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Robust bounded control for nonlinear uncertain systems with inequality constraints
Mechanical Systems and Signal Processing 140 (2020) 106665
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Robust bounded control for nonlinear uncertain systems with inequality constraints Chenming Li a,d, Han Zhao a,b, Hao Sun a,b, Ye-Hwa Chen c,d,⇑ a
School of Mechanical Engineering, Hefei University of Technology, Hefei, Anhui 230009, PR China AnHui Key Laboratory of Digital Design and Manufacturing, Hefei University of Technology, Hefei, Anhui 230009, PR China c National Engineering Laboratory for Highway Maintenance Equipment, Chang’an University, Xi’an, Shanxi 710065, PR China d The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA b
a r t i c l e
i n f o
Article history: Received 9 October 2019 Received in revised form 21 January 2020 Accepted 21 January 2020
Keywords: Bounded control Inequality constraints Uncertainty Nonlinear systems
a b s t r a c t A novel bounded control for nonlinear uncertain systems with inequality constraints is considered. First, a state transformation is applied to satisfy the inequality constraints of the controlled outputs. Therefore, the controlled outputs are within the desired bounds. Next, a diffeomorphism is introduced for the control inputs. Based on that, the control inputs are transformed to be bounded functions. The maximum and minimum of the control inputs can be artificially set by choosing a proper bounding function. This control scheme is able to guarantee uniform boundedness and uniform ultimate boundedness. The simulation shows that the control cost of the novel bounded control and the error of the controlled system are smaller than LQR control, regardless of the uncertainty. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction Constraints are important factors that affect system performance. We usually divide constraints into passive and servo constraints in mechanics. For passive constraints, Papastavridis made important contributions [1]. For servo constraints, many control approaches are proposed for solving the constraints following problem, such as robust control [2–8], adaptive control [9–12], fuzzy logic control [13,14]. In most research works, the proposed control methods are used to deal with equality constraints of controlled outputs [15–19]. Recently, there have been some developments about the control input constraints as well. The distributed constrained containment control problem with nonconvex control input constraints is investigated in [20]. A new control law is designed with input constraints for nonholonomic wheeled mobile robots in [21]. The saturated control scheme is applied in [22] for control input constraints. The control input constraints are considered for linear systems in [23,24]. In [25], a fuzzy control with input constraints is proposed. Despite the success with equality constraints, on the other hand, many inequality constraints exist in actual systems. On one hand, the inequality controlled output constraints should be considered due to the mechanical structure or physical meaning. For instance, the water level height in a hydraulic system, which is always positive, can be treated as an inequality constraint of control outputs [26]. In [27], the inequality constraints of control outputs are solved based on robust control for nonlinear uncertain maglev systems. The inequality constraint is the magnitude limitation on the airgap between the suspended chassis and the guideway. It is used to prevent undesirable contact. On the other hand, the inequality control input constraints can be used to avoid unexpected control inputs because of the presence of the uncertainties or initial condition ⇑ Corresponding author at: National Engineering Laboratory for Highway Maintenance Equipment, Chang’an University, Xi’an, Shanxi 710065, PR China. E-mail address: [email protected] (Y.-H. Chen). https://doi.org/10.1016/j.ymssp.2020.106665 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
deviation [28–30]. For example, the values of a motor’s voltage should be less than the peak voltage. Only a few studies take the inequality constraints of control inputs and outputs into consideration at the same time. In [31], the one-sided robust control is designed to be either positive or negative. In [32], the input/output physically constraints are taken into consideration based on H1 controller. A reference updating method is proposed to keep the response and control effort within the desired bounds. In this paper, we endeavor solving the inequality constraints control design problem. The main contributions are twofold. First, a creative state transformation is proposed and applied to satisfy the state inequality constraints. With this transformation, the states are bounded, regardless of the uncertainty. Second, a novel diffeomorphism is introduced. Based on that, the control input can be regarded as a function of an auxiliary variable. As a result, the value of the control input is limited to be within a prescribed bound. It provides a new way to deal with the inequality control input constraints. The range of the input bounds is related to the uncertainties and can be designed according to the actual situation. The paper is organized as follows. In Section 2, the state transformation approach is introduced. In Section 3, a novel robust control is proposed based on diffeomorphism. We also prove that the control can guarantee uniform boundedness and uniform ultimate boundedness. In Section 4, we summarize the control design for nonlinear uncertain systems. Lastly, two examples are given for demonstration in Sections 5 and 6. 2. The state transformation In most physical systems, the values of state variables should be within the bounds. A large control input may be useful to limit the size of the variables. However, the existence of uncertainty can cause the variables to exceed their limitations. In this section, based on a state transformation, the domains of state variables are transformed, which is able to satisfy the inequality constraints of controlled outputs. Consider a dynamical system
n_ ðtÞ ¼ f ðnðtÞ; rðt Þ; uðtÞ; t Þ; nðt 0 Þ ¼ n0 :
ð1Þ
Suppose that nðtÞ ! nd as t ! þ1 and the inequality constraint is nm < nðt Þ < nM , where nd is the desired value of n; nm and nM are the maximum and minimum of n, respectively. Then, by choosing a proper function aðnÞ, the state transformation can be applied to convert the state n to a new state f without limitation. The function f ¼ aðnÞ should satisfy the condition that f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; nM Þ. Since f ¼ aðnÞ, we have
n ¼ a1 ðfÞ:
ð2Þ
Taking the first order derivative of (2) yields
@ a ðfÞ _ n_ ¼ a_ 1 ðfÞ ¼ f: @f 1
ð3Þ
This means
1 1 @ a ðfÞ f_ ¼ n_ : @f
ð4Þ
Substituting (4) into (1), the dynamical system is transformed to f
1 1 @ a ðfÞ f_ ðtÞ ¼ f a1 ðfðt ÞÞ; rðt Þ; uðtÞ; t : @f
ð5Þ
The second order derivative of (2) is needed when the dynamical system is in the second order form, such that
€nðtÞ ¼ f nðt Þ; n_ ðt Þ; rðtÞ; uðt Þ; t ; nðt 0 Þ ¼ n0 ; n_ ðt 0 Þ ¼ n_ 0 :
ð6Þ
Taking the second order derivative of (2) yields 2 1 1 €n ¼ €f @ a ðfÞ þ f_ 2 @ a ðfÞ : @f @f2
ð7Þ
Thus the dynamic system can be rewritten as
1 1 1 @ a1 ðfÞ _ @ 2 a1 ðfÞ @ a1 ðfÞ €f ¼ @ a ðfÞ f; r; u; t f_ 2 f a1 ðfÞ; : 2 @f @f @f @f
ð8Þ
The following figure shows the state transformation process. The first part ðaÞ of Fig. 1 shows the trajectory of the state variable nðt Þ. In the ideal case, nðt Þ always stays within the bounds nM and nm . It can also get closer to the desired trajectory nd over time. However, the second part ðbÞ shows the
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
3
Fig. 1. The state transformation process.
trajectory of n0ðtÞ in an actual situation. The inequality constraints cannot be easily satisfied due to unknown factors such as uncertainty. Therefore, a state transformation is applied as shown in the third part ðcÞ. The state variable n is transformed to f by the function a. In this process, the constraints can be satisfied based on the property of the transformation function a. Remark 1. This approach is able to satisfy one-sided inequality constraints and two-sided inequality constraints with proper transformation functions. For example, a logarithmic function is applied to solve one-sided inequality constraints. Suppose n > nm and the desired trajectory is nd , then the following transformation function a1 ðnÞ is proposed:
f ¼ a1 ðnÞ ¼ ln
n nm : nd nm
ð9Þ
Thus f n ¼ a1 1 ðfÞ ¼ e ðnd nm Þ þ nm :
ð10Þ
Based on (9) and (10), this logarithmic function satisfies that f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; þ1Þ. Based on a similar conception, a tangent function is applied when there is a two-sided inequality constraint. Suppose that the constraint is nm < n < nM , then the following transformation function a2 ðnÞ is needed:
f ¼ a2 ðnÞ ¼ tan where fd ¼ tan
nd nm nM nm
n ¼ a1 2 ðfÞ ¼
n nm p p fd ; nM nm 2
ð11Þ
p p2 . This means
nM nm
p
arctanðf þ fd Þ þ
nM þ nm : 2
Based on (11) and (12), f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; nM Þ are met.
ð12Þ
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3. Robust bounded control 3.1. Control design Consider the following uncertain system
x_ ðt Þ ¼ f ðxðtÞ; t Þ þ BðxðtÞ; t ÞUðwðtÞ; rðtÞ; t Þ þ BðxðtÞ; t Þeðxðt Þ; rðt Þ; tÞ;
ð13Þ
where t 2 R is the time, xðt Þ 2 R is the state, rðt Þ 2 R R is the time-varying uncertain parameter, wðtÞ 2 R is the input. The function rðÞ is Lebesgue measurable and its value lies within a prescribed compact set R Rp . Moreover, the functions f ðÞ; UðÞ, and eðÞ are Carathéodory, and the function BðÞ is strongly Carathéodory [33]. Suppose in practice the input is required to be bounded: wðt Þ 2 W, where W Rm is a prescribed compact set. Suppose n
p
m
also there is a smooth and bijective (i.e., one-to-one) diffeomorphism XðÞ : W ! Rm ; XðwÞ ¼ u; w ¼ X1 ðuÞ, such that Uðwðt Þ; rðt Þ; tÞ ¼ U X1 ðuðt ÞÞ; rðtÞ; t ¼: /ðuðtÞ; rðt Þ; t Þ. We rewrite the system as
x_ ðt Þ ¼ f ðxðtÞ; t Þ þ BðxðtÞ; t Þ/ðuðt Þ; rðtÞ; t Þ þ BðxðtÞ; t Þeðxðt Þ; rðt Þ; tÞ;
ð14Þ
m
where uðt Þ 2 R is regarded as the new control or auxiliary control from now on. Obviously, a design of uðt Þ corresponds to a design of wðt Þ since the diffeomorphism is bijective. Assumption 1. Suppose there exists a known function qðÞ : Rn R ! Rþ such that for all ðx; t Þ 2 Rn R,
r 2 R,
keðx; r; t Þk 6 qðx; t Þ:
ð15Þ
Here qðx; t Þ is related to the system uncertainty bound. Assumption 2. There are known continuous functions (1) wðÞ : Rþ ! Rþ ; wð0Þ ¼ 0; wð pÞ > 0 /ðÞ : Rþ ! Rþ ; /ð0Þ ¼ 0; /ð pÞ > 0 for p > 0, (3) cðÞ : Rþ ! Rþ ; cð0Þ ¼ 0; cð pÞ > 0 for p > 0, such that 2.a)
cðkukÞ 6 uT /ðu; r; tÞ;
for
p > 0,
(2)
ð16Þ
for all ðu; tÞ 2 Rm R; r 2 R. Furthermore, /ð0; r; tÞ ¼ 0. 2.b)
cðwðqÞÞ P qwðqÞ;
ð17Þ
for all q 0. Assumption 3. The origin x ¼ 0 of the nominal system
x_ ¼ f ðx; t Þ;
ð18Þ 1
f ð0; tÞ ¼ 0, is an uniformly asymptotically stable equilibrium point. Moreover, we can find a C Lyapunov function V ðÞ : Rn R ! Rþ and continuous, strictly increasing functions ci ðÞ; i ¼ 1; 2; 3, belonging to class K, such that
c1 ðkxkÞ 6 V ðx; tÞ 6 c2 ðkxkÞ;
ð19Þ
@V ðx; t Þ þ rx V ðx; tÞf ðx; t Þ 6 c3 ðkxkÞ: @t
ð20Þ
limc3 ðrÞ ¼ l < 1;
ð21Þ
Besides, if limr!1 c3 ðrÞ < 1, then this limit exists; that is, r!1
and the domain of c1 3 ðÞ is ½0; lÞ, so that
c1 3 ðÞ : ½0; lÞ ! R þ
ð22Þ
is defined, continuous and strictly increasing. In this case, c3 ðÞ is bounded, and the following assumption is introduced. Assumption 4. If c3 ðÞ is bounded, that is, if (21) is valid, then
< l; where
> 0 is a constant which is related to the control design (shown later).
ð23Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Now, the control uðt Þ ¼ pðxðt Þ; tÞ is proposed below for
pðx; tÞ ¼
lðx; tÞ wðqðx; t ÞÞ; klðx; t Þk
pðx; tÞkaðx; tÞk ¼ aðx; t Þkpðx; t Þk;
5
> 0:
if klðx; tÞk > if klðx; tÞk 6
ð24Þ
ð25Þ
where
lðx; tÞ ¼ aðx; tÞqðx; tÞ;
ð26Þ
aðx; tÞ ¼ BT ðx; tÞrTx V ðx; tÞ:
ð27Þ
Then, we are ready to state a theorem. Theorem 1. Consider uncertain system (14) with the control uðt Þ ¼ pðxðtÞ; tÞ and satisfying Assumptions 1–4. Then, the following results hold [34]: (i) Uniform boundedness: If xðÞ : ½t 0 ; 1Þ ! Rn ; xðt 0 Þ ¼ x0 , is a solution of the controlled system, then kx0 k 6 r ) kxðtÞk 6 dðr Þ; 8t 2 ½t0 ; 1Þ. (ii) Uniform ultimate boundedness: If xðÞ : ½t0 ; 1Þ ! Rn ; xðt 0 Þ ¼ x0 , is a solution of the controlled system with kx0 k 6 r, then r . > 0; kxðt Þk 6 d; 8t P t0 þ T d; for given d The ci functions and related quantities are shown in Fig. 2 [34].
Fig. 2. The ci functions and related quantities.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Remark 2. Fig. 2 shows the functions ci ; i ¼ 1; 2; 3. It implies that ci ð0Þ ¼ 0 for i ¼ 1; 2; 3, and limr!1 ci ðrÞ ¼ 1 for i ¼ 1; 2. In [34], Corless and Leitman have given a clear explanation for the c functions and Fig. 2. Based on that, Assumption 3 in this paper means that the uncontrolled system without uncertainty is Lyapunov stable with respect to the zero state. As shown in Fig. 2, if there exists a limit l of c3 , then Assumption 4 is applied in the following proof. This assumption can be met by choosing a proper in (23). Similarly, Assumption 2 can be met by choosing suitable functions c and w, which is shown in Remark 4. As for Assumption 1, it means there is a possible size of uncertain elements. The differences between this paper and the previous work [34] are as follows: First, the state transformation approach is applied in this paper to limit the values of control outputs. In this way, a superior system performance can be obtained. Second, based on the previous work, the diffeomorphism approach is introduced. Hence, the novel control scheme is proposed to satisfy the inequality constraints. Proof For a given
rðÞ, the derivative of the Lyapunov function V ðÞ for the closed-loop system is given by
@V @V @V V_ ðx; t Þ ¼ ðx; tÞ þ ðx; tÞf ðx; tÞ þ ðx; t ÞBðx; t Þ½/ð pðx; tÞ; rðtÞ; t Þ þ eðx; rðtÞ; t Þ: @t @x @x
ð28Þ
If klk > , the control is in the form of (24) and can be rewritten as u ¼ ga, where g ¼ wðqÞq=klk. Then, the left side of (16) can be rewritten as
aqwðqÞ ¼ cðwðqÞÞ; cðkukÞ ¼ c l
ð29Þ
and the right side of (16) can be rewritten as
uT /ðu; r; t Þ ¼ gaT /ðga; r; t Þ: Thus, we can obtain
aT /ðga; r; tÞ 6
1
g
cðwðqÞÞ:
ð30Þ
ð31Þ
Obviously, a – 0; u – 0 and wðqÞq > 0 whenever klk > . If klk 6 ; u ¼ ðkuk=kakÞa when a – 0; u – 0. So, by (16), we have
aT /ðu; r; tÞ 6
kak cðkukÞ: kuk
ð32Þ
Besides, if u ¼ 0; ð@V=@xÞB/ ¼ 0 since /ð0; r; t Þ ¼ 0. And if a ¼ 0; ð@V=@xÞB/ ¼ ð@V=@xÞBe ¼ 0. Thus, if klk > , by (15) in Assumption 1, (17) in Assumption 2, (20) in Assumption 3 and Eq. (31), we have
V_ ðx; t Þ 6 c3 ðkxkÞ þ @V B/ðu; r; t Þ þ @V Beðx; r; tÞ @x @x 6 c3 ðkxkÞ þ 1 g cðwðqÞÞ þ kaqk 6 c3 ðkxkÞ wkðlqÞkq cðwðqÞÞ þ klk
ð33Þ
6 c3 ðkxkÞ wkðlqÞkq wðqÞq þ klk ¼ c3 ðkxkÞ: If klk 6 and a – 0; u – 0, by (32), ak V_ ðx; t Þ 6 c3 ðkxkÞ kkuk cðkukÞ þ klk
6 c3 ðkxkÞ þ :
ð34Þ
Also, if u ¼ 0 which implies klk 6 ,
V_ ðx; t Þ 6 c3 ðkxkÞ þ klk 6 c3 ðkxkÞ þ ;
ð35Þ
and if a ¼ 0,
V_ ðx; t Þ 6 c3 ðkxkÞ:
ð36Þ
Therefore, for all ðx; tÞ 2 Rn R, we have
V_ ðx; t Þ 6 c3 ðkxkÞ þ :
ð37Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
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Fig. 3. The design procedure.
Definition 1. The symbol between functions means the composition of function, such as g ð f ðxÞÞ is represented as ð g f ÞðxÞ. If c3 belongs to K1 , then V_ is negative definite for sufficiently large kxk. If not, there exists a limit, that is limr!1 c3 ðr Þ ¼ l < 1 (21). Then, assumption 4 (23) is applied. The same conclusion can be obtained as well. Consequently, V_ is always negative definite for sufficiently large kxk. Following the standard arguments as in Corless and Leitmann (1981), the solution of the controlled system is uniformly bounded and uniformly ultimately bounded with
(
dðr Þ ¼
c1 if r 6 R 1 c2 ðRÞ; 1 c1 c2 ðrÞ; if r > R
ð38Þ
and
R ¼ c1 3 ðÞ;
ð39Þ
is any constant with d
> c1 c ðRÞ; d 2 1 T d; r ¼
and
8 < 0; c ðrÞc1 ðRÞ : 2 ; c3 ðRÞ
R ¼ c1 2 c1 d :
ð40Þ if r 6 R if r > R
ð41Þ
ð42Þ
Q.E.D. 3.2. Design implications The control design is based on uðtÞ, the transformed input. Once it is determined, the actual input action wðtÞ is determined since the diffeomorphism XðÞ is bijective. The advantage of this approach is that since pðx; tÞ (and hence u) is always continuous, wðt Þ is continuous. A direct design approach for wðt Þ, as was adopted in some past research, on the other hand, may trigger a discontinuous control input. We further substantiated the design details by a few remarks below.
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Fig. 4. The quarter vehicle model.
Remark 3. A symmetric structure for pðx; t Þ is implied in (25). In principle, there are infinite choices of pðx; t Þ. A typical choice of pðx; tÞ for klðx; t Þk 6 is
pðx; tÞ ¼
lðx; tÞ wðqðx; t ÞÞ:
ð43Þ
Remark 4. To demonstrate the choices of several functions, let us consider, for example, Uðw; r; t Þ ¼ w, with k < w < k, where k > 0 is a constant. Then one choice of the diffeomorphism XðÞ is XðwÞ ¼ tan p2kw. Since u ¼ tan p2kw, one can show that w ¼ 2k p arctan u and therefore we choose
/ðu; r; tÞ ¼
2k
p
arctan u:
ð44Þ
In order to satisfy (16) in Assumption 2, cðkukÞ can be chosen as
cðkukÞ ¼ juj
2k
p
arctan juj:
ð45Þ
Then (17) in Assumption 2 can be satisfied with wðqÞ ¼ tan p2kq. Remark 5. The magnitude of the control function pðx; tÞ is wðqðx; tÞÞ, while the bound of the uncertainty eðx; r; tÞ is qðx; t Þ. These two magnitudes may not be the same. The reason for this is that the input u acts on the system through the bounding function /ðu; r; t Þ, which may reduce the influence of u. Therefore we need to design the control with the magnitude wðqðx; tÞÞ to pre-compensate the influence due to /ðu; r; tÞ.
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4. Design procedure In Fig. 3, we summarize the proposed control design for the uncertain system with a flowchart. In this paper, the transformed dynamical systems are obtained to meet both the inequality constraints of the control inputs and system states. On one hand, the inequality constraints of control outputs can be satisfied by state transformation. This approach is introduced in Section 2. Fig. 1 clearly shows that the trajectory of control outputs nðt Þ is within bounds. The issue of satisfying the inequality constraints is to find a suitable transformation function a. The domain of state variable can be changed by transformation function. In Remark 1, the logarithmic function and the tangent function are given as examples to deal with one-sided and two-sided inequality constraints. On the other hand, a novel control scheme is proposed to deal with inequality constraints of the control inputs. There are four steps involved in designing this control. First, the control input can be regarded as a function / based on diffeomorphism. Second, a proper control function for different control constraints need be chosen. In this paper, a tangent function (44) is chosen. Thus, the values of control inputs is limited based on the property of the tangent function. Third, the specific value of the bound is to be set by the parameter k. As a result, the inequality constraints can be satisfied. Fourth, the functions c; w and q are selected for meeting the assumptions. We are able to prove that the transformed system state is uniformly bounded and uniformly ultimately bounded. 5. Vehicle active suspension-seat system We apply this control to the vehicle active suspension-seat system shown in Fig. 4. In this case, we take two subsystems into consideration simultaneously. As shown in Fig. 1, there are the active body subsystem and the active seat subsystem. The parameters m1 and m2 are the vehicle body mass and the seat mass, respectively. The displacements of the corresponding masses in the vertical direction are x1 and x2 . The stiffness and damping of the vehicle body suspension (the seat suspension) are k1 and c1 (k2 and c2 ), respectively. The variables u1 and u2 are the control inputs for vehicle body and seat, respectively. The variables z1 ðtÞ and z2 ðtÞ are uncertainties: specifically z1 ðt Þ represents the road excitation displacement, z2 ðt Þ represents the vibration from the vehicle body and the equivalent displacement which is caused by the passenger and/or seat load variation [35]. The constrained dynamic system can be given as [36–38]
8 m1 €x1 ðtÞ þ k1 ðx1 ðt Þ z1 ðt ÞÞ þ c1 x_ 1 ðt Þ þ k2 ðx1 ðt Þ x2 ðt Þ z2 ðt ÞÞ þ c2 ðx_ 1 ðt Þ x_ 2 ðt ÞÞ ¼ u1 ðt Þ; > > > < xm1 < x1 ; > m > 2 €x2 ðtÞ þ k2 ðx2 ðt Þ þ z2 ðt Þ x1 ðt ÞÞ þ c2 ðx_ 2 ðt Þ x_ 1 ðt ÞÞ ¼ u2 ðt Þ; > : xm2 < x2 < xM2 ;
ð46Þ
where xm1 is the lower bound of the vehicle body, xm2 and xM2 are the lower and upper bounds of the seat, respectively. Remark 6. The inequality constraints are used to guarantee the safety and comfort of the vehicle system. The condition xm1 < x1 is proposed to prevent the possibility of collision between the vehicle body and ground. The condition xm2 < x2 < xM2 is proposed to restrict seat shaking. Since the inequality constraints of the vehicle body x1 and seat x2 are one-sided and two-sided, respectively, we can take the state transformations for x1 and x2 as follows:
y1 ¼ ln
x1 xm1 ; xd1 xm1
y2 ¼ tan
x2 xd2 p ; xM2 xm2
ð47Þ ð48Þ
m2 where xd1 ; xd2 are the desired positions of vehicle body and seat and xd2 ¼ xM2 þx . The new state variables are y1 2 ð1; þ1Þ 2 and y2 2 ð1; þ1Þ which are transformed from x1 and x2 . Therefore, x1 ! xm1 as y1 ! 1 and x1 ! xd1 as y1 ! 0. Similarly, x2 ! xm2 as y2 ! 1; x2 ! xM2 as y2 ! þ1 and x2 ! xd2 as y2 ! 0. The process of the state transformations is able to assure that the states x1 and x2 cannot exceed their threshold. Then, we have
x1 ¼ ey1 ðxd1 xm1 Þ þ xm1 ; x2 ¼
xM2 xm2
p
arctan y2 þ xd2 :
ð49Þ ð50Þ
Take the first and second order derivatives of (49) and (50),
x_ 1 ¼ y_ 1 ey1 ðxd1 xm1 Þ;
ð51Þ
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Fig. 5. The displacements of the vehicle body.
Fig. 6. The displacements of the seat.
x_ 2 ¼
xM2 xm2
p
y_ 2 ; 1 þ y22
ð52Þ
€1 ey1 ðxd1 xm1 Þ þ y_ 21 ey1 ðxd1 xm1 Þ; €x1 ¼ y €x2 ¼
xM2 xm2
p
! €2 2y2 y_ 22 y : 2 1 þ y22 1 þ y2 2
Substituting (49), (50), (51), (52), (53) and (54) into (46), we have
ð53Þ ð54Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
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Fig. 7. The velocity of the vehicle body.
Fig. 8. The velocity of the seat.
1 m2 ðk1 þ k2 Þðey1 ðxd1 xm1 Þ þ xm1 Þ ðc1 þ c2 Þðy_ 1 ey1 ðxd1 xm1 ÞÞ þ k2 xM2 x arctan y2 þ xd2 p
xM2 xm2 y_ 2 þ c2 þ k1 z1 þ k2 z2 þ u1 y_ 21 ey1 ðxd1 xm1 Þm1 p 1 þ y22 1 1 x1 þ y 1 u1 ; ¼ g1 þ y e 1 ðxd1 xm1 Þm1 e ðxd1 xm1 Þm1 x x p 1 þ y22 xM2 xm2 y_ 2 M2 m2 €2 ¼ arctan y2 þ xd2 þ c2 k2 ey1 ðxd1 xm1 Þ þ c2 y_ 1 ey1 ðxd1 xm1 Þ k2 y 2 m2 ðxM2 xm2 Þ p p 1 þ y2 p 1 þ y22 p 1 þ y22 m2 ðxM2 xm2 Þ 2y2 y_ 22 ¼ g2 þ x2 þ k2 z2 þ u2 þ u2 ; m2 ðxM2 xm2 Þ m2 ðxM2 xm2 Þ p 1 þ y22 1 þ y22 €1 ¼ ey1 ðx y
d1 xm1 Þm1
ð55Þ
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Fig. 9. The acceleration of the vehicle body.
Fig. 10. The acceleration of the seat.
h y_ 2 m2 m2 ðk1 þ k2 Þðey1 ðxd1 xm1 Þ þ xm1 Þ ðc1 þ c2 Þðy_ 1 ey1 ðxd1 xm1 ÞÞ þ k2 xM2 x arctan y2 þ xd2 þ c2 xM2 x p p 1þy22 h i 2 p 1þy ð Þ _2 m2 m2 y y_ 21 ey1 ðxd1 xm1 Þm1 ; x1 ¼ k1 z1 þ k2 z2 ; g 2 ¼ m2 ðxM2 x2 m2 Þ k2 ey1 ðxd1 xm1 Þ þ c2 y_ 1 ey1 ðxd1 xm1 Þ k2 xM2 x arctan y2 þ xd2 þ c2 xM2 x ; p p 1þy2
where
g 1 ¼ ey1 ðx
1
d1 xm1 Þm1
x2 ¼ k2 z2 . Thus, the transformed system is a nonlinear system with ey1 ; arctan y2 and y22 in g 1 ; g 2 .
2
Choose the state variables as y1 ; y2 ; y3 ¼ y_ 1 ; y4 ¼ y_ 2 . The system (55) can be rewritten as
8 y_1 ¼ y3 ; > > > > > < y_2 ¼ y4 ; 1 1 x1 þ ey1 ðxd1 x u1 ¼ g 1 þ ax1 þ au1 ; y_3 ¼ g 1 þ ey1 ðx x > m1 Þm1 m1 Þm1 d1 > > > 2 2 > p 1þy p 1þy : y_ ¼ g þ ð 2 Þ x þ ð 2 Þ u ¼ g þ bx þ bu : 2 2 2 4 2 2 m2 ðxM2 xm2 Þ m2 ðxM2 xm2 Þ 2
ð56Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
13
Fig. 11. The control input on the vehicle body.
Fig. 12. The control input on the seat.
In an actual vehicle suspension-seat system, the control inputs are limited within a proper range for comfort and safety. In this paper, the control input channel is designed to be a tangent function (44). This system can be rewritten based on (14) and we obtain
2 f ð y; t Þ
y3
2
3
0 0
3
60 07 6y 7 x1 6 7 6 47 ; ¼ 6 7; Bð y; t Þ ¼ 6 7; eð y; r; tÞ ¼ 4a 05 4 g1 5 x2
/ðu; r; t Þ ¼
g2 /1 ðu; r; t Þ /2 ðu; r; t Þ
0 " 2k
¼
b
1
arctan u1
2k2
arctan u2
p p
# ;
ð57Þ
14
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Fig. 13. The relation between k1 ; k2 and the accumulative error of the seat.
Fig. 14. The relation between ; k2 and the accumulative error of the seat.
where k1 > 0 and k2 > 0 are constants. Then the control inputs are properly bounded: /1 ðu; r; t Þ 2 ðk1 ; k1 Þ; /2 ðu; r; t Þ 2 ðk2 ; k2 Þ. The simulations were performed by using m1 ¼ 10; m2 ¼ 3; k1 ¼ 20; k2 ¼ 15; c1 ¼ 1; c2 ¼ 3; xd1 ¼ 0:2; xm1 ¼ 0; xd2 ¼ 0:5; xm2 ¼ 0:49 and xM2 ¼ 0:51. For the uncertainties, we choose z1 ¼ 0:1 sinð4tÞ; z2 ¼ 0:03 cosðt Þ. In simulation, let k1 ¼ 2:7; k2 ¼ 0:5; q1 ¼ 2:5; q2 ¼ 0:48 and ¼ 0:1. Then, Assumption 1 is satisfied because of jjx1 jj 6 q1 and jjx2 jj 6 q2 . As shown in Remark 4, Assumption 2 is also satisfied. In simulation, y_ 3 and y_4 are given as y_ 3 ¼ g1 þ ðg 1 g1 Þ þ ax1 þ a/1 2 þ ðg 2 g2 Þ þ bx2 þ b/b , where g 1 ¼ y1 2y3 and g 2 ¼ y2 2y4 . Thus, for the nominal system, P can be and y_ 4 ¼ g obtained by solving the Lyapunov equation [39] by choosing V ¼ yT Py and Q ¼ I. Assumption 3 is satisfied by choosing
c1 ðjjxjjÞ ¼ Emin ðPÞjjxjj2 ¼ 0:29jjxjj2 ; c2 ðjjxjjÞ ¼ Emax ðPÞjjxjj2 ¼ 1:71jjxjj2 and c3 ðjjxjjÞ ¼ Emax ðQ Þjjxjj2 ¼ jjxjj2 , where EminðmaxÞ ðÞ represents the minimum (maximum) eigenvalue of ðÞ. And Assumption 4 is not applied in this case since limr!1 c3 ðr Þ ¼ 1. For comparisons, the simulations of (i) a bounded control without state transformation and (ii) the LQR control are performed. The matrices Q and R in the LQR control are both selected as I in simulations. This means all states and all controls are equally weighted. The reasons for choosing the LQR are twofold. First, the LQR is a proven optimal control. It minimizes a performance index including the state and control costs. Second, the LQR is a proven robust control. The most common robustness measures attributed to the LQR are a one-half gain reduction in any input channel, an infinite gain amplification in any input channel, or a phase error of plus or minus sixty degrees in any input channel. In addition, there is robustness to uncertainty in the real coefficients of the model and certain nonlinearities, including control switching and saturation [40].
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
15
Fig. 15. The Motor-Elevator model.
Fig. 5 shows the comparison of displacements of the vehicle body under different controls. The blue line represents the displacement of the vehicle body under bounded control with state transformation. The green and red lines represent the displacements under the bounded control without state transformation and LQR control, respectively. The yellow and black lines represent the desired displacement and lower bound of the vehicle body, respectively. From this figure, we can see that there are smaller fluctuations of the displacements under bounded control, represented by the blue and green lines, than that under LQR control which is represented by the red line. In the partial view, we can clearly see that the vehicle body cannot exceed the lower bound under bounded control with state transformation. However, LQR control and bounded control without state transformation cannot assure this. Similarly, Fig. 6 shows the comparison of displacements of the seat under different controls. Blue, green and red lines represent the displacements under bounded control with state transformation, bounded control without state transformation, and LQR control, respectively. Yellow and black lines are the desired displacement and bounds. The best performance is under bounded control with state transformation, which does not exceed the bounds with the smallest displacement fluctuation. The performance of displacement under bounded control without state transformation is better than the performance under LQR control. However, they both exceed the bounds due to the uncertainty. Figs. 7 and 8 show the velocities of vehicle body and seat, respectively. The blue, green and red lines represent the accelerations under bounded control with state transformation, bounded control without state transformation and LQR control. From Figs. 7 and 8, we can see that the fluctuation under bounded control with state transformation is the smallest. Figs. 9 and 10 show the accelerations of vehicle body and seat, respectively. The blue, green and red lines represent the accelerations under the bounded control with state transformation, bounded control without state transformation and LQR control. From these two figures, we find that a better superior was obtained under bounded control (by comparing the green line with red line). At the same time, the state transformation can also be helpful for seeking a better performance by comparing the blue line with green line. In summary, the fluctuation under bounded control with state transformation is the smallest. Figs. 11 and 12 show the control inputs on the vehicle body and seat, respectively. The blue, green and red lines represent the bounded control with state transformation, the bounded control without state transformation, and LQR control. We can see that the bounded control cannot exceed the bounds which are represented by black dotted lines while the LQR control exceeds the bounds. In this case, the parameters of control bounds are set as: k1 ¼ 2:7 and k2 ¼ 0:5. Thus, in Fig. 11, the values of bounded controls, in the blue and green lines, are between ð2:7; 2:7Þ. Similarly, the values of bounded controls in Fig. 12 are between ð0:5; 0:5Þ. These two figures prove that the proposed control method is able to satisfy the control input inequality constraints. Fig. 13 shows the relation between k1 ; k2 , and the accumulative error of the seat. From this figure, we can see that the accumulative error decreases as the two parameters k1 and k2 increase, because these two parameters decide the bounds of control as shown in (44). A larger value of k means that the control can provide a larger control input to obtain a better system performance. Fig. 14 shows the relation between ; k2 and the accumulative error of the seat. We can see the accumulative error is proportional to and inversely proportional to k2 . As for , it implies the neighborhood size of the zero state for all possible system responses. By letting ! 0, an arbitrarily small neighborhood of the zero state can be obtained. Thus, in this figure, the system performance gets better as the parameter decreases. As for k2 , it decides the bound of the seat control. The larger control input can be obtained with a larger k2 . Thus, in this figure, the system performance gets better as the parameter k2 increases.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
6. Motor-elevator system We next apply this control to a first-order system. Consider the elevator system that is shown in Fig. 15, where M a ; x; U are the output torque, speed, and voltage of the motor, respectively, T load is the load torque including the elevator gravity, nonlinear friction, and disturbance. The dynamical model of the motor-elevator system is
U ðtÞ ke xðt Þ Bm xðt Þ T load ðt Þ; R
_ ðt Þ ¼ k t Ma ðt Þ ¼ J x
ð58Þ
where J is the moment of inertia of the motor, Bm is the viscous damping coefficients of the motor, R is the total resistance of the armature circuit, kt and ke are the constants of the motor torque and the electromotive force, respectively. In order to render desirable performances of motor speed and voltage, the inequality constraints xm < x < xM and U < U < U are imposed, where xm and xM are the maximum and minimum of the motor speed, respectively, U is the peak voltage. The speed is constrained: x 2 ðxm ; xM Þ. The state transformation is taken by using a tangent function:
m ¼ tan
2x xM xm p ; 2ðxM xm Þ
ð59Þ
where m 2 ð1; þ1Þ is the transformed state from x. Thus, x ! xm when m ! 1; x ! xM when m ! þ1 and x ! ðxM þ xm Þ=2 ¼ xd when m ! 0. xd is the desired speed. Therefore we have
x¼
xM xm xM þ xm arctan m þ : p 2
ð60Þ
Taking the first order derivative of (60) yields
x_ ¼
xM xm m_ : p 1 þ m2
ð61Þ
Next, the control input is proposed in the form of (44). Substitute (60) and (61) into (58). Let us rewrite it in the form of (14):
m_ ¼
kt ke RJ
kt þ RJ
þ BJm
pð1þm2 Þ xM xm
h
M þxm arctan m þ m2 arctan m þ m2 x xM xm
i
ke ðxM þxm Þ Bm R xM þxm R : arctan u T load 2 2 kt pð1þm Þ kt p pð1þm Þ
2k
Fig. 16. The motor speed.
ð62Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
17
Fig. 17. The motor voltage.
Therefore,
f ðm; t Þ
h i t ke M þxm ¼ kRJ þ BJm arctan m þ m2 arctan m þ m2 x xM xm ;
Bðm; t Þ
¼ kRJt
pð1þm2 Þ xM xm ;
þxm M þxm Þ BkmtR pxM1þ ; ¼ kRt T load kepðx1þ ð m2 Þ ð m2 Þ 2k /ðu; r; t Þ ¼ p arctan u:
eðm; r; tÞ
ð63Þ
Let xd ¼ 0:5; xm ¼ 0:48 and xM ¼ 0:52. The values of the relevant parameters are given as kt ¼ 1; ke ¼ 5; R ¼ 1:2; J ¼ 5 and Bm ¼ 3. Suppose that the uncertainty T load ¼ 10 sinð4t Þ and the bound of control input k ¼ 15. The simulation results are shown as follows. The LQR control is also applied to this model as a comparison. Fig. 16 shows the motor speeds under (i) the bounded control with state transformation, (ii) a bounded control without state transformation, and (ii) the LQR control. They are represented by the blue, green, and red lines, respectively. The yellow and black lines represent the desired speed and the bounds of speed, respectively. From this figure, we can see that the performance is better under the bounded control, comparing with the LQR control. The speed cannot exceed the threshold under the state transformation. Fig. 17 shows the comparison of the motor voltages under three different control schemes. The blue solid line represents the voltage under the bounded control with state transformation. The bounds of blue solid line are represented by the blue dotted lines. The green solid line represents the voltage under a bounded control without state transformation. The bounds of green solid line are represented by the green dotted lines. The red solid line represents the voltage under the LQR control. In these three control schemes, the control cost is the smallest under the bounded control with state transformation. From these two figures, we can see that the bounded control with state transformation exhibits the best performance with the smallest control cost. 7. Conclusion In this paper, the inequality constraints of control inputs and outputs are taken into consideration for nonlinear uncertain systems simultaneously. There are two major contributions. First, the state transformation is applied to convert the bounded states into new states without bounds. The tangent and logarithmic transformed functions are used to deal with one-sided and two-sided inequality constraints. Second, the control input is regarded as a bounding function through a diffeomorphism. This way, the bound of the control input can be artificially set regardless of the uncertainty by choosing a proper bounding function. This control scheme is proven to guarantee uniform boundedness and uniform ultimate boundedness. In the simulations, we show that this control scheme exhibits a superior performance and smaller control cost when compared to the LQR control. We believe that this study opens a new door for dealing with inequality constraints in controlling uncertain dynamical systems. Future explorations along this avenue may include the choices of more transformation functions or the asymmetric bounded problem of inequality constraints.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Conflicts of Interest The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. CRediT authorship contribution statement Chenming Li: Software, Validation, Formal analysis, Investigation, Writing - original draft. Han Zhao: Resources, Project administration. Hao Sun: Data curation, Validation, Funding acquisition. Ye-Hwa Chen: Conceptualization, Methodology, Supervision, Writing - review & editing. Acknowledgements The research is supported by the China Scholarship Council (No. 201806690019). The research is also supported by the Natural Science Foundation of Anhui Province (No. 1908085QE194), the Fundamental Research Funds for the Central Universities of China (Nos. JZ2019HGTA0042, PA2019GDPK0066), and National Natural Science Foundation of China (No. 51905140). The first author would like to thank his wife Qiaozhi Zhang, for her support and company. References [1] J.G. 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Pandey, Robust hybrid control of robot manipulators, in: Proceedings, 1989 International Conference on Robotics and Automation, IEEE, 1989, pp. 236–241. [8] Y.H. Chen, Constraint-following servo control design for mechanical systems, J. Vib. Control 15 (3) (2009) 369–389. [9] C. Hua, J. Chen, Y. Li, L. Liang, Adaptive prescribed performance control of half-car active suspension system with unknown dead-zone input, Mech. Syst. Signal Process. 111 (2018) 135–148. [10] H.N. Rahimi, I. Howard, L. Cui, Neural adaptive tracking control for an uncertain robot manipulator with time-varying joint space constraints, Mech. Syst. Signal Process. 112 (2018) 44–60. [11] Q. Guo, J. Yin, Y. Tian, J. Dan, Saturated adaptive control of electrohydraulic actuator with parametric uncertainty and load disturbance, IEEE Trans. Industr. Electron. 64 (10) (2017) 7930–7941. [12] D. Li, D. Li, Y. Liu, S. Tong, P. 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Niamsup, Stabilization of linear nonautonomous systems with norm-bounded controls, J. Optim. Theory Appl. 131 (1) (2006) 135–149. [23] D.E. Miller, E.J. Davison, An adaptive tracking problem with a control input constraint, Automatica 29 (4) (1993) 877–887. [24] H. Du, N. Zhang, Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint, IEEE Trans. Fuzzy Syst. 17 (2) (2009) 343–356. [25] D. Saifia, M. Chadli, S. Labiod, H.R. Karimi, Fuzzy control of DC-DC converters with input constraint, Math. Probl. Eng. 2012 (2012) (2012) 1–18. [26] Y.H. Chen, C.Y. Kuo, Positive uncertain systems with one-sided robust control, J. Dyn. Syst. Meas. Contr. 119 (4) (1997) 675–684. [27] J. Xu, Y.H. Chen, H. Guo, Robust levitation control for maglev systems with guaranteed bounded airgap, ISA Trans. 59 (1) (2015) 205–214. [28] O. Ellabban, J. Van Mierlo, P. Lataire, A DSP-based dual-loop peak DC-link voltage control strategy of the Z-source inverter, IEEE Trans. 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[32] T. Shaqarin, A. Al-Rawajfeh, M. Hajaya, N. Alshabatat, B. Noack, Model-based robust h1 control of a granulation process using smith predictor with reference updating, J. Process Control 77 (2019) 38–47. [33] Y.H. Chen, Deterministic control for a new class of uncertain dynamical systems, IEEE Trans. Autom. Control 32 (1) (1987) 73–74. [34] M.J. Corless, G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Autom. Control 26 (5) (1981) 1139–1144. [35] A. Alfadhli, J. Darling, A.J. Hillis, The control of an active seat with vehicle suspension preview information, J. Vib. Control 24 (8) (2018) 1412–1426. [36] H. Du, W. Li, N. Zhang, Integrated seat and suspension control for a quarter car with driver model, IEEE Trans. Veh. Technol. 61 (9) (2012) 3893–3980. [37] M.L.H.A. Rahman, K. Hudha, Improvement on ride and handling performance in HMMWV suspension control using dual acting semi-active suspension system, in: 2015 10th Asian Control Conference (ASCC), IEEE, 2015, pp. 1–6. [38] S. Rajendiran, P. Lakshmi, B. Rajkumar, Fractional order fuzzy sliding mode controller for the quarter car with driver model and dual actuators, IET Electr. Syst. Transp. 7 (2) (2016) 145–153. [39] H.K. Khalil, Nonlinear Systems, Prentice hall Upper Saddle River, Bergen, NJ, 2002. [40] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, Dover Publications, Mineloa, NY, 2009.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Robust bounded control for nonlinear uncertain systems with inequality constraints Chenming Li a,d, Han Zhao a,b, Hao Sun a,b, Ye-Hwa Chen c,d,⇑ a
School of Mechanical Engineering, Hefei University of Technology, Hefei, Anhui 230009, PR China AnHui Key Laboratory of Digital Design and Manufacturing, Hefei University of Technology, Hefei, Anhui 230009, PR China c National Engineering Laboratory for Highway Maintenance Equipment, Chang’an University, Xi’an, Shanxi 710065, PR China d The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA b
a r t i c l e
i n f o
Article history: Received 9 October 2019 Received in revised form 21 January 2020 Accepted 21 January 2020
Keywords: Bounded control Inequality constraints Uncertainty Nonlinear systems
a b s t r a c t A novel bounded control for nonlinear uncertain systems with inequality constraints is considered. First, a state transformation is applied to satisfy the inequality constraints of the controlled outputs. Therefore, the controlled outputs are within the desired bounds. Next, a diffeomorphism is introduced for the control inputs. Based on that, the control inputs are transformed to be bounded functions. The maximum and minimum of the control inputs can be artificially set by choosing a proper bounding function. This control scheme is able to guarantee uniform boundedness and uniform ultimate boundedness. The simulation shows that the control cost of the novel bounded control and the error of the controlled system are smaller than LQR control, regardless of the uncertainty. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction Constraints are important factors that affect system performance. We usually divide constraints into passive and servo constraints in mechanics. For passive constraints, Papastavridis made important contributions [1]. For servo constraints, many control approaches are proposed for solving the constraints following problem, such as robust control [2–8], adaptive control [9–12], fuzzy logic control [13,14]. In most research works, the proposed control methods are used to deal with equality constraints of controlled outputs [15–19]. Recently, there have been some developments about the control input constraints as well. The distributed constrained containment control problem with nonconvex control input constraints is investigated in [20]. A new control law is designed with input constraints for nonholonomic wheeled mobile robots in [21]. The saturated control scheme is applied in [22] for control input constraints. The control input constraints are considered for linear systems in [23,24]. In [25], a fuzzy control with input constraints is proposed. Despite the success with equality constraints, on the other hand, many inequality constraints exist in actual systems. On one hand, the inequality controlled output constraints should be considered due to the mechanical structure or physical meaning. For instance, the water level height in a hydraulic system, which is always positive, can be treated as an inequality constraint of control outputs [26]. In [27], the inequality constraints of control outputs are solved based on robust control for nonlinear uncertain maglev systems. The inequality constraint is the magnitude limitation on the airgap between the suspended chassis and the guideway. It is used to prevent undesirable contact. On the other hand, the inequality control input constraints can be used to avoid unexpected control inputs because of the presence of the uncertainties or initial condition ⇑ Corresponding author at: National Engineering Laboratory for Highway Maintenance Equipment, Chang’an University, Xi’an, Shanxi 710065, PR China. E-mail address: [email protected] (Y.-H. Chen). https://doi.org/10.1016/j.ymssp.2020.106665 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
deviation [28–30]. For example, the values of a motor’s voltage should be less than the peak voltage. Only a few studies take the inequality constraints of control inputs and outputs into consideration at the same time. In [31], the one-sided robust control is designed to be either positive or negative. In [32], the input/output physically constraints are taken into consideration based on H1 controller. A reference updating method is proposed to keep the response and control effort within the desired bounds. In this paper, we endeavor solving the inequality constraints control design problem. The main contributions are twofold. First, a creative state transformation is proposed and applied to satisfy the state inequality constraints. With this transformation, the states are bounded, regardless of the uncertainty. Second, a novel diffeomorphism is introduced. Based on that, the control input can be regarded as a function of an auxiliary variable. As a result, the value of the control input is limited to be within a prescribed bound. It provides a new way to deal with the inequality control input constraints. The range of the input bounds is related to the uncertainties and can be designed according to the actual situation. The paper is organized as follows. In Section 2, the state transformation approach is introduced. In Section 3, a novel robust control is proposed based on diffeomorphism. We also prove that the control can guarantee uniform boundedness and uniform ultimate boundedness. In Section 4, we summarize the control design for nonlinear uncertain systems. Lastly, two examples are given for demonstration in Sections 5 and 6. 2. The state transformation In most physical systems, the values of state variables should be within the bounds. A large control input may be useful to limit the size of the variables. However, the existence of uncertainty can cause the variables to exceed their limitations. In this section, based on a state transformation, the domains of state variables are transformed, which is able to satisfy the inequality constraints of controlled outputs. Consider a dynamical system
n_ ðtÞ ¼ f ðnðtÞ; rðt Þ; uðtÞ; t Þ; nðt 0 Þ ¼ n0 :
ð1Þ
Suppose that nðtÞ ! nd as t ! þ1 and the inequality constraint is nm < nðt Þ < nM , where nd is the desired value of n; nm and nM are the maximum and minimum of n, respectively. Then, by choosing a proper function aðnÞ, the state transformation can be applied to convert the state n to a new state f without limitation. The function f ¼ aðnÞ should satisfy the condition that f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; nM Þ. Since f ¼ aðnÞ, we have
n ¼ a1 ðfÞ:
ð2Þ
Taking the first order derivative of (2) yields
@ a ðfÞ _ n_ ¼ a_ 1 ðfÞ ¼ f: @f 1
ð3Þ
This means
1 1 @ a ðfÞ f_ ¼ n_ : @f
ð4Þ
Substituting (4) into (1), the dynamical system is transformed to f
1 1 @ a ðfÞ f_ ðtÞ ¼ f a1 ðfðt ÞÞ; rðt Þ; uðtÞ; t : @f
ð5Þ
The second order derivative of (2) is needed when the dynamical system is in the second order form, such that
€nðtÞ ¼ f nðt Þ; n_ ðt Þ; rðtÞ; uðt Þ; t ; nðt 0 Þ ¼ n0 ; n_ ðt 0 Þ ¼ n_ 0 :
ð6Þ
Taking the second order derivative of (2) yields 2 1 1 €n ¼ €f @ a ðfÞ þ f_ 2 @ a ðfÞ : @f @f2
ð7Þ
Thus the dynamic system can be rewritten as
1 1 1 @ a1 ðfÞ _ @ 2 a1 ðfÞ @ a1 ðfÞ €f ¼ @ a ðfÞ f; r; u; t f_ 2 f a1 ðfÞ; : 2 @f @f @f @f
ð8Þ
The following figure shows the state transformation process. The first part ðaÞ of Fig. 1 shows the trajectory of the state variable nðt Þ. In the ideal case, nðt Þ always stays within the bounds nM and nm . It can also get closer to the desired trajectory nd over time. However, the second part ðbÞ shows the
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
3
Fig. 1. The state transformation process.
trajectory of n0ðtÞ in an actual situation. The inequality constraints cannot be easily satisfied due to unknown factors such as uncertainty. Therefore, a state transformation is applied as shown in the third part ðcÞ. The state variable n is transformed to f by the function a. In this process, the constraints can be satisfied based on the property of the transformation function a. Remark 1. This approach is able to satisfy one-sided inequality constraints and two-sided inequality constraints with proper transformation functions. For example, a logarithmic function is applied to solve one-sided inequality constraints. Suppose n > nm and the desired trajectory is nd , then the following transformation function a1 ðnÞ is proposed:
f ¼ a1 ðnÞ ¼ ln
n nm : nd nm
ð9Þ
Thus f n ¼ a1 1 ðfÞ ¼ e ðnd nm Þ þ nm :
ð10Þ
Based on (9) and (10), this logarithmic function satisfies that f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; þ1Þ. Based on a similar conception, a tangent function is applied when there is a two-sided inequality constraint. Suppose that the constraint is nm < n < nM , then the following transformation function a2 ðnÞ is needed:
f ¼ a2 ðnÞ ¼ tan where fd ¼ tan
nd nm nM nm
n ¼ a1 2 ðfÞ ¼
n nm p p fd ; nM nm 2
ð11Þ
p p2 . This means
nM nm
p
arctanðf þ fd Þ þ
nM þ nm : 2
Based on (11) and (12), f ! 0 as n ! nd and f 2 ð1; þ1Þ as n 2 ðnm ; nM Þ are met.
ð12Þ
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3. Robust bounded control 3.1. Control design Consider the following uncertain system
x_ ðt Þ ¼ f ðxðtÞ; t Þ þ BðxðtÞ; t ÞUðwðtÞ; rðtÞ; t Þ þ BðxðtÞ; t Þeðxðt Þ; rðt Þ; tÞ;
ð13Þ
where t 2 R is the time, xðt Þ 2 R is the state, rðt Þ 2 R R is the time-varying uncertain parameter, wðtÞ 2 R is the input. The function rðÞ is Lebesgue measurable and its value lies within a prescribed compact set R Rp . Moreover, the functions f ðÞ; UðÞ, and eðÞ are Carathéodory, and the function BðÞ is strongly Carathéodory [33]. Suppose in practice the input is required to be bounded: wðt Þ 2 W, where W Rm is a prescribed compact set. Suppose n
p
m
also there is a smooth and bijective (i.e., one-to-one) diffeomorphism XðÞ : W ! Rm ; XðwÞ ¼ u; w ¼ X1 ðuÞ, such that Uðwðt Þ; rðt Þ; tÞ ¼ U X1 ðuðt ÞÞ; rðtÞ; t ¼: /ðuðtÞ; rðt Þ; t Þ. We rewrite the system as
x_ ðt Þ ¼ f ðxðtÞ; t Þ þ BðxðtÞ; t Þ/ðuðt Þ; rðtÞ; t Þ þ BðxðtÞ; t Þeðxðt Þ; rðt Þ; tÞ;
ð14Þ
m
where uðt Þ 2 R is regarded as the new control or auxiliary control from now on. Obviously, a design of uðt Þ corresponds to a design of wðt Þ since the diffeomorphism is bijective. Assumption 1. Suppose there exists a known function qðÞ : Rn R ! Rþ such that for all ðx; t Þ 2 Rn R,
r 2 R,
keðx; r; t Þk 6 qðx; t Þ:
ð15Þ
Here qðx; t Þ is related to the system uncertainty bound. Assumption 2. There are known continuous functions (1) wðÞ : Rþ ! Rþ ; wð0Þ ¼ 0; wð pÞ > 0 /ðÞ : Rþ ! Rþ ; /ð0Þ ¼ 0; /ð pÞ > 0 for p > 0, (3) cðÞ : Rþ ! Rþ ; cð0Þ ¼ 0; cð pÞ > 0 for p > 0, such that 2.a)
cðkukÞ 6 uT /ðu; r; tÞ;
for
p > 0,
(2)
ð16Þ
for all ðu; tÞ 2 Rm R; r 2 R. Furthermore, /ð0; r; tÞ ¼ 0. 2.b)
cðwðqÞÞ P qwðqÞ;
ð17Þ
for all q 0. Assumption 3. The origin x ¼ 0 of the nominal system
x_ ¼ f ðx; t Þ;
ð18Þ 1
f ð0; tÞ ¼ 0, is an uniformly asymptotically stable equilibrium point. Moreover, we can find a C Lyapunov function V ðÞ : Rn R ! Rþ and continuous, strictly increasing functions ci ðÞ; i ¼ 1; 2; 3, belonging to class K, such that
c1 ðkxkÞ 6 V ðx; tÞ 6 c2 ðkxkÞ;
ð19Þ
@V ðx; t Þ þ rx V ðx; tÞf ðx; t Þ 6 c3 ðkxkÞ: @t
ð20Þ
limc3 ðrÞ ¼ l < 1;
ð21Þ
Besides, if limr!1 c3 ðrÞ < 1, then this limit exists; that is, r!1
and the domain of c1 3 ðÞ is ½0; lÞ, so that
c1 3 ðÞ : ½0; lÞ ! R þ
ð22Þ
is defined, continuous and strictly increasing. In this case, c3 ðÞ is bounded, and the following assumption is introduced. Assumption 4. If c3 ðÞ is bounded, that is, if (21) is valid, then
< l; where
> 0 is a constant which is related to the control design (shown later).
ð23Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Now, the control uðt Þ ¼ pðxðt Þ; tÞ is proposed below for
pðx; tÞ ¼
lðx; tÞ wðqðx; t ÞÞ; klðx; t Þk
pðx; tÞkaðx; tÞk ¼ aðx; t Þkpðx; t Þk;
5
> 0:
if klðx; tÞk > if klðx; tÞk 6
ð24Þ
ð25Þ
where
lðx; tÞ ¼ aðx; tÞqðx; tÞ;
ð26Þ
aðx; tÞ ¼ BT ðx; tÞrTx V ðx; tÞ:
ð27Þ
Then, we are ready to state a theorem. Theorem 1. Consider uncertain system (14) with the control uðt Þ ¼ pðxðtÞ; tÞ and satisfying Assumptions 1–4. Then, the following results hold [34]: (i) Uniform boundedness: If xðÞ : ½t 0 ; 1Þ ! Rn ; xðt 0 Þ ¼ x0 , is a solution of the controlled system, then kx0 k 6 r ) kxðtÞk 6 dðr Þ; 8t 2 ½t0 ; 1Þ. (ii) Uniform ultimate boundedness: If xðÞ : ½t0 ; 1Þ ! Rn ; xðt 0 Þ ¼ x0 , is a solution of the controlled system with kx0 k 6 r, then r . > 0; kxðt Þk 6 d; 8t P t0 þ T d; for given d The ci functions and related quantities are shown in Fig. 2 [34].
Fig. 2. The ci functions and related quantities.
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Remark 2. Fig. 2 shows the functions ci ; i ¼ 1; 2; 3. It implies that ci ð0Þ ¼ 0 for i ¼ 1; 2; 3, and limr!1 ci ðrÞ ¼ 1 for i ¼ 1; 2. In [34], Corless and Leitman have given a clear explanation for the c functions and Fig. 2. Based on that, Assumption 3 in this paper means that the uncontrolled system without uncertainty is Lyapunov stable with respect to the zero state. As shown in Fig. 2, if there exists a limit l of c3 , then Assumption 4 is applied in the following proof. This assumption can be met by choosing a proper in (23). Similarly, Assumption 2 can be met by choosing suitable functions c and w, which is shown in Remark 4. As for Assumption 1, it means there is a possible size of uncertain elements. The differences between this paper and the previous work [34] are as follows: First, the state transformation approach is applied in this paper to limit the values of control outputs. In this way, a superior system performance can be obtained. Second, based on the previous work, the diffeomorphism approach is introduced. Hence, the novel control scheme is proposed to satisfy the inequality constraints. Proof For a given
rðÞ, the derivative of the Lyapunov function V ðÞ for the closed-loop system is given by
@V @V @V V_ ðx; t Þ ¼ ðx; tÞ þ ðx; tÞf ðx; tÞ þ ðx; t ÞBðx; t Þ½/ð pðx; tÞ; rðtÞ; t Þ þ eðx; rðtÞ; t Þ: @t @x @x
ð28Þ
If klk > , the control is in the form of (24) and can be rewritten as u ¼ ga, where g ¼ wðqÞq=klk. Then, the left side of (16) can be rewritten as
aqwðqÞ ¼ cðwðqÞÞ; cðkukÞ ¼ c l
ð29Þ
and the right side of (16) can be rewritten as
uT /ðu; r; t Þ ¼ gaT /ðga; r; t Þ: Thus, we can obtain
aT /ðga; r; tÞ 6
1
g
cðwðqÞÞ:
ð30Þ
ð31Þ
Obviously, a – 0; u – 0 and wðqÞq > 0 whenever klk > . If klk 6 ; u ¼ ðkuk=kakÞa when a – 0; u – 0. So, by (16), we have
aT /ðu; r; tÞ 6
kak cðkukÞ: kuk
ð32Þ
Besides, if u ¼ 0; ð@V=@xÞB/ ¼ 0 since /ð0; r; t Þ ¼ 0. And if a ¼ 0; ð@V=@xÞB/ ¼ ð@V=@xÞBe ¼ 0. Thus, if klk > , by (15) in Assumption 1, (17) in Assumption 2, (20) in Assumption 3 and Eq. (31), we have
V_ ðx; t Þ 6 c3 ðkxkÞ þ @V B/ðu; r; t Þ þ @V Beðx; r; tÞ @x @x 6 c3 ðkxkÞ þ 1 g cðwðqÞÞ þ kaqk 6 c3 ðkxkÞ wkðlqÞkq cðwðqÞÞ þ klk
ð33Þ
6 c3 ðkxkÞ wkðlqÞkq wðqÞq þ klk ¼ c3 ðkxkÞ: If klk 6 and a – 0; u – 0, by (32), ak V_ ðx; t Þ 6 c3 ðkxkÞ kkuk cðkukÞ þ klk
6 c3 ðkxkÞ þ :
ð34Þ
Also, if u ¼ 0 which implies klk 6 ,
V_ ðx; t Þ 6 c3 ðkxkÞ þ klk 6 c3 ðkxkÞ þ ;
ð35Þ
and if a ¼ 0,
V_ ðx; t Þ 6 c3 ðkxkÞ:
ð36Þ
Therefore, for all ðx; tÞ 2 Rn R, we have
V_ ðx; t Þ 6 c3 ðkxkÞ þ :
ð37Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
7
Fig. 3. The design procedure.
Definition 1. The symbol between functions means the composition of function, such as g ð f ðxÞÞ is represented as ð g f ÞðxÞ. If c3 belongs to K1 , then V_ is negative definite for sufficiently large kxk. If not, there exists a limit, that is limr!1 c3 ðr Þ ¼ l < 1 (21). Then, assumption 4 (23) is applied. The same conclusion can be obtained as well. Consequently, V_ is always negative definite for sufficiently large kxk. Following the standard arguments as in Corless and Leitmann (1981), the solution of the controlled system is uniformly bounded and uniformly ultimately bounded with
(
dðr Þ ¼
c1 if r 6 R 1 c2 ðRÞ; 1 c1 c2 ðrÞ; if r > R
ð38Þ
and
R ¼ c1 3 ðÞ;
ð39Þ
is any constant with d
> c1 c ðRÞ; d 2 1 T d; r ¼
and
8 < 0; c ðrÞc1 ðRÞ : 2 ; c3 ðRÞ
R ¼ c1 2 c1 d :
ð40Þ if r 6 R if r > R
ð41Þ
ð42Þ
Q.E.D. 3.2. Design implications The control design is based on uðtÞ, the transformed input. Once it is determined, the actual input action wðtÞ is determined since the diffeomorphism XðÞ is bijective. The advantage of this approach is that since pðx; tÞ (and hence u) is always continuous, wðt Þ is continuous. A direct design approach for wðt Þ, as was adopted in some past research, on the other hand, may trigger a discontinuous control input. We further substantiated the design details by a few remarks below.
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C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Fig. 4. The quarter vehicle model.
Remark 3. A symmetric structure for pðx; t Þ is implied in (25). In principle, there are infinite choices of pðx; t Þ. A typical choice of pðx; tÞ for klðx; t Þk 6 is
pðx; tÞ ¼
lðx; tÞ wðqðx; t ÞÞ:
ð43Þ
Remark 4. To demonstrate the choices of several functions, let us consider, for example, Uðw; r; t Þ ¼ w, with k < w < k, where k > 0 is a constant. Then one choice of the diffeomorphism XðÞ is XðwÞ ¼ tan p2kw. Since u ¼ tan p2kw, one can show that w ¼ 2k p arctan u and therefore we choose
/ðu; r; tÞ ¼
2k
p
arctan u:
ð44Þ
In order to satisfy (16) in Assumption 2, cðkukÞ can be chosen as
cðkukÞ ¼ juj
2k
p
arctan juj:
ð45Þ
Then (17) in Assumption 2 can be satisfied with wðqÞ ¼ tan p2kq. Remark 5. The magnitude of the control function pðx; tÞ is wðqðx; tÞÞ, while the bound of the uncertainty eðx; r; tÞ is qðx; t Þ. These two magnitudes may not be the same. The reason for this is that the input u acts on the system through the bounding function /ðu; r; t Þ, which may reduce the influence of u. Therefore we need to design the control with the magnitude wðqðx; tÞÞ to pre-compensate the influence due to /ðu; r; tÞ.
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4. Design procedure In Fig. 3, we summarize the proposed control design for the uncertain system with a flowchart. In this paper, the transformed dynamical systems are obtained to meet both the inequality constraints of the control inputs and system states. On one hand, the inequality constraints of control outputs can be satisfied by state transformation. This approach is introduced in Section 2. Fig. 1 clearly shows that the trajectory of control outputs nðt Þ is within bounds. The issue of satisfying the inequality constraints is to find a suitable transformation function a. The domain of state variable can be changed by transformation function. In Remark 1, the logarithmic function and the tangent function are given as examples to deal with one-sided and two-sided inequality constraints. On the other hand, a novel control scheme is proposed to deal with inequality constraints of the control inputs. There are four steps involved in designing this control. First, the control input can be regarded as a function / based on diffeomorphism. Second, a proper control function for different control constraints need be chosen. In this paper, a tangent function (44) is chosen. Thus, the values of control inputs is limited based on the property of the tangent function. Third, the specific value of the bound is to be set by the parameter k. As a result, the inequality constraints can be satisfied. Fourth, the functions c; w and q are selected for meeting the assumptions. We are able to prove that the transformed system state is uniformly bounded and uniformly ultimately bounded. 5. Vehicle active suspension-seat system We apply this control to the vehicle active suspension-seat system shown in Fig. 4. In this case, we take two subsystems into consideration simultaneously. As shown in Fig. 1, there are the active body subsystem and the active seat subsystem. The parameters m1 and m2 are the vehicle body mass and the seat mass, respectively. The displacements of the corresponding masses in the vertical direction are x1 and x2 . The stiffness and damping of the vehicle body suspension (the seat suspension) are k1 and c1 (k2 and c2 ), respectively. The variables u1 and u2 are the control inputs for vehicle body and seat, respectively. The variables z1 ðtÞ and z2 ðtÞ are uncertainties: specifically z1 ðt Þ represents the road excitation displacement, z2 ðt Þ represents the vibration from the vehicle body and the equivalent displacement which is caused by the passenger and/or seat load variation [35]. The constrained dynamic system can be given as [36–38]
8 m1 €x1 ðtÞ þ k1 ðx1 ðt Þ z1 ðt ÞÞ þ c1 x_ 1 ðt Þ þ k2 ðx1 ðt Þ x2 ðt Þ z2 ðt ÞÞ þ c2 ðx_ 1 ðt Þ x_ 2 ðt ÞÞ ¼ u1 ðt Þ; > > > < xm1 < x1 ; > m > 2 €x2 ðtÞ þ k2 ðx2 ðt Þ þ z2 ðt Þ x1 ðt ÞÞ þ c2 ðx_ 2 ðt Þ x_ 1 ðt ÞÞ ¼ u2 ðt Þ; > : xm2 < x2 < xM2 ;
ð46Þ
where xm1 is the lower bound of the vehicle body, xm2 and xM2 are the lower and upper bounds of the seat, respectively. Remark 6. The inequality constraints are used to guarantee the safety and comfort of the vehicle system. The condition xm1 < x1 is proposed to prevent the possibility of collision between the vehicle body and ground. The condition xm2 < x2 < xM2 is proposed to restrict seat shaking. Since the inequality constraints of the vehicle body x1 and seat x2 are one-sided and two-sided, respectively, we can take the state transformations for x1 and x2 as follows:
y1 ¼ ln
x1 xm1 ; xd1 xm1
y2 ¼ tan
x2 xd2 p ; xM2 xm2
ð47Þ ð48Þ
m2 where xd1 ; xd2 are the desired positions of vehicle body and seat and xd2 ¼ xM2 þx . The new state variables are y1 2 ð1; þ1Þ 2 and y2 2 ð1; þ1Þ which are transformed from x1 and x2 . Therefore, x1 ! xm1 as y1 ! 1 and x1 ! xd1 as y1 ! 0. Similarly, x2 ! xm2 as y2 ! 1; x2 ! xM2 as y2 ! þ1 and x2 ! xd2 as y2 ! 0. The process of the state transformations is able to assure that the states x1 and x2 cannot exceed their threshold. Then, we have
x1 ¼ ey1 ðxd1 xm1 Þ þ xm1 ; x2 ¼
xM2 xm2
p
arctan y2 þ xd2 :
ð49Þ ð50Þ
Take the first and second order derivatives of (49) and (50),
x_ 1 ¼ y_ 1 ey1 ðxd1 xm1 Þ;
ð51Þ
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Fig. 5. The displacements of the vehicle body.
Fig. 6. The displacements of the seat.
x_ 2 ¼
xM2 xm2
p
y_ 2 ; 1 þ y22
ð52Þ
€1 ey1 ðxd1 xm1 Þ þ y_ 21 ey1 ðxd1 xm1 Þ; €x1 ¼ y €x2 ¼
xM2 xm2
p
! €2 2y2 y_ 22 y : 2 1 þ y22 1 þ y2 2
Substituting (49), (50), (51), (52), (53) and (54) into (46), we have
ð53Þ ð54Þ
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11
Fig. 7. The velocity of the vehicle body.
Fig. 8. The velocity of the seat.
1 m2 ðk1 þ k2 Þðey1 ðxd1 xm1 Þ þ xm1 Þ ðc1 þ c2 Þðy_ 1 ey1 ðxd1 xm1 ÞÞ þ k2 xM2 x arctan y2 þ xd2 p
xM2 xm2 y_ 2 þ c2 þ k1 z1 þ k2 z2 þ u1 y_ 21 ey1 ðxd1 xm1 Þm1 p 1 þ y22 1 1 x1 þ y 1 u1 ; ¼ g1 þ y e 1 ðxd1 xm1 Þm1 e ðxd1 xm1 Þm1 x x p 1 þ y22 xM2 xm2 y_ 2 M2 m2 €2 ¼ arctan y2 þ xd2 þ c2 k2 ey1 ðxd1 xm1 Þ þ c2 y_ 1 ey1 ðxd1 xm1 Þ k2 y 2 m2 ðxM2 xm2 Þ p p 1 þ y2 p 1 þ y22 p 1 þ y22 m2 ðxM2 xm2 Þ 2y2 y_ 22 ¼ g2 þ x2 þ k2 z2 þ u2 þ u2 ; m2 ðxM2 xm2 Þ m2 ðxM2 xm2 Þ p 1 þ y22 1 þ y22 €1 ¼ ey1 ðx y
d1 xm1 Þm1
ð55Þ
12
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Fig. 9. The acceleration of the vehicle body.
Fig. 10. The acceleration of the seat.
h y_ 2 m2 m2 ðk1 þ k2 Þðey1 ðxd1 xm1 Þ þ xm1 Þ ðc1 þ c2 Þðy_ 1 ey1 ðxd1 xm1 ÞÞ þ k2 xM2 x arctan y2 þ xd2 þ c2 xM2 x p p 1þy22 h i 2 p 1þy ð Þ _2 m2 m2 y y_ 21 ey1 ðxd1 xm1 Þm1 ; x1 ¼ k1 z1 þ k2 z2 ; g 2 ¼ m2 ðxM2 x2 m2 Þ k2 ey1 ðxd1 xm1 Þ þ c2 y_ 1 ey1 ðxd1 xm1 Þ k2 xM2 x arctan y2 þ xd2 þ c2 xM2 x ; p p 1þy2
where
g 1 ¼ ey1 ðx
1
d1 xm1 Þm1
x2 ¼ k2 z2 . Thus, the transformed system is a nonlinear system with ey1 ; arctan y2 and y22 in g 1 ; g 2 .
2
Choose the state variables as y1 ; y2 ; y3 ¼ y_ 1 ; y4 ¼ y_ 2 . The system (55) can be rewritten as
8 y_1 ¼ y3 ; > > > > > < y_2 ¼ y4 ; 1 1 x1 þ ey1 ðxd1 x u1 ¼ g 1 þ ax1 þ au1 ; y_3 ¼ g 1 þ ey1 ðx x > m1 Þm1 m1 Þm1 d1 > > > 2 2 > p 1þy p 1þy : y_ ¼ g þ ð 2 Þ x þ ð 2 Þ u ¼ g þ bx þ bu : 2 2 2 4 2 2 m2 ðxM2 xm2 Þ m2 ðxM2 xm2 Þ 2
ð56Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
13
Fig. 11. The control input on the vehicle body.
Fig. 12. The control input on the seat.
In an actual vehicle suspension-seat system, the control inputs are limited within a proper range for comfort and safety. In this paper, the control input channel is designed to be a tangent function (44). This system can be rewritten based on (14) and we obtain
2 f ð y; t Þ
y3
2
3
0 0
3
60 07 6y 7 x1 6 7 6 47 ; ¼ 6 7; Bð y; t Þ ¼ 6 7; eð y; r; tÞ ¼ 4a 05 4 g1 5 x2
/ðu; r; t Þ ¼
g2 /1 ðu; r; t Þ /2 ðu; r; t Þ
0 " 2k
¼
b
1
arctan u1
2k2
arctan u2
p p
# ;
ð57Þ
14
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
Fig. 13. The relation between k1 ; k2 and the accumulative error of the seat.
Fig. 14. The relation between ; k2 and the accumulative error of the seat.
where k1 > 0 and k2 > 0 are constants. Then the control inputs are properly bounded: /1 ðu; r; t Þ 2 ðk1 ; k1 Þ; /2 ðu; r; t Þ 2 ðk2 ; k2 Þ. The simulations were performed by using m1 ¼ 10; m2 ¼ 3; k1 ¼ 20; k2 ¼ 15; c1 ¼ 1; c2 ¼ 3; xd1 ¼ 0:2; xm1 ¼ 0; xd2 ¼ 0:5; xm2 ¼ 0:49 and xM2 ¼ 0:51. For the uncertainties, we choose z1 ¼ 0:1 sinð4tÞ; z2 ¼ 0:03 cosðt Þ. In simulation, let k1 ¼ 2:7; k2 ¼ 0:5; q1 ¼ 2:5; q2 ¼ 0:48 and ¼ 0:1. Then, Assumption 1 is satisfied because of jjx1 jj 6 q1 and jjx2 jj 6 q2 . As shown in Remark 4, Assumption 2 is also satisfied. In simulation, y_ 3 and y_4 are given as y_ 3 ¼ g1 þ ðg 1 g1 Þ þ ax1 þ a/1 2 þ ðg 2 g2 Þ þ bx2 þ b/b , where g 1 ¼ y1 2y3 and g 2 ¼ y2 2y4 . Thus, for the nominal system, P can be and y_ 4 ¼ g obtained by solving the Lyapunov equation [39] by choosing V ¼ yT Py and Q ¼ I. Assumption 3 is satisfied by choosing
c1 ðjjxjjÞ ¼ Emin ðPÞjjxjj2 ¼ 0:29jjxjj2 ; c2 ðjjxjjÞ ¼ Emax ðPÞjjxjj2 ¼ 1:71jjxjj2 and c3 ðjjxjjÞ ¼ Emax ðQ Þjjxjj2 ¼ jjxjj2 , where EminðmaxÞ ðÞ represents the minimum (maximum) eigenvalue of ðÞ. And Assumption 4 is not applied in this case since limr!1 c3 ðr Þ ¼ 1. For comparisons, the simulations of (i) a bounded control without state transformation and (ii) the LQR control are performed. The matrices Q and R in the LQR control are both selected as I in simulations. This means all states and all controls are equally weighted. The reasons for choosing the LQR are twofold. First, the LQR is a proven optimal control. It minimizes a performance index including the state and control costs. Second, the LQR is a proven robust control. The most common robustness measures attributed to the LQR are a one-half gain reduction in any input channel, an infinite gain amplification in any input channel, or a phase error of plus or minus sixty degrees in any input channel. In addition, there is robustness to uncertainty in the real coefficients of the model and certain nonlinearities, including control switching and saturation [40].
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15
Fig. 15. The Motor-Elevator model.
Fig. 5 shows the comparison of displacements of the vehicle body under different controls. The blue line represents the displacement of the vehicle body under bounded control with state transformation. The green and red lines represent the displacements under the bounded control without state transformation and LQR control, respectively. The yellow and black lines represent the desired displacement and lower bound of the vehicle body, respectively. From this figure, we can see that there are smaller fluctuations of the displacements under bounded control, represented by the blue and green lines, than that under LQR control which is represented by the red line. In the partial view, we can clearly see that the vehicle body cannot exceed the lower bound under bounded control with state transformation. However, LQR control and bounded control without state transformation cannot assure this. Similarly, Fig. 6 shows the comparison of displacements of the seat under different controls. Blue, green and red lines represent the displacements under bounded control with state transformation, bounded control without state transformation, and LQR control, respectively. Yellow and black lines are the desired displacement and bounds. The best performance is under bounded control with state transformation, which does not exceed the bounds with the smallest displacement fluctuation. The performance of displacement under bounded control without state transformation is better than the performance under LQR control. However, they both exceed the bounds due to the uncertainty. Figs. 7 and 8 show the velocities of vehicle body and seat, respectively. The blue, green and red lines represent the accelerations under bounded control with state transformation, bounded control without state transformation and LQR control. From Figs. 7 and 8, we can see that the fluctuation under bounded control with state transformation is the smallest. Figs. 9 and 10 show the accelerations of vehicle body and seat, respectively. The blue, green and red lines represent the accelerations under the bounded control with state transformation, bounded control without state transformation and LQR control. From these two figures, we find that a better superior was obtained under bounded control (by comparing the green line with red line). At the same time, the state transformation can also be helpful for seeking a better performance by comparing the blue line with green line. In summary, the fluctuation under bounded control with state transformation is the smallest. Figs. 11 and 12 show the control inputs on the vehicle body and seat, respectively. The blue, green and red lines represent the bounded control with state transformation, the bounded control without state transformation, and LQR control. We can see that the bounded control cannot exceed the bounds which are represented by black dotted lines while the LQR control exceeds the bounds. In this case, the parameters of control bounds are set as: k1 ¼ 2:7 and k2 ¼ 0:5. Thus, in Fig. 11, the values of bounded controls, in the blue and green lines, are between ð2:7; 2:7Þ. Similarly, the values of bounded controls in Fig. 12 are between ð0:5; 0:5Þ. These two figures prove that the proposed control method is able to satisfy the control input inequality constraints. Fig. 13 shows the relation between k1 ; k2 , and the accumulative error of the seat. From this figure, we can see that the accumulative error decreases as the two parameters k1 and k2 increase, because these two parameters decide the bounds of control as shown in (44). A larger value of k means that the control can provide a larger control input to obtain a better system performance. Fig. 14 shows the relation between ; k2 and the accumulative error of the seat. We can see the accumulative error is proportional to and inversely proportional to k2 . As for , it implies the neighborhood size of the zero state for all possible system responses. By letting ! 0, an arbitrarily small neighborhood of the zero state can be obtained. Thus, in this figure, the system performance gets better as the parameter decreases. As for k2 , it decides the bound of the seat control. The larger control input can be obtained with a larger k2 . Thus, in this figure, the system performance gets better as the parameter k2 increases.
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6. Motor-elevator system We next apply this control to a first-order system. Consider the elevator system that is shown in Fig. 15, where M a ; x; U are the output torque, speed, and voltage of the motor, respectively, T load is the load torque including the elevator gravity, nonlinear friction, and disturbance. The dynamical model of the motor-elevator system is
U ðtÞ ke xðt Þ Bm xðt Þ T load ðt Þ; R
_ ðt Þ ¼ k t Ma ðt Þ ¼ J x
ð58Þ
where J is the moment of inertia of the motor, Bm is the viscous damping coefficients of the motor, R is the total resistance of the armature circuit, kt and ke are the constants of the motor torque and the electromotive force, respectively. In order to render desirable performances of motor speed and voltage, the inequality constraints xm < x < xM and U < U < U are imposed, where xm and xM are the maximum and minimum of the motor speed, respectively, U is the peak voltage. The speed is constrained: x 2 ðxm ; xM Þ. The state transformation is taken by using a tangent function:
m ¼ tan
2x xM xm p ; 2ðxM xm Þ
ð59Þ
where m 2 ð1; þ1Þ is the transformed state from x. Thus, x ! xm when m ! 1; x ! xM when m ! þ1 and x ! ðxM þ xm Þ=2 ¼ xd when m ! 0. xd is the desired speed. Therefore we have
x¼
xM xm xM þ xm arctan m þ : p 2
ð60Þ
Taking the first order derivative of (60) yields
x_ ¼
xM xm m_ : p 1 þ m2
ð61Þ
Next, the control input is proposed in the form of (44). Substitute (60) and (61) into (58). Let us rewrite it in the form of (14):
m_ ¼
kt ke RJ
kt þ RJ
þ BJm
pð1þm2 Þ xM xm
h
M þxm arctan m þ m2 arctan m þ m2 x xM xm
i
ke ðxM þxm Þ Bm R xM þxm R : arctan u T load 2 2 kt pð1þm Þ kt p pð1þm Þ
2k
Fig. 16. The motor speed.
ð62Þ
C. Li et al. / Mechanical Systems and Signal Processing 140 (2020) 106665
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Fig. 17. The motor voltage.
Therefore,
f ðm; t Þ
h i t ke M þxm ¼ kRJ þ BJm arctan m þ m2 arctan m þ m2 x xM xm ;
Bðm; t Þ
¼ kRJt
pð1þm2 Þ xM xm ;
þxm M þxm Þ BkmtR pxM1þ ; ¼ kRt T load kepðx1þ ð m2 Þ ð m2 Þ 2k /ðu; r; t Þ ¼ p arctan u:
eðm; r; tÞ
ð63Þ
Let xd ¼ 0:5; xm ¼ 0:48 and xM ¼ 0:52. The values of the relevant parameters are given as kt ¼ 1; ke ¼ 5; R ¼ 1:2; J ¼ 5 and Bm ¼ 3. Suppose that the uncertainty T load ¼ 10 sinð4t Þ and the bound of control input k ¼ 15. The simulation results are shown as follows. The LQR control is also applied to this model as a comparison. Fig. 16 shows the motor speeds under (i) the bounded control with state transformation, (ii) a bounded control without state transformation, and (ii) the LQR control. They are represented by the blue, green, and red lines, respectively. The yellow and black lines represent the desired speed and the bounds of speed, respectively. From this figure, we can see that the performance is better under the bounded control, comparing with the LQR control. The speed cannot exceed the threshold under the state transformation. Fig. 17 shows the comparison of the motor voltages under three different control schemes. The blue solid line represents the voltage under the bounded control with state transformation. The bounds of blue solid line are represented by the blue dotted lines. The green solid line represents the voltage under a bounded control without state transformation. The bounds of green solid line are represented by the green dotted lines. The red solid line represents the voltage under the LQR control. In these three control schemes, the control cost is the smallest under the bounded control with state transformation. From these two figures, we can see that the bounded control with state transformation exhibits the best performance with the smallest control cost. 7. Conclusion In this paper, the inequality constraints of control inputs and outputs are taken into consideration for nonlinear uncertain systems simultaneously. There are two major contributions. First, the state transformation is applied to convert the bounded states into new states without bounds. The tangent and logarithmic transformed functions are used to deal with one-sided and two-sided inequality constraints. Second, the control input is regarded as a bounding function through a diffeomorphism. This way, the bound of the control input can be artificially set regardless of the uncertainty by choosing a proper bounding function. This control scheme is proven to guarantee uniform boundedness and uniform ultimate boundedness. In the simulations, we show that this control scheme exhibits a superior performance and smaller control cost when compared to the LQR control. We believe that this study opens a new door for dealing with inequality constraints in controlling uncertain dynamical systems. Future explorations along this avenue may include the choices of more transformation functions or the asymmetric bounded problem of inequality constraints.
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Conflicts of Interest The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. CRediT authorship contribution statement Chenming Li: Software, Validation, Formal analysis, Investigation, Writing - original draft. Han Zhao: Resources, Project administration. Hao Sun: Data curation, Validation, Funding acquisition. Ye-Hwa Chen: Conceptualization, Methodology, Supervision, Writing - review & editing. Acknowledgements The research is supported by the China Scholarship Council (No. 201806690019). The research is also supported by the Natural Science Foundation of Anhui Province (No. 1908085QE194), the Fundamental Research Funds for the Central Universities of China (Nos. JZ2019HGTA0042, PA2019GDPK0066), and National Natural Science Foundation of China (No. 51905140). The first author would like to thank his wife Qiaozhi Zhang, for her support and company. References [1] J.G. 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