed as ηðtÞ ¼ 0:5 sin ð2tÞ þ 0:5 cos ð5tÞ. The complete
Fig. 1. Comparison between the proposed PDSTA controller, a classical PD controller and a twisting controller. (a) Pendulum position and (b) pendulum velocity.
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Then, the positive definite solution of (23) was obtained as P2 ¼
2:8655
0:0023
0:0023
0:0497
ð48Þ
In Fig. 1, the trajectories of the pendulum system are depicted. The control signal given by the PD evolved slowly in comparison with the other control techniques. The PDSTA and the twisting controller (TWT) reached the desired trajectory. By the perturbation defined by ηðtÞ, neither the PD nor the PDSTA reached the zero tracking error. However, by the characteristics offered by the second order sliding modes, the twisting controller produced a zero tracking response, as it can be seen in Fig. 2. However, the energy required by this controller was much higher than the energy required by the PD and the PDSTA. Then, the more efficient technique was the PDSTA. Even when this controller did not reach the zero tracking error, the energy in comparison with the TWT was significantly smaller. The reaching time of the PD controller with a similar gain was bigger (four times) than the other controllers. In Fig. 3, the Euclidean norm for each controller was shown. The PDSTA converged firstly, however, a state steady error can be appreciated. The best convergence performance was obtained with the TWT. Fig. 4. Control signal obtained from the PDSTA, PD and twisting controllers.
Fig. 2. Zoom to the trajectories of the pendulum stabilized at π rad. Fig. 5. Comparison between the PDSTA controller and the PDSTA controller with an extra discontinuous term (SPDSTA).
The control input obtained by each controller is shown in Fig. 4. The PDSTA controller presented higher overshoot, but in steady state, this controller offered less energy to control the nonlinear system (15). According to Lemma 1, the results obtained where the control law is modified showed that the perturbation ηðtÞ was rejected. The trajectories for the pendulum position converged faster to the reference (Fig. 5).
4.2. Tracking problem of a two-link robot manipulator
Fig. 3. Time evolution of the tracking error norm for each control.
Consider the 4-dimensional nonlinear system depicted in Fig. 6. The nonlinear system has a classical representation given by € þ CðqðtÞ; qðtÞÞ _ _ þ GðqðtÞÞ þ pðtÞ ¼ uðtÞ with the followMðqðtÞÞqðtÞ qðtÞ ing parameters [21]: " # α þ 2β c 2 δ þ β c 2 MðqÞ ¼ ð49Þ δ þ β c2 δ
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Fig. 6. 2-link robot manipulator [21]. Fig. 7. Comparison between the PD controller and the PDSTA controller. (a) First link position and (b) second link position. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Table 1 Simulation parameters. Mass ðmi Þ (kg) Inertia ðIzi Þ (kg m2) Length ðli Þ (m) Distance ðr i Þ (m)
" _ ¼ Cðq; qÞ
β s2 θ_ 2 β s θ_ 2
" GðqÞ ¼
3 0.231 0.3 0.15
β s2 ðθ_ 1 þ θ_ 2 Þ
2 0.0071 0.2 0.1
# ð50Þ
0
1
m1 gr 1 c1 þ m2 gðl1 c1 þ r 2 c12 Þ
# ð51Þ
m2 gr 2 c12
where α ¼ I z1 þI z2 þ m1 r 21 þ m2 ðl1 þ r 22 Þ, β ¼ m2 l1 r 2 , δ ¼ I z2 þ m2 r 22 , c12 ¼ cos ðθ1 þ θ2 Þ, c2 ¼ cos θ2 , s2 ¼ sin θ2 , g is a gravity acceleration and τ ¼ ½τ1 τ2 > are the control action to be applied. θi is the joint angle of joint i, mi is the mass of link i, I zi is the inertia moment of link i about the axis that passes through the center of mass and is parallel to the z-axis, li is the length of link i, ri is the distance between joint i and the center of mass of link i (the center of mass is assumed to be on the straight line connecting the two joints). The parameters of the model are summarized in Table 1. To evaluate the efficiency of the PD controlled efficiency supplied with the STA, several simulations were performed. The PDSTA controller was compared with a classical PD and the twisting (TWT) second order sliding mode controllers, both of them supplied with an Euler numerical derivation algorithm. The complete control laws for the PDSTA, the PD and the TWT for simulation proposes were selected as 2
Controller PDSTA
Controller law " u ¼ MðqðtÞÞ "
PD
u ¼ MðqðtÞÞ "
TWT
u ¼ MðqðtÞÞ "
SPDSTA
u ¼ MðqðtÞÞ
150e1 ðtÞ þ 50d1 ðtÞ
#
190e2 ðtÞ þ 60d2 ðtÞ 180e1 ðtÞ þ 100d1 ðtÞ
#
165e2 ðtÞ þ 97d2 ðtÞ 55 signðe1 ðtÞÞ þ 40 signðd1 ðtÞÞ
#
55 signðe2 ðtÞÞ þ 41 signðd2 ðtÞÞ 21e1 ðtÞ þ85 signðe1 ðtÞÞ þ 50d1 ðtÞ
#
36e2 ðtÞ þ95 signðe2 ðtÞÞ þ 60d2 ðtÞ ð52Þ
Fig. 8. Comparison between the PDSTA controller and the twisting controller. (a) First link position and (b) second link position.
The desired trajectories were x1d ¼ sin ðtÞ x2d ¼ sin ðtÞ
ð53Þ
In Fig. 7, the PDSTA is compared with the classical PD controller. The available output was contaminated with a band-limited noise. For the first link position, under the noisy output, the classical PD controller cannot reach the desired trajectory (red line). Nonetheless, in the PDSTA after a short period of time, the first link reached the desired trajectory. The same phenomena were appreciated for the second link. For a noisy output a classical PD cannot accomplish the tracking task. A second simulation was designed to prove the PDSTA with the TWT controller. In Fig. 8, the behavior of the PDSTA and the TWT is shown. Both controllers reached the desired trajectory, however, the STA working as a differentiator
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Fig. 9. Velocities behavior for each control strategy. (a) First link velocity using the PDSTA controller, (b) first link velocity using the PD controller, (c) first link velocity using the twisting controller, (d) second link velocity using the PDSTA controller, (e) second link velocity using the PD controller, and (f) second link velocity using the twisting controller.
Fig. 10. Euclidean norm of the tracking error for each control strategy. (a) PDSTA controller, (b) PD controller, and (c) twisting controller.
improved significantly the robustness of the PD controller. The velocities of the robot manipulator for each control strategy are depicted in Fig. 9. For a full comparative study, the Euclidean norm for the tracking error was obtained. The zone of convergence was reduced with the PDSTA implementation. This aspect could be appreciated
Fig. 11. Energy used for each control technique, ‖uðtÞ‖2 . (a) PDSTA controller, (b) PD controller, and (c) twisting controller.
in Fig. 10. Finally, the required energy for each control strategy is described in Fig. 11. Applying Lemma 1, the control law was modified for implementing a discontinuous gain inside the structure of the PDSTA controller. The complete control law was given in Eq. (52). For the length of the paper, only the simulations for the position of the
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Note that V 1;i ðξi Þ is continuous but not differentiable in δ1;i ¼ 0. The ideas given in [19] are used here to handle this no classical type of Lyapunov functions. If the time derivative along the > trajectories of (21) is V_ i ðtÞ ¼ 2ξi ðtÞP 1;i ξ_ i ðtÞ þ 2Ei> ðtÞP 2;i ðd=dtÞEi ðtÞ. Here,
ξ_ i ðtÞ ¼ ½12 jδ1;i ðtÞj 1=2 ðδ_ 1;i ðtÞÞ δ_ 2;i ðtÞ >
ð55Þ
and E_ i ðtÞ ¼ ½e_ i ðtÞ e_ n þ i ðtÞ >
Fig. 12. Comparison between the PDSTA controller and the PDSTA controller with an extra discontinuous term (SPDSTA). (a) First link position and (b) second link position.
robot manipulator are shown. In Fig. 12, the PDSTA and the SPDSTA are depicted. The tracking for the links position can be appreciated, the discontinuous term produces a faster convergence to the reference trajectories. In both states, the SPDSTA in a time period less than 5 s, the robot manipulator reaches the desired trajectories.
5. Conclusions In this paper, a PD controller supplied with the STA seems to be a better option when the available output is contaminated with noise. The control scheme proposed in this paper increased the performance of a PD controller as the simulation showed for two nonlinear systems, the inverted pendulum and a 2-link robot manipulator. Several comparisons against a simple PD controller and the twisting sliding mode controller showed the advantages using the so-called PDSTA. For the tracking objective, the PDSTA and the twisting controller reached the desired trajectory. However, the energy required by the twisting controller due to its switching terms was bigger than the energy used by the PDSTA. The discontinuous term introduced in the control law (SPDSTA) improved the convergence for the control scheme. The advantages given by the STA in close-loop suggest that a similar Lyapunov function can be used to improve the performance of a PID controller with the STA applied as a RED. Additionally, this scheme can be extended to control larger class of systems that includes the so-called chain of integrators type plants.
ð56Þ
In [20], it was explained that all the functions V 1;i ðξi Þ ði ¼ 1; nÞ are continuous but not locally Lipschitz. Then, the usual second method of Lyapunov is no longer valid to analyze the convergence of the close loop system (21). However, in the same paper all the conditions required by Zubov's theorem [24, Theorem 20.2, p. 568] were got for the same system and the proposed Lyapunov candidate function: (a) Each V 1;i ðξi Þ is differentiable almost everywhere (then the derivative can be calculated applying the chain rule in all the point where the differentiability property holds), (b) if the full time derivative of V 1;i ðξi Þ is negative definite almost everywhere, then V 1;i ðξi Þ is monotone decreasing and converges to zero. Then we can continue the Lyapunov analysis as usual and in those points where the derivative does not exist we are using the argument recently presented. Therefore, the full-time derivative of (54) is formally calculated as n
n
i¼1
i¼1
> V_ ðtÞ ¼ ∑ V_ i ðξi ðtÞ; Ei ðtÞÞ ¼ ∑ ð2ξi ðtÞP 1;i ξ_ i ðtÞ þ 2Ei> ðtÞP 2;i E_ i ðtÞÞ
ð57Þ In the paper [19], it has been proved that due the conditions (2) for f i ð; Þ and ηi ð; ; Þ, the trajectories of tracking error cannot escape to infinity in finite time. Therefore we can claim that je€ i j remains þ bounded (with a bound je€ i j r ed;i ) in a certain bounded time interval ½0; T a;i , T a;i 4 0. Under this condition it is not so difficult to prove that
ξ_ i ðtÞ ¼ jδ1;i ðtÞj 1=2 A1;i ξi ðtÞ
ð58Þ
Then > V_ 1;i ðtÞ r 12 jδ1;i ðtÞj 1=2 ξi ðtÞðAi> P 1;i þ P 1;i Ai Þξi ðtÞ
ð59Þ
Because the gains λ1;i and λ2;i are positive, then Ai has two eigenvalues with negative real part, then > V_ 1;i ðtÞ r 12 jδ1;i ðtÞj 1=2 ξi ðtÞQ 1;i ξi ðtÞ
ð60Þ
2n2n
with Q 1;i A R a positive definite matrix. Following the technique described in [20], one can show that qffiffiffiffiffiffiffiffiffiffiffiffiffi ð61Þ V_ 1;i ðtÞ r κ 1;i V 1;i ðtÞ Taking the worst case when the equality is satisfied in the previous differential inclusion, one can show that qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1;i ðtÞ ¼ V 1;i ð0Þ 2κ 1;i t ð62Þ
Appendix A By the comparison lemma [10], one finally got Proof of the main theorem. Consider the following Lyapunov functions as a candidate one: n
Vðξ; EÞ ¼ ∑ V i ðξi ; Ei Þ i¼1
V i ðξi ; Ei Þ ¼ V 1;i ðξi Þ þV 2;i ðEi Þ >
V 1;i ðξi Þ ¼ ξi P 1;i ξi ;
V 2;i ðEi Þ ¼ Ei> P 2;i Ei
with ξi ≔½jδ1;i j1=2 signðδ1;i Þ δ2;i > and Ei ¼ ½ei ei þ n > .
ð54Þ
V 1;i ðtÞ ¼ 0;
8 t Z T a;i ð63Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where T a;i ¼ V 1;i ð0Þ=2κ 1;i . Now, if one takes into account the inclusion (61) and the result got in (63), the following equation can be introduced: di ðtÞ ¼ ei þ n ðtÞ þ ζ i ðtÞ
ð64Þ þ i
with ζ i ðtÞ fulfilling J ζ i ðtÞ J r ζ 8 t r T a;i and J ζ i ðtÞ J ¼ 0 8 t 4 T a;i . Once this result is got, the second part of the Lyapunov function
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Ei> ðtÞP 2;i E_ i ðtÞ satisfies 0 1 V_ 2;i ðtÞ r 2Ei> ðtÞP 2;i Ei ðtÞ þ 2Ei> ðtÞP 2;i N1 ðxðtÞ; xn ðtÞÞ 0 0
Once again, by the comparison lemma, it can be proved that V 2;i ðtÞ r V 2;i ð0Þe 2αi t þ
γi ð1 e 2αi t Þ αi
ð76Þ
þ 2Ei> ðtÞP 2;i N 2 ðxðtÞ; xn ðtÞÞ þ 2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ
ð65Þ where
"
# 0 n N 1 ðxðtÞ; x ðtÞÞ ¼ f i ðxðtÞÞ f i ðxn ðtÞÞ " # 0 N 2 ðxðtÞ; xn ðtÞÞ ¼ f i ðxn ðtÞÞ hi ðxn ðtÞÞ þ ηi ðxðtÞ; tÞ " # 0 N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ ðtÞÞ ¼ g i ðxðtÞÞðk1;i ei ðtÞ þ k2;i ei þ n ðtÞ þk2;i ζ ðtÞÞ ð66Þ Applying the so-called lambda inequality X > Y þ XY > r 1 > X Λ X þ Y > ΛY X A Rqs , Y A Rsq with 0 o Λ ¼ Λ A Rss [24] one gets >
Ei> ðtÞP 2;i N 1 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞP 2;i Λa;i P 2;i Ei ðtÞ 1
þ N 1 ðxðtÞ; xn ðtÞÞ > Λa;i N1 ðxðtÞ; xn ðtÞÞ
ð67Þ
Because each component fi for the vector function f ðxÞ is Lipschitz, one has 1
N 1 ðxðtÞ; xn ðtÞÞ > Λa;i N 1 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞΛ a;i Ei ðtÞ 1 a;i
ð68Þ
1 a;i
where Λ ¼ Li Λ according to assumption (4). Equally, with the application of the lambda inequality one can get the following result: Ei> ðtÞP 2;i N 2 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞP 2;i Λb;i P 2;i Ei ðtÞ 1
þ N2 ðxðtÞ; xn ðtÞÞ > Λb;i N 2 ðxðtÞ; xn ðtÞÞ
ð69Þ
By the conditions given in (2) and because both fi and hi are Lipschitz evaluated in a bounded value, one gets 1
N 2 ðxðtÞ; xn ðtÞÞ > Λb;i N 2 ðxðtÞ; xn ðtÞÞ 1
r λmax fΛb;i gð2‖f i ðxn ðtÞÞ hi ðxn ðtÞÞ‖2 þ 2 J ηi ðxðtÞ; tÞ J Þ
ð70Þ
By assumptions (8) and (9) the following result is obtained: 1
þ
‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 r 2λmax fΛb;i gh þ 2λmax ðη0 þ η1 ‖xðtÞ‖2 Þ b;i
1
þ
r 2λmax fΛb;i gh þ 2λmax ðη0 þ η1 ‖xðtÞ xn ðtÞ þ xn ðtÞ‖2 Þ
ð71Þ
Finally for ‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 the next bound is got b;i
1
þ
1
‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 r 2λmax fΛb;i gh þ 2λmax fΛb;i gη0 b;i
1
1
þ 4λmax fΛb;i gE > ðtÞEðtÞ þ 4λmax fΛb;i gX nþ ð72Þ The term
2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞÞ
for 8 t rT a;i "
2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ;
ζ i ðtÞÞ
¼ 2Ei> ðtÞP 2;i
0
0
k 1;i
k 2;i
# Ei ðtÞ ð73Þ
Bringing all these results together, one can show that V_ 2;i ðtÞ r γ i 2αi Ei> ðtÞP 2;i Ei> ðtÞ þ Ei> ðtÞðP 2;i R2;i P 2;i þ Q 2;i ÞEi ðtÞ Ei> ðtÞðP 2;i ðA2;i þ αi IÞ þ ðA2;i þ αi IÞ > P 2;i ÞEi ðtÞ
ð74Þ
By the assumption given in (23), the previous inequality yields V_ 2;i ðtÞ r γ i 2αi V 2;i ðtÞ
ð75Þ
With the result obtained in (62) and (76), one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ V i ðtÞ r V 1;i ð0Þ 2κ 1;i t þ V 2;i ð0Þe 2αi t þ i ð1 e 2αi t Þ
αi
Then n
VðtÞ r ∑
i¼1
ð77Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n γ V 1;i ð0Þ 2t ∑ κ 1;i þ ∑ V 2;i ð0Þe αi t þ i ð1 e αi t Þ i¼1
αi
i¼1
ð78Þ Clearly, when t ZT with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1;i ð0Þ ∑n T ¼ i ¼ 1n 2∑i ¼ 1 κ 1;i
ð79Þ
the following inequality is valid: n
n
i¼1
i¼1
VðtÞ r ∑ ðV 2;i ðTÞe αi ðt TÞ Þ þ ∑
γi ð1 e αi ðt TÞ Þ; αi
8t ZT
ð80Þ
And if we take the upper limit when t-1 in the previous inequality, one gets n
lim VðtÞ r ∑
t-1
i¼1
γi αi
This result finished the proof of this theorem.
ð81Þ □
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I. Salgado et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [21] Murray R, Li Z, Sastry S. A mathematical introduction to robotic manipulation. Florida, USA: CRC Press; 1994. [22] Picó J, Picó-Marco E, Vignoni A, De Battista H. Stability preserving maps for finite-time convergence: super-twisting sliding-mode algorithm. Automatica 2013;49(2):534–9. [23] Pisano A, Usai E. Sliding mode control: a survey with applications in math. Math Comput Simul 2011;81:954–79. [24] Poznyak A. Advanced mathematical tools for automatic control engineers: deterministic systems, vol. 1. Amsterdam, the Netherlands: Elsevier Science; 2008. [25] Poznyak A, Sánchez E, Wen Y. Differential neural networks for robust nonlinear control (identification, state estimation and trajectory tracking). Massachusetts, USA: World Scientific Press; 2001. [26] Santiesteban R, Fridman L, Moreno A. Finite-time convergence analysis for “twisting” controller via a strict Lyapunov function. In: 2010 11th international workshop on variable structure systems (VSS); 2010.
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Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems Ivan Salgado a, Isaac Chairez b,n, Oscar Camacho a, Cornelio Yañez a a
Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico city, Mexico Unidad Profesional Interdisciplinaria de Biotecnología, Instituto Politécnico Nacional, Av. Acueducto de Guadalupe s/n. Col. La Laguna Ticoman, Mexico city, Mexico
b
art ic l e i nf o
a b s t r a c t
Article history: Received 28 December 2013 Received in revised form 14 March 2014 Accepted 3 April 2014 This paper was recommended for publication by Jeff Pieper
Designing a proportional derivative (PD) controller has as main problem, to obtain the derivative of the output error signal when it is contaminated with high frequency noises. To overcome this disadvantage, the supertwisting algorithm (STA) is applied in closed-loop with a PD structure for multi-input multioutput (MIMO) second order nonlinear systems. The stability conditions were analyzed in terms of a strict non-smooth Lyapunov function and the solution of Riccati equations. A set of numerical test was designed to show the advantages of implementing PD controllers that used STA as a robust exact differentiator. The first numerical example showed the stabilization of an inverted pendulum. The second example was designed to solve the tracking problem of a two-link robot manipulator. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Signal differentiation Sliding modes PD controller
1. Introduction 1.1. Motivation The classical proportional-derivative (PD) and proportionalintegral-derivative (PID) controllers are the most successful techniques implemented in real applications [27,31]. Unfortunately, for unknown plants or systems with high frequency noises at the output, the PD controllers present an evident disadvantage regarding the need for calculating the error derivative [1]. Several approaches have been designed for improving the robustness of classical PD and PID control schemes. These researches have been oriented in two directions: (a) modifying the complete structure of the PD controller and (b) proposing robust techniques for numerical differentiation or state estimation [30]. For the first case, in [2], the classical PD controller structure was tuned with reinforcement learning based on intelligent techniques like fuzzy logic and neural networks. Robust techniques based on sliding modes have also been proposed [26]. The twisting algorithm is a discontinuous second order sliding mode controller that forces finite time convergence for nonlinear systems with a sliding
n
Corresponding author. E-mail addresses: [email protected] (I. Salgado), [email protected] (I. Chairez), [email protected] (O. Camacho), [email protected] (C. Yañez).
surface that has relative degree two with respect to the input signal. The second approach requires the use of state estimators to reconstruct the unmeasurement states (velocity for this case). State estimators based on the Luenberger structure are the most popular solution. However, the exact description of the model associated with the signal is demanded and the observer parameters cannot be tuned easily for reducing sensitivity under measurement noises or perturbations [17,6]. Observers based on fuzzy logic techniques provide robust estimation for uncertain nonlinear systems, see for example [15,28] where small gain observers were designed. On the other hand, high gain observers and extended observers were compared in [16]. When the observer uses position measurements for estimating velocity, the estimator turns to signal differentiators. The most useful methods to approximate the numerical derivative are based on linear filters [32,6]. The main approach for designing a linear differentiator is to approximate a transfer function within a finite frequency band [12]. For nonlinear signal differentiators, a complete review can be found in [18]. For uncertain, perturbed and even unknown systems, sliding modes based differentiators are a suitable option for overcoming the main drawbacks exhibited by other differentiators [29,12]. 1.2. Sliding mode theory for numerical differentiation The main characteristics offered by sliding modes are robustness against parametric uncertainties, external perturbations and finite
http://dx.doi.org/10.1016/j.isatra.2014.04.003 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
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time convergence [29,23]. The first order signal differentiator based on second order sliding modes is the well known super-twisting algorithm (STA) [13]. A complete analysis on second order sliding mode theory was described in [11,14]. The STA has been successfully applied as a controller [8], state estimator [5] and robust exact differentiator (RED) [12]. The stability and finite-time convergence analysis for the STA has been jointly addressed from different points of view, most of them based on the use of scaling symmetries (homogeneity), nonsmooth Lyapunov functions based on the Zubov theorem [25,3,20] and proposing a change of coordinates to analyze the STA with conventional techniques [22]. Moreover, by means of a nonsmooth Lyapunov function the gain selection can be easily obtained for ensuring the STA finite-time convergence [19]. In this way, the emerging of these Lyapunov functions brings out new opportunities to develop several novel applications of the STA that can include observers [19] and adaptive gain versions [8]. In particular, these Lyapunov functions have also been successfully applied to obtain uniform convergence for the STA [4].
1.3. Contribution of the paper In this paper, a novel scheme for controlling partially unknown second order non-linear systems was designed using a basic PD structure in the closed-loop with the STA applied as a RED [12]. The stability of this controller was analyzed using the ideas proposed in [20,19]. This paper analyzed a different contribution from the one presented in [20,19], where only the stability of the differentiator was considered. In this paper, the STA was implemented in the closed-loop with a classical PD scheme. The use of non-smooth Lyapunov functions allowed a constructive method to select an adequate combination of gains for the modified PD controller. The Lyapunov function used to complete the stability analysis considered the full extended system formed by the STA and the PD controller. The contribution proposed in this paper did not use the concept of separation principle when the RED converged. The approach considered the transient period that provides a more accurate solution for the close-loop problem [1]. The behavior of a PD controller can provide asymptotic stability in steady state only if the controlled system is not exhibiting any perturbation in the output signal. Otherwise, for a bounded perturbation, the tracking error will exhibit only ultimately boundedness that coincides with the result obtained in this study. In [9] a first order sliding mode term was added to a PD structure for controlling robot manipulators. In this paper, a second control structure was designed in order to enforce finite time convergence. The classical PD controller structure was slightly modified to include a first order discontinuous term [9]. The Lyapunov analysis for this structure was also studied and the corresponding gains for the controller and the differentiator were obtained. The proposed controllers were evaluated in a simple pendulum and in a two link robot manipulator in order to illustrate the benefits and drawbacks offered by them. Several comparisons were made with a classical PD controller with a linear filter derivative and second order sliding mode controllers [7]. The rest of the paper is organized as follows: in Section 2 the class of nonlinear systems to deal with is introduced as well as the STA working as a RED. In the same section, the extended system that incorporates the STA to estimate the derivative of the error signal in the closed loop with the PD controller is given. In Section 3 the main result is introduced. Numerical results are presented in Section 4 and finally in Section 5, some conclusions are given.
1.4. Notation In this paper, the following notation is used: Rn represents the vector space with n-components. As usual, > is used to define the transpose operation. J z J is used to define the Euclidean norm of z A Rn . ‖z‖2H ≔z > Hz is the weighted norm of the real-valued vector
z A Rn with weight matrix H 4 0, H ¼ H > , H A Rnn . If two matrices N A Rnn and M A Rnn fulfill M 4 Nð Z Þ that means that M N is a positive definite (semidefinite) matrix. The symbol R þ represents the set of positive real scalars. The symbols I nn and 0nn are used to represent the identity matrix and the matrix formed with zeroes of dimension n n.
2. Super-twisting PD controller 2.1. Class of nonlinear systems Consider the nonlinear system described by the following second order nonlinear differential equation: z€ ðtÞ ¼ f ðzðtÞ; z_ ðtÞÞ þ gðzðtÞÞuðtÞ þ ηðz_ ðtÞ; zðtÞ; uðtÞ; tÞ yðtÞ ¼ zðtÞ zð0Þ ¼ z0 and z_ ð0Þ ¼ zd0 given z0 ; zd0 A Rn
ð1Þ
Here z A R and z_ A R , zð0Þ and z_ ð0Þ, are the initial conditions for the differential equation. The drift term f : R2n -Rn is a Lipschitz function and the input associated term g : Rn -Rnn is bounded as it will be explained later. The nonlinear function η : R2n þ 1 -Rn represents some uncertainties affecting the nonlinear system satisfying 2 z þ ‖η‖2 r η0 þ η1 ð2Þ z_ ; η0 ; η1 A R n
n
The signal y A Rn is the available output vector. The control action is represented by u A Rn . The class of systems considered in (1) is a rough generalization of many mechanic, electromechanical, electric, thermodynamic and hydrodynamic systems. The system presented in (1) can be represented as (with the selection of xa ¼ z and xb ¼ z_ ) x_ a ðtÞ ¼ xb ðtÞ x_ b ðtÞ ¼ f ðxðtÞÞ þ gðxa ðtÞÞuðtÞ þ ηðxðtÞ; uðtÞ; tÞ yðtÞ ¼ CxðtÞ x ¼ ½xa>
ð3Þ xb> >
2n
A R and C ¼ ½I nn ; 0nn . where Throughout the paper, the following assumptions are assumed to be fulfilled. Proposition 1. The nonlinear function f ðÞ is unknown but satisfies the Lipschitz condition J f ðxÞ f ðx0 Þ J r L1 J x x0 J ;
8 x; x0 A R2n ; L1 A R þ
ð4Þ
Proposition 2. The nonlinear system (3) is controllable, therefore the function gðxÞ is known and it satisfies 0 o g r J gðxa Þ J r g þ o1;
8 xa A Rn ;
g ; g þ ARþ
ð5Þ
By this assumption, the matrix gðxa Þ is invertible 8 t Z 0. Proposition 3. The control input belongs to the set Uadm defined as U adm ≔fu : ‖uðtÞ‖2 r u0 þ u1 ‖x‖2 g
ð6Þ
þ
with u0 ; u1 A R . The previous condition includes several control techniques such as classical PD controllers and even discontinuous controllers such as sliding modes. 2.2. Nonlinear reference system The problem considered in this paper was to complete the trajectory tracking between the states of (1) and the stable
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The previous differential equation is a state representation of the signal rðtÞ. The scalar STA to obtain the derivative of rðtÞ looks like
reference model given by n z€ ðtÞ ¼ hðz_ n ðtÞ; zn ðtÞÞ; zn ð0Þ; z_ n ð0Þ are given
yn ðtÞ ¼ zn ðtÞ
ð7Þ
where hðzn ; z_ n Þðzn A Rn Þ is a Lipschitz function. Again, system (7) can be transformed using the state space method with the change of variables xna ¼ zn and xnb ¼ z_ n . The reference system (7) has a stable equilibrium point and by the converse Lyapunov theorem [10], one can ensure that the system in the new coordinates xn ¼ ½ðxna Þ > ðxnb Þ > > satisfies ‖xn ðtÞ‖2 r X nþ ;
xn ¼ ½ðzn Þ > ; ðz_ n Þ > >
ð8Þ
then, the next inequalities are assumed to be valid þ
n
Λ ¼ Λ > 40;
Λ A Rnn
ð9Þ
2n
solution of (7), the last inequality is valid because 8x AR functions f ðxn Þ and hðxn Þ are Lipschitz (continuous) functions and (7) has a stable equilibrium point.
2.3. PD controller design In general, a PD controller is designed using the following structure: _ uðtÞ ¼ k1 eðtÞ k2 eðtÞ
d w 1 ðtÞ ¼ w 2 ðtÞ λ1 jw 1 ðtÞ w1 ðtÞj1=2 signðw 1 ðtÞ w1 ðtÞÞ dt d w 2 ðtÞ ¼ λ2 signðw 1 ðtÞ w1 ðtÞÞ dt dðtÞ ¼
d w 1 ðtÞ dt
ð10Þ
ð13Þ
where λ1 ; λ2 40 are the STA gains. Here d(t) is the output of the differentiator [12]. In this equation 8 if z 4 0 > <1 signðzÞ≔ A ½ 1; 1 > : 1
X nþ A R þ ; 8 t Z 0
‖f ðxn Þ hðxn Þ‖2Λ r h ;
3
if z ¼ 0
To apply the STA as a differentiator, let us represent the uncertain system (3) as the composition of the following n second order systems: x_ a;i ðtÞ ¼ xb;i ðtÞ x_ b;i ðtÞ ¼ f i ðxðtÞÞ þ g i ðxa ðtÞÞui ðtÞ þ η i ðxðtÞ; uðtÞ; tÞ ð15Þ
i ¼ 1; n
where xa;i and xb;i are the i-th and ðn þ iÞth states of (3), respectively. The nonlinear functions f i ðÞ and g i ðÞ are the functions associated with the states xa;i and xb;i . Similarly, ηi ð; Þ is the corresponding uncertainty that affects the same subsystem. In this paper, x_ b;i ðtÞ could be affected by every component of the input vector uðtÞ. The following proposition explaining how the product gðÞuðtÞ is handled. Proposition 4. For the system given in (15), the product of elements g i;j uj for i,j ¼1:n are included in η i ðxðtÞ; uðtÞ; tÞ, that is
η i ðxðtÞ; uðtÞ; tÞ ¼ ηi ðxðtÞ; tÞ þ
n
where k1 ; k2 A R are the controller gains that must be adjusted and e A Rn is the output error given by eðtÞ≔CxðtÞ Cxn ðtÞ
ð11Þ
However, this controller is hardly to be implemented considering _ that eðtÞ and eðtÞ are rarely measured simultaneously without an important resources investment. Therefore, in the classical literature, one can find two important solutions: to construct an observer or using a first order filter to approximate the error derivative. The first one requires the system structure (that is in this paper is assumed to be unknown) and in the second case, the derivative approximation is usually poor specially if the output information is contaminated with noises. One additional option is considering a class of RED that can provide a suitable and accurate approximation of the error derivative despite the presence of noises. STA has demonstrated to be one of the best RED several times.
2.3.1. Super-twisting algorithm The STA application as a RED is described as follows. If w1 ðtÞ ¼ rðtÞ where rðtÞ A R is the signal to be differentiated, w2 ðtÞ ¼ r_ ðtÞ represents its derivative and under the assumption of jr€ ðtÞj r r þ , the following auxiliary differential equation is got: _ 1 ðtÞ ¼ w2 ðtÞ w _ 2 ðtÞ ¼ r€ ðtÞ w
ð14Þ
if z o 0
n
∑
j ¼ 1;j a i
g i;j ðxa ðtÞÞuj
with i a j and j ¼ 1 : n. The term ηi ðxðtÞ; uðtÞ; tÞ is the i-th component of ηðxðtÞ; uðtÞ; tÞ. By Propositions 2 and 3, each element g i;j ðxa ðtÞÞuj is bounded as follows: jg i;j ðxa ðtÞÞuj j2 rg þ ðu0 þ u1 ‖x‖2 Þ
8t Z0
Considering that the kind of perturbations that can affect the nonlinear system (15) satisfies jηi ðxðtÞ; uðtÞÞj r η0 þ η1 ‖x‖2 Then, the joint perturbation η i ðxðtÞ; uðtÞ; tÞ is bounded by jη i ðxðtÞ; uðtÞ; tÞj r η 0;i þ η 1;i ‖x‖2
η 0 ¼ η0 þ g þ u0 η 1 ¼ η1 þ g þ u1
ð16Þ
Before the STA can be applied, the dynamics of the tracking error also must be treated as in (15). The tracking error is defined as e ¼ x xn . The definition for the individual elements of the vector e is given by e_ i ¼ x_ a;i x_ na;i
ð17Þ
and the complete dynamics for the tracking error are composed by the following differential equations: e_ i ðtÞ ¼ ei þ n ðtÞ e_ i þ n ðtÞ ¼ f i ðxðtÞÞ þ g i ðxa ðtÞÞui ðtÞ þ η i ðxðtÞ; uðtÞ; tÞ hi ðxn ðtÞÞ
ð12Þ
ð18Þ
n
where the function hi ðx ðtÞÞ is the i-th component of the vector field hðxn ðtÞÞ.
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2.3.2. PD controller enhancement with the super-twisting algorithm Using the STA, the i-element of the PD controller was proposed as ui ðtÞ ¼ k1;i ðtÞei ðtÞ k2;i ðtÞ di ðtÞ
ð19Þ
where di is the output of the corresponding RED applied over ei. The gains for the i-element of the PD controller were selected as k1;i ðtÞ ¼ k2;i ðtÞ ¼
g i 1 ðxa ðtÞÞk 1;i g i 1 ðxa ðtÞÞk 2;i
ð20Þ
with k 1;i and k 2;i positive scalars. The following extended system describes the complete dynamics of the error signal in the close-loop with the implementation of (13) e_ i ðtÞ ¼ ei þ n ðtÞ e_ i þ n ðtÞ ¼ f i ðxðtÞÞ hi ðxn ðtÞÞ g i ðxÞðk1;i ðtÞei ðtÞþ k2;i ðtÞdi ðtÞÞ þ η i ðxðtÞ; uðtÞ; tÞ
δ_ 1;i ðtÞ ¼ δ2;i ðtÞ λ1;i jδ1;i ðtÞj1=2 signðδ1;i ðtÞÞ δ_ 2;i ðtÞ ¼ λ2;i signðδ1;i ðtÞÞ e€ i ðtÞ
ð21Þ
Theorem 1. Consider the nonlinear system given in (1), supplied with the control law (19) that is adjusted with the gains given in (20) and the derivative of the error signal obtained by (13). If there exist a positive scalar αi and if the gains λ1;i , λ2;i are positive, such that the next Lyapunov inequalities have positive definite solutions P1,i > A1;i P 1;i þ P 1;i A1;i r Q 1;i
2λ2;i
P 2;1
6 6 022 P2 ¼ 6 6 ⋮ 4 022
022
⋯
P 2;2
⋯
⋮
⋱
022
⋯
022
3
7 022 7 7 ⋮ 7 5 P 2;n
ð25Þ
and 1 1 1 þ γ i ¼ 2λmax fΛb;i ghi þ 2λmax fΛb;i gη 0;i þ4λmax fΛb;i gX nþ ;i
ð26Þ
The last theorem was useful to prove practical stability for the tracking error despite the exact convergence provided by the STA of the tracking error derivative. Interestingly, a simple modification of the control action yields to a even more robust result with better convergence properties. The following lemma gives the result associated with this modification. Lemma. If each controller ui ðtÞ is modified to ui ðtÞ ¼ k1;i ðtÞei ðtÞ k2;i ðtÞ di ðtÞ k3;i ðtÞ
si ðtÞ jsi ðtÞj
ð27Þ
where si ¼ C i> Ei with C i A R2 , C i> ¼ ½c1 ; c2 with the control gains ks;i , s¼1, 2, 3 selected as k2;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 2;i
The main result in this paper can be considered as the form of coupling the STA with a classical PD controller with a non-smooth Lyapunov function. In this way, the problem statement to deal with, is To select, an adequate combinations of gains λ1;i ; λ2;i for the STA and k 1;i , k 2;i for the PD controller described in (19) and (20), such that, the error derivatives δ1;i and the output tracking errors ei converge into a boundary layer characterized by the power of uncertainties and noises. The main result of this paper is summarized in the following theorem:
A1;i ¼
where 2
ð24Þ
k1;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 1;i
3. Main result
λ1;i
γi α i¼1 i n
lim E > ðtÞP 2 EðtÞ r ∑
t-1
Proof. The proof is given in the Appendix.
where δ1;i ¼ w 1;i ei and δ2;i ðtÞ ¼ δ_ 1;i ðtÞ. The term di refers to the approximation of ei þ n which is obtained by means of the second order STA presented in (13) and the variable w 1;i is the first state of the corresponding STA implemented as the differentiator. The constants λ1;i , λ2;i were selected according the main theorem that will be introduced below. A set of n differentiators were used to reconstruct the information of ei þ n . The following section shows the main result of this paper. The theorem introduced here gives a constructive way to select the gains of the STA and the PD controller.
"
bound
1 0
#
;
> 4 0; Q 1;i ¼ Q 1;i
Q 1;i A R22
ð22Þ
k3;i ðtÞ ¼ g i 1 ðxa ðtÞÞk 3;i
ð28Þ
þ
k 3;i ¼ μi þ k 3;i
μi ≔4ðhiþ Þ1=2 þ 4η01=2 þ 4ðη1 X þþ Þ1=2 ;
þ
k 3;i 4 0
ð29Þ
and if there exists a positive semidefinite solution for the following matrix inequalities: 1
> P 3;i A3;i þ A3;i P 3;i þ P 3;i Ξ a;i P 3;i þ Ξ a;i r 0
ð30Þ
> a;i 4 0,
with Ξ a;i ¼ Li Ξ a;i and Ξ a;i ¼ Ξ Ξ a;i A R then the trajectories of si ðtÞ converge in finite time Tci to the origin with a convergence time of pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2;i ð0Þ rT ci ð31Þ þ 2k 3;i J C i J 2n2n
>
where V 2;i ð0Þ ¼ s2i ð0Þ and the extended vector z ¼ ½ξ ; s > > converges in finite time to the origin with a convergence time T given by n
n
i¼1
i¼1
∑ T ci þ ∑ T ai rT
ð32Þ
and if for every positive value of L1 satisfying Eq. (4) and a positive þ value h defined in (9), there exist positive gains k 1;i , k 2;i such that the Riccati equations given by
where Tai was defined in the main theorem of this study.
P 2;i ðA2;i þ αi IÞ þ ðA2;i þ αi IÞ > P 2;i þ P 2;i R2;i P 2;i þ Q 2;i r0 " # 0 1 A2;i ¼ ; R2;i ¼ Λa;i þ Λb;i k 1;i k 2;i
Vðξ; EÞ ¼ ∑ V i ðξi ; si Þ
Proof. Consider the Lyapunov candidate function given by n
i¼1
V i ðξi ; Ei Þ ¼ V 1;i ðξi Þ þ V 2;i ðsi Þ
1
Q 2;i ¼ 4λmax fΛb;i gI 22 þ Λ a;i ;
Λ a;i ¼ Li Λa;i ; Λa;i ; Λb;i ; Λc;i 40 and symmetric; Λa;i ; Λb;i ; Λc;i A R22 ; αi A R þ ð23Þ have positive definite solutions P 2;i . Thus, the trajectories of E > ¼ ½e1 ; …; en ; en þ 1 ; …; e2n are globally ultimately bounded with
ð33Þ
The first section of the with V 1;i ðξi Þ ¼ ðξi Þ P 1;i ξi and V 2;i ðsi Þ ¼ previous function V 1;i ðξi Þ is analyzed in the same way as the previous theorem. Now, continuing the analysis for the second section of the candidate Lyapunov function V 2;i ðsi Þ, its full time derivative is >
V_ 2;i ðtÞ ¼ 2si ðtÞs_ i ðtÞ
s2i .
ð34Þ
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where s_ i ðtÞ ¼ C i> E_ i . The substitution of E_ i in the full time derivative of V 2;i ðsi Þ yields to 0 1 V_ 2;i ðtÞ ¼ 2si ðtÞC i> Ei ðtÞ þ 2si ðtÞC i> N 1 ðxðtÞ; xn ðtÞÞ 0 0 þ 2si ðtÞC i> N 2 ðxðtÞ; xn ðtÞÞ þ 2si ðtÞC i> N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ ð35Þ n
n
Considering the definitions of N 1 ðxðtÞ; x ðtÞÞ, N 2 ðxðtÞ; x ðtÞÞ and N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ used in the proof of the main theorem of this paper, one gets ! " # 0 1 1 V_ 2;i ðtÞ r 2si ðtÞC i> Ei ðtÞ þ si ðtÞC i> Ξ a C i si þ Ei> ðtÞΞ a Ei ðtÞ k 1;i k 2;i þ μi J C i J jsi ðtÞj þ 4η1 J Ei ðtÞ J J C i si ðtÞ J 2si> ðtÞC i> k 3;i 1=2
si ðtÞ jsi ðtÞj
ð36Þ
with μi defined as in (29). This last inclusion can be arranged as follows: 1 > V_ 2;i ðtÞ r Ei> ðtÞ½P 3;i A3;i þ A3;i P 3;i þ P 3;i Ξ a P 3;i þ Ξ a Ei ðtÞ þ ðμi k 3;i ÞJ C i J jsi ðtÞj
A3;i ¼ A2;i þ π I 2n2n
ð37Þ
with P 3;i ¼ C i C i> and π Z 2ðη1 X nþ Þ1=2 . By the assumption given in the corollary statement (30) and considering the gain selection given in (29), then þ V_ 2;i ðtÞ r k 3;i J C i J jsi ðtÞj
ð38Þ
Now, taking into consideration the structure of V 2;i , one gets qffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð39Þ V_ 2;i ðtÞ r k 3;i J C i J V 2;i ðtÞ Taking just the case when the previous inclusion became into an ordinary differential equation and redefining V 2;i ðtÞ as V eq 2;i ðtÞ, one can prove that qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ V eq ð40Þ V eq 2;i ðtÞ ¼ 2;i ð0Þ 2k 3;i J C i J t By the comparison lemma [19] and due to the function V 2;i ðtÞ Z 0, one can prove that 0 r V 2;i ðtÞ r V eq 2;i ðtÞ and V 2;i ðtÞ ¼ 0;
8 t Z T ci
This result finished the proof of this lemma.
ð41Þ □
5
control strategies were selected as Controller
Controller law
PDSTA
u ¼ Jð60e1 ðtÞ þ 49d1 ðtÞÞ
PD
u ¼ Jð110e1 ðtÞ þ 85d1 ðtÞÞ
TWT
u ¼ Jð90 signðe1 ðtÞÞ þ 75 signðd1 ðtÞÞÞ
SPDSTA
u ¼ Jð60e1 ðtÞ þ 20 signðe1 ðtÞÞ þ 49d1 ðtÞÞ
ð43Þ
and the gains for the STA were chosen as λ1 ¼ 10 and λ2 ¼ 9. The solutions for the Lyapunov and Riccati inequalities given in (22) and (23) were obtained according to the assumptions described by Eqs. (4)–(6). Eq. (22) with the values showed in (43) and the matrix Q1 defined as 10 1 A1 ¼ 18 0 " # 590 50 λ1 2λ2 þ λ21 λ1 ¼ ð44Þ Q1 ¼ 2 50 5 λ1 1 The positive definite solution for the matrix Lyapunov inequality with the previous parameters was 29:5139 0:1389 P1 ¼ ð45Þ 0:1389 582:639 For the case of the Riccati equation, the reference signal for the inverted pendulum was selected as a constant, that is xn ðtÞ ¼ π =2. Then, hðxn Þ ¼ 0. Taking into account the worst case when the pendulum position is in 0 rad, and if a DC motor with maximum speed at a free load of 57 rpm was used. The variable L defined in (4) was obtained as follows: 2 3 π 0 4 5 jf ðxÞ f ðxn Þj rL ð46Þ 1:9π 2 ; L Z 0:6822 0 Using this value, selecting Λa ¼ Λb ¼ I 22 and α ¼1, the parameters of the Riccati inequality (23) became 0 1 A2;i ¼ ; R2;i ¼ 2I 22 ; 60 49 Q 2;i ¼ 4:6828I 22
ð47Þ
4. Numerical results 4.1. Stabilization of an inverted pendulum As an illustration of the results presented in this paper, the PD control was applied to control a nonlinear pendulum system. The performance of the regular PD controller and the modified PD controller (supplied by the STA) was presented. Consider a pendulum represented by [20] x_ 1 ðtÞ ¼ x2 ðtÞ 1 MgL Vs sin ðx1 ðtÞÞ x2 ðtÞ þ ηðtÞ x_ 2 ðtÞ ¼ uðtÞ J 2J J y ¼ x1 ðtÞ
ð42Þ
where x1 ¼ θ is the angle of oscillation (rad), x2 ¼ θ_ is the angular velocity (rad/s), M is the pendulum mass, g is the gravitational force, L is the pendulum length, J is the inertia arm, Vs is the pendulum viscous friction coefficient and ηðtÞ is a bounded perturbation. The initial conditions were chosen as x1;0 ¼ 1 and x2;0 ¼ 3. The following numeric values were applied to simulate the pendulum model: M ¼1.1 kg, L¼1 m, g ¼ 9:81 ðm=s2 Þ and V s ¼ 0:18kg m=s2 . For simulation proposes the bounded perturbation was expressed as ηðtÞ ¼ 0:5 sin ð2tÞ þ 0:5 cos ð5tÞ. The complete
Fig. 1. Comparison between the proposed PDSTA controller, a classical PD controller and a twisting controller. (a) Pendulum position and (b) pendulum velocity.
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Then, the positive definite solution of (23) was obtained as P2 ¼
2:8655
0:0023
0:0023
0:0497
ð48Þ
In Fig. 1, the trajectories of the pendulum system are depicted. The control signal given by the PD evolved slowly in comparison with the other control techniques. The PDSTA and the twisting controller (TWT) reached the desired trajectory. By the perturbation defined by ηðtÞ, neither the PD nor the PDSTA reached the zero tracking error. However, by the characteristics offered by the second order sliding modes, the twisting controller produced a zero tracking response, as it can be seen in Fig. 2. However, the energy required by this controller was much higher than the energy required by the PD and the PDSTA. Then, the more efficient technique was the PDSTA. Even when this controller did not reach the zero tracking error, the energy in comparison with the TWT was significantly smaller. The reaching time of the PD controller with a similar gain was bigger (four times) than the other controllers. In Fig. 3, the Euclidean norm for each controller was shown. The PDSTA converged firstly, however, a state steady error can be appreciated. The best convergence performance was obtained with the TWT. Fig. 4. Control signal obtained from the PDSTA, PD and twisting controllers.
Fig. 2. Zoom to the trajectories of the pendulum stabilized at π rad. Fig. 5. Comparison between the PDSTA controller and the PDSTA controller with an extra discontinuous term (SPDSTA).
The control input obtained by each controller is shown in Fig. 4. The PDSTA controller presented higher overshoot, but in steady state, this controller offered less energy to control the nonlinear system (15). According to Lemma 1, the results obtained where the control law is modified showed that the perturbation ηðtÞ was rejected. The trajectories for the pendulum position converged faster to the reference (Fig. 5).
4.2. Tracking problem of a two-link robot manipulator
Fig. 3. Time evolution of the tracking error norm for each control.
Consider the 4-dimensional nonlinear system depicted in Fig. 6. The nonlinear system has a classical representation given by € þ CðqðtÞ; qðtÞÞ _ _ þ GðqðtÞÞ þ pðtÞ ¼ uðtÞ with the followMðqðtÞÞqðtÞ qðtÞ ing parameters [21]: " # α þ 2β c 2 δ þ β c 2 MðqÞ ¼ ð49Þ δ þ β c2 δ
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Fig. 6. 2-link robot manipulator [21]. Fig. 7. Comparison between the PD controller and the PDSTA controller. (a) First link position and (b) second link position. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Table 1 Simulation parameters. Mass ðmi Þ (kg) Inertia ðIzi Þ (kg m2) Length ðli Þ (m) Distance ðr i Þ (m)
" _ ¼ Cðq; qÞ
β s2 θ_ 2 β s θ_ 2
" GðqÞ ¼
3 0.231 0.3 0.15
β s2 ðθ_ 1 þ θ_ 2 Þ
2 0.0071 0.2 0.1
# ð50Þ
0
1
m1 gr 1 c1 þ m2 gðl1 c1 þ r 2 c12 Þ
# ð51Þ
m2 gr 2 c12
where α ¼ I z1 þI z2 þ m1 r 21 þ m2 ðl1 þ r 22 Þ, β ¼ m2 l1 r 2 , δ ¼ I z2 þ m2 r 22 , c12 ¼ cos ðθ1 þ θ2 Þ, c2 ¼ cos θ2 , s2 ¼ sin θ2 , g is a gravity acceleration and τ ¼ ½τ1 τ2 > are the control action to be applied. θi is the joint angle of joint i, mi is the mass of link i, I zi is the inertia moment of link i about the axis that passes through the center of mass and is parallel to the z-axis, li is the length of link i, ri is the distance between joint i and the center of mass of link i (the center of mass is assumed to be on the straight line connecting the two joints). The parameters of the model are summarized in Table 1. To evaluate the efficiency of the PD controlled efficiency supplied with the STA, several simulations were performed. The PDSTA controller was compared with a classical PD and the twisting (TWT) second order sliding mode controllers, both of them supplied with an Euler numerical derivation algorithm. The complete control laws for the PDSTA, the PD and the TWT for simulation proposes were selected as 2
Controller PDSTA
Controller law " u ¼ MðqðtÞÞ "
PD
u ¼ MðqðtÞÞ "
TWT
u ¼ MðqðtÞÞ "
SPDSTA
u ¼ MðqðtÞÞ
150e1 ðtÞ þ 50d1 ðtÞ
#
190e2 ðtÞ þ 60d2 ðtÞ 180e1 ðtÞ þ 100d1 ðtÞ
#
165e2 ðtÞ þ 97d2 ðtÞ 55 signðe1 ðtÞÞ þ 40 signðd1 ðtÞÞ
#
55 signðe2 ðtÞÞ þ 41 signðd2 ðtÞÞ 21e1 ðtÞ þ85 signðe1 ðtÞÞ þ 50d1 ðtÞ
#
36e2 ðtÞ þ95 signðe2 ðtÞÞ þ 60d2 ðtÞ ð52Þ
Fig. 8. Comparison between the PDSTA controller and the twisting controller. (a) First link position and (b) second link position.
The desired trajectories were x1d ¼ sin ðtÞ x2d ¼ sin ðtÞ
ð53Þ
In Fig. 7, the PDSTA is compared with the classical PD controller. The available output was contaminated with a band-limited noise. For the first link position, under the noisy output, the classical PD controller cannot reach the desired trajectory (red line). Nonetheless, in the PDSTA after a short period of time, the first link reached the desired trajectory. The same phenomena were appreciated for the second link. For a noisy output a classical PD cannot accomplish the tracking task. A second simulation was designed to prove the PDSTA with the TWT controller. In Fig. 8, the behavior of the PDSTA and the TWT is shown. Both controllers reached the desired trajectory, however, the STA working as a differentiator
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Fig. 9. Velocities behavior for each control strategy. (a) First link velocity using the PDSTA controller, (b) first link velocity using the PD controller, (c) first link velocity using the twisting controller, (d) second link velocity using the PDSTA controller, (e) second link velocity using the PD controller, and (f) second link velocity using the twisting controller.
Fig. 10. Euclidean norm of the tracking error for each control strategy. (a) PDSTA controller, (b) PD controller, and (c) twisting controller.
improved significantly the robustness of the PD controller. The velocities of the robot manipulator for each control strategy are depicted in Fig. 9. For a full comparative study, the Euclidean norm for the tracking error was obtained. The zone of convergence was reduced with the PDSTA implementation. This aspect could be appreciated
Fig. 11. Energy used for each control technique, ‖uðtÞ‖2 . (a) PDSTA controller, (b) PD controller, and (c) twisting controller.
in Fig. 10. Finally, the required energy for each control strategy is described in Fig. 11. Applying Lemma 1, the control law was modified for implementing a discontinuous gain inside the structure of the PDSTA controller. The complete control law was given in Eq. (52). For the length of the paper, only the simulations for the position of the
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Note that V 1;i ðξi Þ is continuous but not differentiable in δ1;i ¼ 0. The ideas given in [19] are used here to handle this no classical type of Lyapunov functions. If the time derivative along the > trajectories of (21) is V_ i ðtÞ ¼ 2ξi ðtÞP 1;i ξ_ i ðtÞ þ 2Ei> ðtÞP 2;i ðd=dtÞEi ðtÞ. Here,
ξ_ i ðtÞ ¼ ½12 jδ1;i ðtÞj 1=2 ðδ_ 1;i ðtÞÞ δ_ 2;i ðtÞ >
ð55Þ
and E_ i ðtÞ ¼ ½e_ i ðtÞ e_ n þ i ðtÞ >
Fig. 12. Comparison between the PDSTA controller and the PDSTA controller with an extra discontinuous term (SPDSTA). (a) First link position and (b) second link position.
robot manipulator are shown. In Fig. 12, the PDSTA and the SPDSTA are depicted. The tracking for the links position can be appreciated, the discontinuous term produces a faster convergence to the reference trajectories. In both states, the SPDSTA in a time period less than 5 s, the robot manipulator reaches the desired trajectories.
5. Conclusions In this paper, a PD controller supplied with the STA seems to be a better option when the available output is contaminated with noise. The control scheme proposed in this paper increased the performance of a PD controller as the simulation showed for two nonlinear systems, the inverted pendulum and a 2-link robot manipulator. Several comparisons against a simple PD controller and the twisting sliding mode controller showed the advantages using the so-called PDSTA. For the tracking objective, the PDSTA and the twisting controller reached the desired trajectory. However, the energy required by the twisting controller due to its switching terms was bigger than the energy used by the PDSTA. The discontinuous term introduced in the control law (SPDSTA) improved the convergence for the control scheme. The advantages given by the STA in close-loop suggest that a similar Lyapunov function can be used to improve the performance of a PID controller with the STA applied as a RED. Additionally, this scheme can be extended to control larger class of systems that includes the so-called chain of integrators type plants.
ð56Þ
In [20], it was explained that all the functions V 1;i ðξi Þ ði ¼ 1; nÞ are continuous but not locally Lipschitz. Then, the usual second method of Lyapunov is no longer valid to analyze the convergence of the close loop system (21). However, in the same paper all the conditions required by Zubov's theorem [24, Theorem 20.2, p. 568] were got for the same system and the proposed Lyapunov candidate function: (a) Each V 1;i ðξi Þ is differentiable almost everywhere (then the derivative can be calculated applying the chain rule in all the point where the differentiability property holds), (b) if the full time derivative of V 1;i ðξi Þ is negative definite almost everywhere, then V 1;i ðξi Þ is monotone decreasing and converges to zero. Then we can continue the Lyapunov analysis as usual and in those points where the derivative does not exist we are using the argument recently presented. Therefore, the full-time derivative of (54) is formally calculated as n
n
i¼1
i¼1
> V_ ðtÞ ¼ ∑ V_ i ðξi ðtÞ; Ei ðtÞÞ ¼ ∑ ð2ξi ðtÞP 1;i ξ_ i ðtÞ þ 2Ei> ðtÞP 2;i E_ i ðtÞÞ
ð57Þ In the paper [19], it has been proved that due the conditions (2) for f i ð; Þ and ηi ð; ; Þ, the trajectories of tracking error cannot escape to infinity in finite time. Therefore we can claim that je€ i j remains þ bounded (with a bound je€ i j r ed;i ) in a certain bounded time interval ½0; T a;i , T a;i 4 0. Under this condition it is not so difficult to prove that
ξ_ i ðtÞ ¼ jδ1;i ðtÞj 1=2 A1;i ξi ðtÞ
ð58Þ
Then > V_ 1;i ðtÞ r 12 jδ1;i ðtÞj 1=2 ξi ðtÞðAi> P 1;i þ P 1;i Ai Þξi ðtÞ
ð59Þ
Because the gains λ1;i and λ2;i are positive, then Ai has two eigenvalues with negative real part, then > V_ 1;i ðtÞ r 12 jδ1;i ðtÞj 1=2 ξi ðtÞQ 1;i ξi ðtÞ
ð60Þ
2n2n
with Q 1;i A R a positive definite matrix. Following the technique described in [20], one can show that qffiffiffiffiffiffiffiffiffiffiffiffiffi ð61Þ V_ 1;i ðtÞ r κ 1;i V 1;i ðtÞ Taking the worst case when the equality is satisfied in the previous differential inclusion, one can show that qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1;i ðtÞ ¼ V 1;i ð0Þ 2κ 1;i t ð62Þ
Appendix A By the comparison lemma [10], one finally got Proof of the main theorem. Consider the following Lyapunov functions as a candidate one: n
Vðξ; EÞ ¼ ∑ V i ðξi ; Ei Þ i¼1
V i ðξi ; Ei Þ ¼ V 1;i ðξi Þ þV 2;i ðEi Þ >
V 1;i ðξi Þ ¼ ξi P 1;i ξi ;
V 2;i ðEi Þ ¼ Ei> P 2;i Ei
with ξi ≔½jδ1;i j1=2 signðδ1;i Þ δ2;i > and Ei ¼ ½ei ei þ n > .
ð54Þ
V 1;i ðtÞ ¼ 0;
8 t Z T a;i ð63Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where T a;i ¼ V 1;i ð0Þ=2κ 1;i . Now, if one takes into account the inclusion (61) and the result got in (63), the following equation can be introduced: di ðtÞ ¼ ei þ n ðtÞ þ ζ i ðtÞ
ð64Þ þ i
with ζ i ðtÞ fulfilling J ζ i ðtÞ J r ζ 8 t r T a;i and J ζ i ðtÞ J ¼ 0 8 t 4 T a;i . Once this result is got, the second part of the Lyapunov function
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Ei> ðtÞP 2;i E_ i ðtÞ satisfies 0 1 V_ 2;i ðtÞ r 2Ei> ðtÞP 2;i Ei ðtÞ þ 2Ei> ðtÞP 2;i N1 ðxðtÞ; xn ðtÞÞ 0 0
Once again, by the comparison lemma, it can be proved that V 2;i ðtÞ r V 2;i ð0Þe 2αi t þ
γi ð1 e 2αi t Þ αi
ð76Þ
þ 2Ei> ðtÞP 2;i N 2 ðxðtÞ; xn ðtÞÞ þ 2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ i ðtÞÞ
ð65Þ where
"
# 0 n N 1 ðxðtÞ; x ðtÞÞ ¼ f i ðxðtÞÞ f i ðxn ðtÞÞ " # 0 N 2 ðxðtÞ; xn ðtÞÞ ¼ f i ðxn ðtÞÞ hi ðxn ðtÞÞ þ ηi ðxðtÞ; tÞ " # 0 N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ; ζ ðtÞÞ ¼ g i ðxðtÞÞðk1;i ei ðtÞ þ k2;i ei þ n ðtÞ þk2;i ζ ðtÞÞ ð66Þ Applying the so-called lambda inequality X > Y þ XY > r 1 > X Λ X þ Y > ΛY X A Rqs , Y A Rsq with 0 o Λ ¼ Λ A Rss [24] one gets >
Ei> ðtÞP 2;i N 1 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞP 2;i Λa;i P 2;i Ei ðtÞ 1
þ N 1 ðxðtÞ; xn ðtÞÞ > Λa;i N1 ðxðtÞ; xn ðtÞÞ
ð67Þ
Because each component fi for the vector function f ðxÞ is Lipschitz, one has 1
N 1 ðxðtÞ; xn ðtÞÞ > Λa;i N 1 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞΛ a;i Ei ðtÞ 1 a;i
ð68Þ
1 a;i
where Λ ¼ Li Λ according to assumption (4). Equally, with the application of the lambda inequality one can get the following result: Ei> ðtÞP 2;i N 2 ðxðtÞ; xn ðtÞÞ r Ei> ðtÞP 2;i Λb;i P 2;i Ei ðtÞ 1
þ N2 ðxðtÞ; xn ðtÞÞ > Λb;i N 2 ðxðtÞ; xn ðtÞÞ
ð69Þ
By the conditions given in (2) and because both fi and hi are Lipschitz evaluated in a bounded value, one gets 1
N 2 ðxðtÞ; xn ðtÞÞ > Λb;i N 2 ðxðtÞ; xn ðtÞÞ 1
r λmax fΛb;i gð2‖f i ðxn ðtÞÞ hi ðxn ðtÞÞ‖2 þ 2 J ηi ðxðtÞ; tÞ J Þ
ð70Þ
By assumptions (8) and (9) the following result is obtained: 1
þ
‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 r 2λmax fΛb;i gh þ 2λmax ðη0 þ η1 ‖xðtÞ‖2 Þ b;i
1
þ
r 2λmax fΛb;i gh þ 2λmax ðη0 þ η1 ‖xðtÞ xn ðtÞ þ xn ðtÞ‖2 Þ
ð71Þ
Finally for ‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 the next bound is got b;i
1
þ
1
‖N 2 ðxðtÞ; xn ðtÞÞ‖Λ 1 r 2λmax fΛb;i gh þ 2λmax fΛb;i gη0 b;i
1
1
þ 4λmax fΛb;i gE > ðtÞEðtÞ þ 4λmax fΛb;i gX nþ ð72Þ The term
2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞÞ
for 8 t rT a;i "
2Ei> ðtÞP 2;i N 3 ðxðtÞ; ei ðtÞ; ei þ n ðtÞ;
ζ i ðtÞÞ
¼ 2Ei> ðtÞP 2;i
0
0
k 1;i
k 2;i
# Ei ðtÞ ð73Þ
Bringing all these results together, one can show that V_ 2;i ðtÞ r γ i 2αi Ei> ðtÞP 2;i Ei> ðtÞ þ Ei> ðtÞðP 2;i R2;i P 2;i þ Q 2;i ÞEi ðtÞ Ei> ðtÞðP 2;i ðA2;i þ αi IÞ þ ðA2;i þ αi IÞ > P 2;i ÞEi ðtÞ
ð74Þ
By the assumption given in (23), the previous inequality yields V_ 2;i ðtÞ r γ i 2αi V 2;i ðtÞ
ð75Þ
With the result obtained in (62) and (76), one has qffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ V i ðtÞ r V 1;i ð0Þ 2κ 1;i t þ V 2;i ð0Þe 2αi t þ i ð1 e 2αi t Þ
αi
Then n
VðtÞ r ∑
i¼1
ð77Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n γ V 1;i ð0Þ 2t ∑ κ 1;i þ ∑ V 2;i ð0Þe αi t þ i ð1 e αi t Þ i¼1
αi
i¼1
ð78Þ Clearly, when t ZT with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1;i ð0Þ ∑n T ¼ i ¼ 1n 2∑i ¼ 1 κ 1;i
ð79Þ
the following inequality is valid: n
n
i¼1
i¼1
VðtÞ r ∑ ðV 2;i ðTÞe αi ðt TÞ Þ þ ∑
γi ð1 e αi ðt TÞ Þ; αi
8t ZT
ð80Þ
And if we take the upper limit when t-1 in the previous inequality, one gets n
lim VðtÞ r ∑
t-1
i¼1
γi αi
This result finished the proof of this theorem.
ð81Þ □
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Please cite this article as: Salgado I, et al. Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.04.003i
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Please cite this article as: Salgado I, et al. Super-twisting sliding mode differentiation for improving PD controllers performance of second order systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.04.003i