Sage Journals: Discover world-class research

Abstract

In this article, a dynamic output feedback based sliding mode learning control is proposed for uncertain mechanical system. After giving the model of uncertain mechanical system, the uncertainty and disturbance of it are discussed and they are assumed to be mismatched. The velocity of the uncertain mechanical system is assumed to be unmeasurable, and then a dynamic output feedback control strategy is utilized here. A dynamic output feedback-based sliding surface is constructed. The parameters of the designed surface are solved by Lyapunov function approach. Then a sliding mode learning controller is proposed for uncertain mechanical system to overcome the chattering of traditional sliding mode control. Finally, a numerical simulation is given to show the effectiveness of the proposed controller.

Keywords

Uncertain mechanical system dynamic output feedback sliding mode learning control

Introduction

In actual production process, a lot of industrial system can be represented by a mechanical system, such as active suspension of vehicle,¹ serial robot arm,² vessel-riser system,³ planar three-link mechanical system⁴ and rotational mechanical system.⁵ For the successful application of industrial system, the controller design of mechanical system has drawn much attention in recent years.^6,7 However, for an actual industrial system, a specific sensor is needed if we want to get a certain variable, for example, a velocity sensor is need if we want to measure the velocity of the industrial system timely. Unfortunately, adding a specific sensor to the industrial system will certainly increase the cost. Simultaneously, the velocity information measured by the sensor is often affected by noise, so it is inconvenient to be applied directly. If we can design a controller without the sensor of velocity, the cost of the industrial system will certainly decrease, and the control performance will be improved. In this case, output feedback based controller design method is an effective way for the control of mechanical system.^8–10

Because of the change of working environment and the existence of various external disturbances, uncertainty and disturbance are inevitable in mechanical system; in this case, a robust controller is needed. Sliding mode control (SMC) is a kind of robust control since it has great robustness for system disturbance.¹¹ For the controller design of SMC, a reduced order sliding surface is constructed first in which the reduced dynamics are stable. Then a discontinuous control law, which can force the system dynamics to the sliding surface, is designed. Based on the designed sliding surface and controller, the controlled system is unaffected by the disturbance.¹² SMC has been applied to the robust control of uncertain system, such as adaptive control of hypersonic flight vehicle¹³ and active suspension vehicle systems,¹⁰ adaptive control of fuzzy system,^14,15 and sliding mode control of master-slave time-delay systems.⁶ Output feedback based SMC has also been widely studied, such as static output feedback (SOF) based SMC,^16,17 and has been successfully utilized in Markovian jump systems¹⁸ and affine nonlinear system.¹⁹ But the information of SOF based SMC is one-sided, and partial states of the original system is missed. In this case, dynamic output feedback (DOF) based SMC is needed.

Meanwhile, when utilizing SMC, the upper bound of the interference is assumed to be known.²⁰ While for a real system, the interference and uncertainty are difficult to be modeled or measured.²¹ An effective way for this question is choosing a big enough upper bound for the interference and uncertainty, but this will cause chattering in the control input.¹¹ Chattering is an obvious shortcoming of SMC, and is harmful to the stability of practical system, so it must be avoided or at least reduced. For reducing chattering, a lot of results have been listed in literature, such as adaptive law based SMC,^22,23 but in real application, chattering is still an open problem, and is the main difficulty for the application of SMC. In this case, a novel DOF based SMC, which can greatly reduce the chattering of traditional SMC, is needed for the control of mechanical system.

Recently, sliding mode learning control (SMLC) has been proposed and successfully applied on the controller design of uncertain system.²⁴ Similar with traditional SMC, a sliding surface is first constructed, and it’s stability is guaranteed by correctly selecting the parameter of the sliding surface. Then, a learning controller is introduced. The learning controller can greatly reduce chattering of traditional SMC on the basis of guaranteeing SMC’s robustness, so it is more practical in real application.²⁵ But for the output feedback based SMLC, there are really few results can be found in literatures, let alone DOF-based SMLC. So for the control of mechanical system, a novel DOF-SMLC should be proposed first.

Motivated by the above discussions, a DOF-based improved SMLC is proposed for the robust control of mechanical system in this paper. An uncertain model of mechanical system is proposed first, and then the model is transformed into a standard one, more specifically, a linear uncertain system with disturbances. A DOF controller is designed, then a DOF-based sliding surface is designed and the stability of it is guaranteed by selecting parameters appropriately. After getting the sliding surface, a learning controller is proposed for the uncertain mechanical system. Finally, a solving algorithm is given for the proposed DOF-SMLC. The proposed DOF based SMLC is confirmed by a numerical example.

The novelties and main contributions of the paper can be summarized and listed as:

For an uncertain mechanical system with unknown parameter uncertainty and disturbance, a learning controller is designed.

A DOF-based sliding surface is proposed for uncertain mechanical system;

A novel SMLC is proposed and the proposed controller can greatly reduce the chattering of traditional SMC.

This paper is organized as follows. The uncertain mechanical system model is listed in section “Problem Formulation,” and the main results are given in section “Main Results.” Numerical simulation results are given in section “Simulation results.” This paper is summarized in section “Conclusion.”

Problem formulation

Model of mechanical system

The mechanical system considered in this paper is a 2 degree-of-freedom mechanical system, and a sketch of the mechanical system is given in Figure 1. The system equation described in physical coordinate is listed as following:

\begin{matrix} M^{'} \overset{\cdot\cdot}{r} (t) + (G^{'} + Δ G^{'}) \overset{\cdot}{r} (t) \\ + (K^{'} + Δ K^{'}) r (t) = B^{'} u (t) + E^{'} d (t) \end{matrix}

(1)

where $r (t) = {[\begin{matrix} r_{1} (t) & r_{2} (t) \end{matrix}]}^{T}$ represents the position vector of the mechanical system, correspondingly, $\overset{\cdot}{r} (t)$ represents the velocity vector, and $\overset{\cdot\cdot}{r} (t)$ represents the acceleration vector. $M^{'}$ is the mass of the system, and $G^{'}$ represents the gyroscopic/dissipation characteristics, $K^{'}$ represents the stiffness characteristics, respectively, $Δ G^{'}$ and $Δ K^{'}$ are unknown uncertainties in gyroscopic/dissipation characteristics and stiffness characteristics, respectively, and

\begin{matrix} M^{'} = [\begin{matrix} m_{s 1} & 0 \\ 0 & m_{s 2} \end{matrix}], G^{'} = [\begin{matrix} c_{s} & - c_{s} \\ - c_{s} & c_{s} \end{matrix}], \\ K^{'} = [\begin{matrix} k_{s} & - k_{s} \\ - k_{s} & k_{s} \end{matrix}] \end{matrix}

Δ G^{'} = [\begin{matrix} Δ c_{s} & - Δ c_{s} \\ - Δ c_{s} & Δ c_{s} \end{matrix}], Δ K^{'} = [\begin{matrix} Δ k_{s} & - Δ k_{s} \\ - Δ k_{s} & Δ k_{s} \end{matrix}]

B^{'} = [\begin{matrix} 1 \\ 0 \end{matrix}], E^{'} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

Figure 1.

Geometry of the mechanical system.

In equation (1), $M^{'}$ , $G^{'}$ and $K^{'}$ are all known. $u (t)$ is the control input and $d (t)$ is unknown disturbance. $B^{'}$ is the input matrix, and $E^{'}$ is the disturbance matrix. Then equation (1) can be rewrote as

\begin{matrix} \overset{\cdot\cdot}{r} (t) = - M^{' - 1} (G^{'} + Δ G^{'}) \overset{\cdot}{r} (t) - M^{' - 1} (K^{'} + Δ K^{'}) r (t) \\ + M^{' - 1} B^{'} u (t) + M^{' - 1} E^{'} d (t) \\ = - M^{' - 1} G^{'} \overset{\cdot}{r} (t) - M^{' - 1} K^{'} r (t) + M^{' - 1} B^{'} u (t) \\ - M^{' - 1} Δ G^{'} \overset{\cdot}{r} (t) - M^{' - 1} Δ K^{'} r (t) + M^{' - 1} E^{'} d (t) \end{matrix}

(2)

For mechanical system model (2), choosing the state as

x (t) = [\begin{matrix} r_{1} (t) \\ r_{2} (t) \\ {\overset{\cdot}{r}}_{1} (t) \\ {\overset{\cdot}{r}}_{2} (t) \end{matrix}]

then system (2) can be rewrote as

\overset{\cdot}{x} (t) = (A + Δ A) x (t) + Bu (t) + Ed (t)

(3)

where

\begin{matrix} A = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - \frac{k_{s}}{m_{s 1}} & \frac{k_{s}}{m_{s 1}} & - \frac{c_{s}}{m_{s 1}} & \frac{c_{s}}{m_{s 1}} \\ \frac{k_{s}}{m_{s 2}} & - \frac{k_{s}}{m_{s 2}} & \frac{c_{s}}{m_{s 2}} & - \frac{c_{s}}{m_{s 2}} \end{matrix}], \\ Δ A = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - \frac{Δ k_{s}}{m_{s 1}} & \frac{Δ k_{s}}{m_{s 1}} & - \frac{Δ c_{s}}{m_{s 1}} & \frac{Δ c_{s}}{m_{s 1}} \\ \frac{Δ k_{s}}{m_{s 2}} & - \frac{Δ k_{s}}{m_{s 2}} & \frac{Δ c_{s}}{m_{s 2}} & - \frac{Δ c_{s}}{m_{s 2}} \end{matrix}] \end{matrix}

B = [\begin{matrix} 0 \\ 0 \\ \frac{1}{m_{s 1}} \\ 0 \end{matrix}], E = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ \frac{1}{m_{s 1}} & 0 \\ 0 & \frac{1}{m_{s 2}} \end{matrix}]

Control objective

For the original system equation (1), assuming that, the measurable output of the system (3) is

y (t) = Cx (t)

then the equation of mechanical system can be summarized as

\begin{matrix} \overset{\cdot}{x} (t) = (A + Δ A) x (t) + Bu (t) + Ed (t), \\ y (t) = Cx (t) \end{matrix}

(4)

For the uncertain matrix $Δ A$ , since $Δ G^{'}$ and $Δ K^{'}$ are all unknown, $Δ A$ is also unknown. Similarly, the disturbance $d (t)$ is unknown. In equation (3), $rank (Δ A) = 2$ , $rank (B) = 1$ , $Δ A$ doesn’t satisfy the matched condition,¹³ so it is mismatched. Similarly, $d (t)$ is also mismatched. For $Δ A$ and $d (t)$ , the following assumption are made for the controller design of this paper.

Assumption 1

$Δ A$ is unknown but bounded, which means that,

‖ Δ A ‖ \leq \bar{A}

where $\bar{A}$ is a known constant vector.

Assumption 2

$d (t)$ is unknown but bounded, which means that, there exist an known scalar $\bar{d}$ ,

‖ d (t) ‖ \leq \bar{d}

Remark 1

In Assumptions 1 and 2, the unbound of $Δ A$ and $d (t)$ are known. This is reasonable since the uncertainties of gyroscopic/dissipation characteristics and stiffness characteristics can be modeled according to the limitation of physical system.

Considering the existence of $Δ A$ and $d (t)$ , a robust controller is needed for equation (3). In consideration of the advantage of SMC in robust control, SMC is utilized here for the robust controller design. Also taking into account that, only $y (t)$ is measurable in equation (4), an output feedback controller is needed here. Then the control goal for mechanical system in this paper is: Designing an output feedback-based SMC, which can guarantee the stability of the closed system in the existence of mismatched $Δ A$ and $d (t) .$

Main results

Since only the output of equation (4) is measurable, output feedback-based controller design strategy is utilized here. Also considering the robust requirement for uncertainty and disturbance, SMC strategy is utilized here. In this case, DOF-based SMC approach is utilized here for the robust controller design of mechanical system.

DOF based sliding surface design

For system equation (4), a full-order DOF controller is constructed as

{\begin{matrix} \overset{\cdot}{\bar{x}} (t) = A_{D} \bar{x} (t) + B_{D} y (t), \\ u (t) = C_{D} \bar{x} (t) + u_{r} (t) \end{matrix}

(5)

where $\bar{x} (t)$ is the state of the new system, $y (t)$ is the measured output of equation (4), $A_{D}$ , $B_{D}$ and $C_{D}$ are matrices needed to be determined, $u (t)$ is the designed controller, $u_{r} (t)$ is the nonlinear part related to SMC designing strategy, and needed to be designed later. Applying the above controller to system (4), the overall system can be obtained as

\begin{matrix} \overset{\cdot}{χ} (t) = A_{e} χ (t) + Δ A_{e} χ (t) + B_{e} u_{r} (t) + E_{e} d (t), \\ y_{χ} (t) = C_{e} χ (t) \end{matrix}

(6)

where

\begin{matrix} χ (t) = [\begin{matrix} x (t) \\ \bar{x} (t) \end{matrix}], A_{e} = [\begin{matrix} A & B C_{D} \\ B_{D} C & A_{D} \end{matrix}], \\ Δ A_{e} = [\begin{matrix} Δ A & 0 \\ 0 & 0 \end{matrix}], \\ B_{e} = [\begin{matrix} B \\ 0 \end{matrix}], E_{e} = [\begin{matrix} E \\ 0 \end{matrix}], C_{e} = [\begin{matrix} C & 0 \\ 0 & I \end{matrix}] \end{matrix}

Then, the control of equation (4) is converted into the closed-loop stability of equation (6). According to the SMC designing strategy, an output feedback based sliding surface for equation (6) is defined as

\begin{matrix} S (t) = K y_{χ} (t) = K C_{e} χ (t) = [\begin{matrix} K_{1} & K_{2} \end{matrix}] [\begin{matrix} C & 0 \\ 0 & I \end{matrix}] [\begin{matrix} x (t) \\ \bar{x} (t) \end{matrix}] \\ = K_{1} Cx (t) + K_{2} \bar{x} (t) = K_{1} y (t) + K_{2} \bar{x} (t) \end{matrix}

(7)

where $K$ is a matrix needed to be designed.

For system (6), $u_{r} (t)$ is the discontinuous part of SMC, so equation (6) is an autonomous system and the equivalent control is not in existence. In this case, what we should do is just discussing the stability of $A_{e}$ . Similarly, for the designed sliding surface, we should only analysis the stability of $S (t)$ without other influence. Then for $S (t)$ , we have

\begin{matrix} \overset{\cdot}{S} (t) = K {\overset{\cdot}{y}}_{χ} (t) = K C_{e} \overset{\cdot}{χ} (t) \\ = K C_{e} A_{e} χ (t) + K C_{e} Δ A_{e} χ (t) \\ + K C_{e} B_{e} u_{r} (t) + K C_{e} E_{e} d (t) \end{matrix}

(8)

For analyzing the stability of $S$ , we define the Lyapunov function for system (8) as

V_{s} (t) = \frac{S^{T} (t) S (t)}{2}

Then

\begin{matrix} {\overset{\cdot}{V}}_{s} (t) = \frac{S^{T} (t) \overset{\cdot}{S} (t) + {\overset{\cdot}{S}}^{T} (t) S (t)}{2} \\ = \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} \overset{\cdot}{χ} (t) + {\overset{\cdot}{χ}}^{T} (t) C_{e}^{T} K^{T} K C_{e} χ (t)) \\ = \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} A_{e} χ (t) + χ^{T} (t) A_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \\ + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} B_{e} u_{r} (t) + u_{r}^{T} (t) B_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \\ + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} D_{e} d (t) + d^{T} (t) D_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \\ + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} Δ A_{e} χ (t) + χ^{T} (t) Δ A_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \end{matrix}

For the stability analysis of $S (t)$ , $Δ A_{e}$ , $u_{r} (t)$ , and $d (t)$ can all be viewed as disturbance, the stability of $S (t)$ is decided on the value of $χ^{T} (t) C_{e}^{T} K^{T} K C_{e} A_{e} χ (t) + χ^{T} (t) A_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)$ . Then we can decisively make the following conclusion: If the following inequation holds, the reduced dynamic on sliding surface equation (7) is stable

C_{e}^{T} K^{T} K C_{e} A_{e} + A_{e}^{T} C_{e}^{T} K^{T} K C_{e} < - Q

(9)

where $Q$ is a positive matrix. If equation (9) is hold, ${\overset{\cdot}{V}}_{s} (t)$ is reduced to

\begin{array}{l} {\dot{V}}_{s} (t) < - e_{ζ}^{T} Q e_{ζ} + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} B_{e} u_{r} (t) \\ + u_{r}^{T} (t) B_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \\ + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} D_{e} d + d^{T} D_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \\ + \frac{1}{2} (χ^{T} (t) C_{e}^{T} K^{T} K C_{e} Δ A_{e} χ (t) + χ^{T} (t) Δ A_{e}^{T} C_{e}^{T} K^{T} K C_{e} χ (t)) \end{array}

Since $u_{r} (t)$ is the discontinuous part of SMC, $u_{r} (t)$ is bounded. From Assumption 1 and 2, $Δ A_{e}$ , and $d (t)$ are all bounded. If

‖ Q ‖ > ‖ B_{e} u_{r} (t) ‖ + ‖ D_{e} d ‖ + ‖ Δ A_{e} ‖

${\overset{\cdot}{V}}_{s}$ is reduced to

{\overset{\cdot}{V}}_{s} (t) < 0

in this case, the stability of the reduced dynamics on sliding surface equation (7) is guaranteed. Then the stability of equation (7) can be summered as the following question

Finding a matrix $K$ and the output feedback gain $A_{D}$ , $B_{D}$ and $C_{D}$ , which can guaranteed the following inequation: $C_{e}^{T} K^{T} K C_{e} A_{e} + A_{e}^{T} C_{e}^{T} K^{T} K C_{e} < - Q$

The matrix $K$ should satisfied $‖ Q ‖ > ‖ B_{e} u_{r} (t) ‖ + ‖ D_{e} d ‖ + ‖ Δ A_{e} ‖$ .

Traditional discontinuous controller design

If equation (9) is held, the stability of the sliding surface $S (t)$ is guaranteed. For the stability of (6), the reachability of the sliding surface should also be guaranteed. For the reachability of $S (t)$ , the following inequation should be satisfied

S^{T} (t) \overset{\cdot}{S} (t) < 0

which means that

\begin{matrix} S^{T} (t) K C_{e} A_{e} χ (t) + S^{T} (t) K C_{e} Δ A_{e} χ (t) \\ + S^{T} (t) K C_{e} B_{e} u_{r} (t) + S^{T} (t) K C_{e} E_{e} d (t) < 0 \end{matrix}

(10)

For equation (10), it is difficult to get a feasible $u_{r}$ since $K$ is unknown. For the designed matrix $K$

K C_{e} B_{e} = [\begin{matrix} K_{1} & K_{2} \end{matrix}] [\begin{matrix} C & 0 \\ 0 & I \end{matrix}] [\begin{matrix} B \\ 0 \end{matrix}] = K_{1} CB

and $K_{1} \in R^{1 \times 6}$ , $C$ and $B$ are all known, then for the convenience of control design, we choose the value of $K_{1}$ to make that, $K_{1} CB = I$ , then

K C_{e} B_{e} = I

(11)

and equation (10) can be reduced to

\begin{matrix} S^{T} K C_{e} A_{e} χ (t) + S^{T} K C_{e} Δ A_{e} χ (t) \\ + S^{T} u_{r} (t) + S^{T} K C_{e} E_{e} d (t) < 0 \end{matrix}

Since

S^{T} K C_{e} A_{e} χ (t) < λ_{max} (A_{e}) S^{T} S

and

S^{T} K C_{e} E_{e} d (t) < λ_{max} (D_{e}) d_{max} S^{T} S

S^{T} K C_{e} Δ A_{e} χ (t) < \bar{A} S^{T} S

the discontinuous controller can be chosen as

u_{r} = {\begin{matrix} - ρ \frac{S}{‖ S ‖} & If S \neq 0, \\ 0 & If S = 0 \end{matrix}

where

ρ > λ_{max} (A_{e}) + λ_{max} (D_{e}) d_{max} ‖ S ‖

Remark 2

For solving $u_{r}$ , we make the assumption that $K_{1} \bar{C} \bar{B} = I$ , then $K C_{e} B_{e} = I$ . $K \in R^{1 \times 5}$ , and $K_{1} \in R^{1 \times 2}$ , in this case, we lost $2$ dimension degree in the design of $K$ , but we still have $4$ dimension degree since $K_{2} \in R^{1 \times 4}$ .

Then for the designed sliding surface equation (7), the SMC controller is

u (t) = C_{D} \bar{x} (t) + u_{r} (t)

u_{r} = {\begin{matrix} - ρ \frac{S}{‖ S ‖} & If S \neq 0, \\ 0 & If S = 0 \end{matrix}

SMLC design

In subsection “Traditional discontinuous controller design,” a discontinuous controller $u_{r} (t)$ is designed for mechanical system (3). But when utilized the discontinuous controller $u_{r} (t)$ , chattering is inevitable. Obviously, chattering is unexpected for a real system, then a novel controller, which can retain the advantage of SMC while reduced chattering, is need. In this subsection, a sliding mode learning controller is proposed. The given learning controller is

u_{r} (t) = u_{r} (t - τ) + Δ u_{r} (t)

(12)

where $τ$ is a time delay, $Δ u_{r} (t)$ is an adaptation term needed to be designed

Δ u_{r} (t) = {\begin{matrix} - (α {\hat{\overset{\cdot}{V}}}_{3} (t - τ) + β V_{3} (t)) & for & S (t) \neq 0, \\ 0 & for & S (t) = 0 \end{matrix}

(13)

where $V_{3} (t)$ is a Lyapunov function which will be given later, $α$ and $β$ are designed parameters. ${\hat{\overset{\cdot}{V}}}_{3} (t - τ)$ is the numerical approximating of ${\overset{\cdot}{V}}_{3} (t - τ)$ .

Assumption 3

The numerical approximating error is reasonable small, which means that, the sign of ${\hat{\overset{\cdot}{V}}}_{3} (t - τ)$ and ${\overset{\cdot}{V}}_{3} (t - τ)$ is the same, and

| {\overset{\cdot}{V}}_{3} (t - τ) - {\hat{\overset{\cdot}{V}}}_{3} (t - τ) | < ϰ | {\hat{\overset{\cdot}{V}}}_{3} (t - τ) |

where $ϰ$ is a positive constant, and $ϰ << 1$ .

Theorem 1

For system (6), if Assumption 3 is hold, and $u (t)$ are chosen as equations (12) and (13), the parameters $α$ and $β$ are

\frac{1}{M} < α < 1 - \frac{1}{M} - γ, β > 0

(14)

then equation (6) is stable.

Proof

Constructing Lyapunov function for equation (6) as

V_{3} (t) = \frac{1}{2} S^{T} (t) S (t)

For sliding surface (7)

\begin{matrix} \overset{\cdot}{S} (t) = K {\overset{\cdot}{y}}_{χ} (t) = K C_{e} \overset{\cdot}{χ} (t) \\ = K C_{e} A_{e} χ (t) + K C_{e} Δ A_{e} χ (t) + K C_{e} B_{e} u_{r} (t) \\ + K C_{e} E_{e} d (t) \\ = F (t) + u_{r} (t) \end{matrix}

where $F (t) = K C_{e} A_{e} χ (t) + K C_{e} Δ A_{e} χ (t) + K C_{e} E_{e} d (t)$ .

Then for $V_{3} (t)$

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = S^{T} (t) \overset{\cdot}{S} (t) \\ = S^{T} (t) (F (t) + u_{r} (t)) \end{matrix}

Using the designed controller (12) we have

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = S^{T} (t) F (t) + S^{T} (t) u_{r} (t - τ) \\ + S^{T} (t) Δ u_{r} (t) \end{matrix}

with the expression of $Δ u (t)$ we have

S^{T} (t) Δ u_{r} (t) = - α {\hat{\overset{\cdot}{V}}}_{3} (t - τ) - β V_{3} (t)

Then

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = S^{T} (t) F (t) + S^{T} (t) u (t - τ) \\ - α {\hat{\overset{\cdot}{V}}}_{3} (t - τ) - β V_{3} (t) \end{matrix}

Since $τ$ is reasonable small, if ${\overset{\cdot}{V}}_{3} (t, t - τ) \neq 0$ , ${\overset{\cdot}{V}}_{3} (t - τ) \neq 0$ and ${\hat{\overset{\cdot}{V}}}_{3} (t - τ) \neq 0,$ then

| {\overset{\cdot}{V}}_{3} (t, t - τ) - {\overset{\cdot}{V}}_{3} (t - τ) | < \frac{1}{M} {\hat{\overset{\cdot}{V}}}_{3} (t - τ)

where $M >> 1$ . Then

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) \leq {\overset{\cdot}{V}}_{3} (t, t - τ) - {\overset{\cdot}{V}}_{3} (t - τ) + {\overset{\cdot}{V}}_{3} (t - τ) \\ - α {\hat{\overset{\cdot}{V}}}_{3} (t - τ) - β V_{3} (t) \\ < \frac{1}{M} {\hat{\overset{\cdot}{V}}}_{3} (t - τ) + {\overset{\cdot}{V}}_{3} (t - τ) - α {\hat{\overset{\cdot}{V}}}_{3} (t - τ) \\ - β V_{3} (t) \end{matrix}

Form the proof of Theorem 2 in Hu et al.,²⁵ whether ${\hat{\overset{\cdot}{V}}}_{3} (t - τ) > 0$ or ${\hat{\overset{\cdot}{V}}}_{3} (t - τ) < 0$ , we can always get that, ${\overset{\cdot}{V}}_{3} (t)$ are always reducing and from (14), $\frac{1}{M} + γ - 1 + α < 0$ , then

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) < - | \frac{1}{M} + γ - 1 + α | | {\hat{\overset{\cdot}{V}}}_{3} (t - τ) | - β V_{3} (t) \\ < - β V_{3} (t) \end{matrix}

Since $β > 0$ , the closed loop system is stable. The proof is completed.

Solving algorithm

For the successfully application of the proposed DOF-SMLC, a solving algorithm is presented in this subsection. In subsection 3.2, the following equations should be

\begin{matrix} C_{e}^{T} K^{T} K C_{e} A_{e} + A_{e}^{T} C_{e}^{T} K^{T} K C_{e} < - Q, \\ K C_{e} B_{e} = I \end{matrix}

Associated with the Lyapunov stability theory, the following inequation should also be hold

C_{e}^{T} K^{T} K C_{e} > 0

Then

\begin{matrix} C_{e}^{T} K^{T} K C_{e} = {[\begin{matrix} C & 0 \\ 0 & I \end{matrix}]}^{T} {[\begin{matrix} K_{1} & K_{2} \end{matrix}]}^{T} [\begin{matrix} K_{1} & K_{2} \end{matrix}] [\begin{matrix} C & 0 \\ 0 & I \end{matrix}] \\ = [\begin{matrix} C^{T} K_{1}^{T} K_{1} C_{e} & C^{T} K_{1}^{T} K_{2} \\ K_{2}^{T} K_{1} C & K_{2}^{T} K_{2} \end{matrix}] \end{matrix}

the solving algorithm can be listed as follows

Step 1: Associated with (11), the value of $K_{1}$ can be determined and

K_{1} CB = I

(15)

Step 2: Find the matrix $K_{2}$ by solving the inequation

[\begin{matrix} C^{T} K_{1}^{T} K_{1} C_{e} & C^{T} K_{1}^{T} K_{2} \\ K_{2}^{T} K_{1} C & K_{2}^{T} K_{2} \end{matrix}] > 0

(16)

Step 3: Getting $A_{D}$ , $B_{D}$ and $C_{D}$ by solving the equation

C_{e}^{T} K^{T} K C_{e} A_{e} + A_{e}^{T} C_{e}^{T} K^{T} K C_{e} < - Q

(17)

For the solving details of equations (15), (16), and (17), corresponding supporting materials can be found in Hu et al.²⁶ Then by solving equations (15), (16), and (17), the DOF-SMLC can be constructed.

Simulation results

For demonstrating the effectiveness of the proposed DOF-SMLC, a numerical simulation example is considered in this section. The parameters of equation (1) are chosen as following

m_{s 1} = 1 kg, m_{s 2} = 2 kg, c_{s} = 3 Ns / m, k_{s} = 5 N / m

Δ c_{s} = 0.5 \cos (t), Δ k_{s} = 0.5 \sin (t)

d (t) = [\begin{matrix} 0.2 \exp (- 0.01 t) \sin (2 t) \\ 0.2 \exp (- 0.01 t) \cos (2 t) \end{matrix}]

then the matrices of (3) are

A = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 5 & 5 & - 3 & 3 \\ 2.5 & - 2.5 & 1.5 & - 1.5 \end{matrix}]

Δ A = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - 0.5 \sin (t) & 0.5 \sin (t) & - 0.5 \cos (t) & 0.5 \cos (t) \\ 0.25 \sin (t) & - 0.25 \sin (t) & 0.25 \cos (t) & - 0.25 \cos (t) \end{matrix}]

B = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], E = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & 0.5 \end{matrix}], C = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}]

Then according to the proposed DOF-SMLC, $A_{D}$ , $B_{D}$ and $C_{D}$ can be computed and

A_{D} = [\begin{matrix} 158 & - 122 & - 703 & - 674 \\ 14.16 & - 4.10 & - 15.3 & - 28.1 \\ 4.19 & 1.31 & - 0.411 & 0.72 \\ - 13.1 & 0.78 & 8.93 & 14.86 \end{matrix}]

B_{D} = [\begin{matrix} - 10.86 \\ 3.45 \\ 3.89 \\ - 6.77 \end{matrix}]

C_{D} = [\begin{matrix} - 276 & - 194 & - 293 & - 369 \end{matrix}]

Then the sliding surface is constructed, and $K = [\begin{matrix} 1 & 0.94 & 2.31 & 13.64 & 70.8 \end{matrix}]$ .

In the leaning controller, we choose $α = 0.3$ , $β = 2$ . For testing the performance of the proposed SMLC, the improved SMLC is marked as $u_{dofsmlc}$ and is implemented on the 2 degree-of-freedom mechanical system together with a traditional sliding model control $u_{smc}$ . The initial states of mechanical system is set to be $x (0) = {[\begin{matrix} 1 & - 0.8 & 0 & 0 \end{matrix}]}^{T}$ , and the simulation results are listed in Figures 2 and 3, and the dashdot line represents the responses under controller $u_{dofsmlc}$ , and solid line is for traditional controller SMC $u_{smc}$ . Figures 4 and Figures 5 are the input of $u_{smc}$ and $u_{dofsmlc}$ . From Figures 2 and 3, we can see that, the system controlled by the proposed SMLC and SMC are all stable. But from equations (4) and (5) we can see that, the input of $u_{dofsmlc}$ is smooth while the input of $u_{smc}$ is chattering. The control performance of $u_{dofsmlc}$ is obviously better than traditional controller $u_{smc}$ .

Figure 2.

Position $r_{1}$ .

Figure 3.

Position $r_{2}$ .

Figure 4.

Input of $u_{smc}$ .

Figure 5.

Input of $u_{dofsmlc}$ .

Conclusion

In this paper, a DOF-SMLC has been proposed for mechanical system. A DOF-based sliding mode surface is designed and then the stability of the designed sliding surface is discussed. A learning controller is designed instead of the discontinuous controller to guarantee the stability of the closed loop system. Finally, a numerical simulation example is given to show the good performance of the proposed strategy.

The proposed control method can be utilized by industrial system without enough sensor, or practical system which only partly states can be measured. But the upper bounds of parameter uncertainty and disturbance are assumed to be known in this paper, while in most instance, these special bounds are hard to be got. Further work is to design a controller without any priori information of them.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the Fundamental Research Funds for the Central Universities (3102019ZDHQD03, 3102019ZDHKY06), Natural Science Basic Research Plan in Shanxi Province of China (2019JM-127), Aeronautical Science Foundation of China (201707U8003), and National Natural Science Foundation of China (61503392, 61304001, 61773386, 61673386).

ORCID iD

Xiaoxiang Hu

References

Sun

Fault-tolerant finite frequency H control for uncertain mechanical system with input delay and constraint. Asian J Control 2017; 19(2): 765–780.

Xiao

Yin

Gao

Reconfigurable tolerant control of uncertain mechanical systems with actuator faults: a sliding mode observer-based approach. IEEE Trans Control Syst Technol 2018; 26(4): 1249–1258.

Zhao

, et al. Adaptive inverse control of a vibrating coupled vessel-riser system with input backlash. IEEE Trans Syst Man Cybern Syst. Epub ahead of print 17 October 2019. DOI: 10.1109/TSMC.2019.2944999.

Lai

Wang

, et al. Stable control strategy for planar three-link underactuated mechanical system. IEEE/ASME Trans Mechatron 2016; 21(3): 1345–1356.

Shao

Chen

, et al. Adaptive neural control for an uncertain fractional-order rotational mechanical system using disturbance observer. IET Control Theor Appl 2016; 10(16): 1972–1980.

Wang

Jing

Karimi

, et al. Robust fault-tolerant H control of active suspension systems with finite-frequency constraint. Mech Syst Signal Process 2015; 62: 341–355.

Kong

Zhang

Chen

, et al. Synchronization analysis and control of three eccentric rotors in a vibrating system using adaptive sliding mode control algorithm. Mech Syst Signal Process 2016; 72: 432–450.

Tong

Sui

Adaptive fuzzy output feedback control for switched nonstrict-feedback nonlinear systems with input nonlinearities. IEEE Trans Fuzzy Syst 2016; 24(6): 1426–1440.

Pan

Sun

Nonlinear output feedback finite-time control for vehicle active suspension systems. IEEE Trans Ind Inf 2019; 15(4): 2073–2082.

10.

Rath

Defoort

Karimi

, et al. Output feedback active suspension control with higher order terminal sliding mode. IEEE Trans Ind Electron 2017; 64(2): 1392–1403.

11.

Jiang

Karimi

Kao

, et al. A novel robust fuzzy integral sliding mode control for nonlinear semi-Markovian jump T–S fuzzy systems. IEEE Trans Fuzzy Syst 2018; 26(6): 3594–3604.

12.

Xiao

Yang

Karimi

, et al. Asymptotic tracking control for a more representative class of uncertain nonlinear systems with mismatched uncertainties. IEEE Trans Ind Electron 2019; 66(12): 9417–9427.

13.

, et al. Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle. J Frankl Inst 2012; 349(2): 559–577.

14.

Wang

Shen

Karimi

, et al. Dissipativity-based fuzzy integral sliding mode control of continuous-time TS fuzzy systems. IEEE Trans Fuzzy Syst 2018; 26(3): 1164–1176.

15.

Xiao

Yin

Exponential tracking control of robotic manipulators with uncertain kinematics and dynamics. IEEE Trans Ind Inf 2019; 15(2): 689–698.

16.

Wang

Shi

Output-feedback based sliding mode control for fuzzy systems with actuator saturation. IEEE Trans Fuzzy Syst 2016; 24(6): 1282–1293.

17.

Song

Wang

Niu

Static output-feedback sliding mode control under round-robin protocol. Int J Robust Nonlinear Control 2018; 28(18): 5841–5857.

18.

Wei

Park

Qiu

, et al. Sliding mode control for semi-Markovian jump systems via output feedback. Automatica 2017; 81: 133–141.

19.

Qiu

, et al. Fuzzy-affine-model-based output feedback dynamic sliding mode controller design of nonlinear systems. IEEE Trans Syst Man Cybern Syst. Epub ahead of print 12 March 2019. DOI: 10.1109/TSMC.2019.2900050.

20.

Feng

Shi

Sliding mode control of singular stochastic Markov jump systems. IEEE Trans Autom Control 2017; 62(8): 4266–4273.

21.

Chen

, et al. Reduced-order disturbance observer design for discrete-time linear stochastic systems. Trans Inst Meas Control 2016; 38(6): 657–664.

22.

Zhou

Yao

Wang

, et al. Robust control of uncertain semi-Markovian jump systems using sliding mode control method. Appl Math Comput 2016; 286: 72–87.

23.

Yang

, et al. Passivity-based asynchronous sliding mode control for delayed singular Markovian jump systems. IEEE Trans Autom Control 2018; 63(8): 2715–2721.

24.

Manh Tuan

Man

Zhang

, et al. Robust sliding mode-based learning control for steer-by-wire systems in modern vehicles. IEEE Trans Veh Technol 2014; 63(2): 580–590.

25.

, et al. Robust sliding mode-based learning control for MIMO nonlinear nonminimum phase system in general form. IEEE Trans Cybern 2018; 49(10): 3793–3805.

26.

, et al. Dynamic output feedback control of a flexible air-breathing hypersonic vehicle via T–S fuzzy approach. Int J Syst Sci 2014; 45(8): 1740–1756.

Sliding mode learning control for uncertain mechanical system: A dynamic output feedback approach

Abstract

Keywords

Introduction

Problem formulation

Model of mechanical system

Control objective

Assumption 1

Assumption 2

Remark 1

Main results

DOF based sliding surface design

Traditional discontinuous controller design

Remark 2

SMLC design

Assumption 3

Theorem 1

Proof

Solving algorithm

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References