Robust Speed Tracking of Networked PMSM Servo Systems with Uncertain Feedback Delay and Load Torque Disturbance

Abstract

This paper considers a class of networked Permanent Magnet Synchronous Motors (PMSMs), whose feedback loops are closed over a shared data network. Although the installation and maintenance cost of the networked PMSM system can be lowered by replacing the conventional point-to-point feedback cables with a network, the network packet dropouts and transmission delay may degrade the system’s performance and even destabilize it. The load torque disturbance is another source to deteriorate the PMSM system’s performance. To investigate the effects of the data network and the torque disturbance on the speed tracking of a PMSM system is one major task of this paper. In particular, we derive a sufficient stability condition for this system in an LMI (linear matrix inequality) form and provides a way to bound the system’s 𝐻_∞ performance. Moreover, we adopt an iterative LMI method to design the speed controller of the PMSM system, which can robustly guarantee stability and performance against the network-induced delays, packet dropouts, and the torque disturbance. Simulations are done to verify the effectiveness of the obtained results.

1. Introduction

In the high-performance applications, such as robotics, aeronautic devices, and precision machine tools, the positioning accuracy is required to be higher and higher so that an alternative better than the traditional induction motors is needed. The permanent magnet synchronous motor (PMSM) is one wise choice to meet this accuracy challenge in the low-to-medium power servo systems. Because the PMSM's rotor is a permanent magnet and the flux linkage is constant [1], it possesses many advantages, like superior power density, large torque-to-inertia ratio, and high efficiency. Consequently, the PMSM has received widespreading acceptance in industrial applications and is recognized as one of the key components in automation applications.

In PMSM applications, speed tracking is of great importance [2]. The existence of the load torque disturbance, however, makes speed tracking a rather tough task. In fact, it is impossible to measure the disturbance directly in real application. One way is to estimate the disturbance and actively cancel it [3]. Such methods may complicate the system design and increase the system's cost. We take another way, that is, designing a robust controller that can guarantee the system's performance is less sensitive to the disturbance.

Conventionally, a PMSM servo system includes a small number of PMSMs, which are connected to controllers by point-to-point cables to close the control loops. The modern industry, however, demands more and more PMSMs that could be located in geographically separated areas. Thus the direct cable connection architecture is inconvenient for the installation and maintenance. Instead, we can close the feedback loops of PMSMs over a digital network to form a networked control system (NCS), which is a newly developed technology being able to reduce the system's wiring cost, simplify the system's diagnosis and maintenance, and improve the system's agility [4]. The use of network will, however, induce intermittent losses and delays of the feedback information, may deteriorate the system's performance, and even cause instability [5]. There are many results on NCSs, some of which are introduced here (see more in the survey [6]). Reference [7] investigates the problem of robust stabilization and disturbance for the NCSs with random communication network-induced delays. Reference [8] Considers 𝐻_∞ output tracking for NCSs with delays and data packet dropouts. In [9], the discrete-time NCSs is studied and a state feedback controller with less conservatism is given. Moreover, a modified optimization algorithm is also proposed in [9] to cope with Bilinear Matrix Inequalities (BMIs).

As an important field of NCSs, networked servo systems are also well studied in recent years. In [10], fuzzy logic control is used to enhance the performance of networked servo systems with delays. To compensate the noise in the transmission channels, two-degrees-of-freedom control is utilized in [11]. Predictive control [12] is implemented to suppress the negative effects of delays and dropouts induced by the network. These results mainly focus on servo systems with DC motors or AC induction motors. Nevertheless, PMSM, as one of the main actuators in servo systems, is rarely mentioned in the previous NCS research. Due to its great potential in the industry, the networked PMSM servo systems could be one of the hot research topics in the future. Though a networked PMSM servo system has the same problems of data packet dropouts and transmission delay as typical networked servo systems, it has its own characteristics due to the employment of field-orientation mechanism and the special motor structures [13]. So when we design a controller for such a system, the specific properties should be taken into account to achieve less conservative results. The load torque disturbance, as a common disturbance in applications, should also be considered in the networked PMSM servo system.

This paper is organized as follows. In Section 2, we establish the mathematical model for the networked PMSM servo systems, which suffers the uncertain network-induced delay and data dropouts as well as the load torque disturbance. A sufficient stability condition of the system is given and proven in Section 3 based on the Lyapunov-Krasovskii techniques. Moreover, a bound on the system's 𝐻_∞ performance is also provided in Section 3. An iterative LMI method is proposed to design a robust controller in Section 4, which can guarantee both the stability and performance under the torque perturbation and the network-induced delay and dropouts. Simulations are done in Section 5 to verify the effectiveness of the obtained results. Some final remarks are placed in Section 6.

2. Problem Formulation

2.1. PMSM Dynamics

In the synchronously rotating rotor 𝑑-𝑞 coordinate, a PMSM drive can be modeled as [2]𝑢_𝑑=𝑅𝑖_𝑑+𝐿_𝑑𝑑𝑖_𝑑𝑑𝑡−𝑛_𝑝𝜔𝐿_𝑞𝑖_𝑞,𝑢_𝑞=𝑅𝑖_𝑞+𝐿_𝑞𝑑𝑖_𝑞𝑑𝑡+𝑛_𝑝𝜔𝐿_𝑑𝑖_𝑑+𝑛_𝑝𝜔𝜙_𝑎,𝐽𝑑𝜔𝑑𝑡+𝐵₀𝜔+𝑇_𝑙=𝑛_𝑝𝜙_𝑎, (2.1)

where 𝜔 is the rotating speed; 𝑖_𝑑 and 𝑖_𝑞 are the 𝑑- and 𝑞-axes stator currents, respectively; 𝑢_𝑑 and 𝑢_𝑞 are the 𝑑- and 𝑞-axes stator voltages, respectively; 𝑅 is the stator resistance per phase; 𝐿_𝑑 and 𝐿_𝑞 are the 𝑑- and 𝑞-axes stator inductances, respectively, and 𝐿_𝑑=𝐿_𝑞=𝐿 in the surface-mounted PMSM; 𝑛_𝑝 is the number of poles; 𝜙_𝑎=3/2𝜙_𝑓, with 𝜙_𝑓 being the flux linkage of the permanent magnet rotor; 𝐽 is the total moment of inertia of the motor and load, and 𝐵₀ is the friction coefficient of the motor; 𝑇_𝑙 is the load torque.

A well-known strategy for a PMSM drive is the field-oriented vector control approach. Under this scheme, a practical structure of cascaded control loops, including a speed loop and two current loops, is usually employed [14]. In order to approximately eliminate the coupling between the 𝑑- and 𝑞-axes currents, the 𝑑-axis reference current 𝑖^∗_𝑑 is set at zero and 𝑖_𝑑 is regulated via a PI controller, as is shown in Figure 1.

Figure 1 

Configuration of a PMSM Servo System.

Design the current controller of the 𝑞-axis as𝑢_𝑞=𝑅𝑖_𝑞𝑟+𝑛_𝑝𝜙_𝑎𝜔, (2.2)

where 𝑖_𝑞𝑟 is the reference current of the 𝑞-axis, which is computed by the speed controller. Then the state-space equation of a PMSM can be represented as𝑑𝑖_𝑞𝑑𝑡𝑑𝜔=−𝑅𝑑𝑡𝐿0𝑛_𝑝𝜙_𝑎𝐽−𝐵₀𝐽𝑖_𝑞𝜔+𝑅𝐿0𝑖_𝑞𝑟+0−𝑇_𝑙𝐽. (2.3)

To proceed further, we need to make the following two definitions.

Definition 2.1. The tracking errors of speed, the 𝑞-axis current 𝑖_𝑞, and the reference current 𝑖_𝑞𝑟 are defined as 𝑒(𝑡)=𝜔^∗−𝜔,𝑒_𝑞(𝑡)=𝑖^∗_𝑞−𝑖_𝑞,𝑒_𝑞𝑟(𝑡)=𝑖^∗_𝑞𝑟−𝑖_𝑞𝑟, (2.4) where 𝜔^∗, 𝑖^∗_𝑞, 𝑖^∗_𝑞𝑟 are the corresponding reference values.
By Definition 2.1, the state-space equation (2.3) can be rewritten into a compact form as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵(𝑢(𝑡)+𝑑(𝑡)), (2.5) where 𝑥(𝑡)=𝑒_𝑞(𝑡)𝑒(𝑡)^𝑇,𝑢(𝑡)=𝑒_𝑞𝑟(𝑡)0^𝑇,𝑑(𝑡)=0Δ𝑇_𝑙(𝑡)^𝑇,−𝑅𝐴=𝐿0𝑛_𝑝𝜙_𝑎𝐽−𝐵₀𝐽𝑅,𝐵=𝐿001𝐽. (2.6)Δ𝑇_𝑙(𝑡) represents the load torque disturbance, which is bounded and perhaps time-varying.

Definition 2.2. The disturbance suppressing performance of the system is evaluated by the following signal: 𝑧(𝑡)=𝑒(𝑡). (2.7) We want to design a robust controller for a networked PMSM to satisfy the following requirements.
(1)When the load torque disturbance 𝑑(𝑡)=0, the system is asymptotically stable under any initial state, namely,lim_𝑡→∞𝑧(𝑡)=0. (2.8)(2)When the load torque disturbance 𝑑(𝑡)≠0, the closed-loop system has the ability to suppress disturbance, namely, ‖𝑧(𝑡)‖₂≤𝛾‖𝑑(𝑡)‖₂, (2.9) where ‖⋅‖₂ stands for the 𝐿₂ norm of a continuous-time signal and 𝛾 is a quantitative measure of the disturbance attenuation. The smaller 𝛾, the better disturbance attenuation.

2.2. Structure of Networked PMSM Servo Systems

A typical networked servo system is shown in Figure 2, which can be divided into three parts:

Figure 2 

The diagram of a typical networked servo system.

(1) the remote unit containing a remote controller and a remote motor,

(2) the central controller,

(3) the communication network.

The remote unit and the central controller exchange feedback information through the communication network.

When the remote motors are PMSMs, we get a networked PMSM servo system as shown in Figure 3. Each distributed remote controller receives control signals from the communication network and then convert them into PWM signals to drive the motor. It also sends local measurements, such as rotating speed, motor current, and local environment information, back to the central controller via the shared data network. The central controller is usually a sophisticated controller and can provide advanced real-time control strategies to the remote units.

Figure 3 

The diagram of a networked PMSM servo system.

2.3. System Modeling

Because of the limited bandwidth of the network, data packet dropouts are unavoidable. When a dropout occurs, it might be more advantageous to drop the old packet and transmit a new one than to retransmit the old one [15]. Network-induced delays are also considered here. The model of the concerned networked PMSM servo system is shown in Figure 4.

Figure 4 

The model of the networked PMSM servo system.

The following assumptions are placed on the networked PMSM servo system. (1)The sensor is clock driven, and the controller and actuator are event driven.(2)The sampling period is a positive constant scalar ℎ.(3)The controller-to-actuator and the sensor-to-controller delays are denoted as 𝜏_𝑐𝑎 and 𝜏_𝑠𝑐, respectively. The state feedback controller is static (see (2.14)). So these two delays can be lumped as 𝜏_𝑘=𝜏_𝑠𝑐+𝜏_𝑐𝑎. (2.10) Moreover, 𝜏_𝑘 is less than ℎ, that is, 0≤𝜏_𝑘≤𝜏≤ℎ, where 𝜏 is the upper bound of delays. Due to the static feedback controller, we can assume that the transmission from the controller to the actuator is delay free and all delays come from the transmissions from the sensor to the controller, that is, 𝜏_𝑠𝑐=𝜏_𝑘 and 𝜏_𝑐𝑎=0, in Figure 4.(4)The maximum numbers of the consecutive controller-to-actuator and sensor-to-controller data dropouts are denoted as 𝑑_𝑐𝑎 and 𝑑_𝑠𝑐, respectively. They can also be lumped as 𝑑_𝑘=𝑑_𝑐𝑎+𝑑_𝑠𝑐. (2.11)𝑑_𝑘 is bounded as 0≤𝑑_𝑘≤𝑑. Similar to 𝜏_𝑘, 𝑑_𝑘 is also assumed to only come from the transmissions from the sensor to the controller, that is, 𝑑_𝑠𝑐=𝑑_𝑘 and 𝑑_𝑐𝑎=0.

If only delays exist in the system, the sampled signal at 𝑘ℎ(∀𝑘∈𝒩) will arrive at the controller at the time 𝑘ℎ+𝜏_𝑘. So the delay is 𝜂(𝑡)=𝑡−𝑘ℎ and its range is 𝜂(𝑡)∈[0,ℎ+𝜏], (2.12)

When the data dropouts also exist, they can be treated as delays and yield the following overall delay range:𝜂(𝑡)∈0,𝑑+1ℎ+𝜏=0,𝜂. (2.13)

It is well known that time-varying delay is more difficult to handle than constant delay from the control system's perspective. The actuator can know the total delay 𝜂(𝑡) by the time stamping technique. In the present paper, the actuator is assumed to purposefully postpone to implement the received control variable by the time of 𝜂−𝜂(𝑡) and yields a constant overall delay of 𝜂, which is easier to deal with. We choose a static state feedback controller. Due to the constant delay strategy, our controller takes the following form:𝑢(𝑡)=𝐾𝑥𝑡−𝜂, (2.14)

where 𝐾 is the feedback gain to be designed. By substituting (2.14) into (2.5), we get the following state-space equation of the networked PMSM servo system:𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾𝑥𝑡−𝜂𝑧(𝑡)(𝑡).+𝐵𝑑,=𝐶𝑥 (2.15)

3. Analysis of Networked PMSM Servo Systems

3.1. A Sufficient Stability Condition

In industry applications, the stability of a servo system is crucial. So we first have to guarantee that the system is stable. Here stability means the asymptotic stability when the disturbance is zero, that is, 𝑑(𝑡)=0. Under the delay and dropout conditions in (2.13), we get the following stability condition, which is expressed in an LMI (linear matrix inequality) form and easy to verify.

Theorem 3.1. Under the given controller gain 𝐾 and the upper bound 𝜂>0 (in (2.13)), the system (2.15) is asymptotically stable if there exist matrices 𝑃>0, 𝑄>0, 𝑍>0, 𝑌 and 𝑊 such that the following matrix inequality (3.1) holds: 𝑊₁𝑊₂−𝜂𝑌𝐴^𝑇∗𝑊₃−𝜂𝑊𝐾^𝑇𝐵^𝑇∗∗−𝜂𝑍0∗∗∗−𝜂⁻¹𝑍⁻¹<0, (3.1) where 𝑊₁=𝑃𝐴+𝐴^𝑇𝑃+𝑌+𝑌^𝑇+𝑄, 𝑊₂=𝑃𝐵𝐾−𝑌+𝑊^𝑇, and 𝑊₃=−𝑄−𝑊−𝑊^𝑇.

Proof. We construct the following Lyapunov-Krasovskii function: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (3.2) where 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡), 𝑉₂(𝑡)=∫^𝑡_𝑡−𝜂𝑥^𝑇(𝛼)𝑄𝑥(𝛼)𝑑𝛼, and 𝑉₃(𝑡)=∫⁰_−𝜂∫^𝑡_𝑡+𝛽𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝛽. By the Newton-Leibniz formula, we get 𝑥𝑡−𝜂=𝑥(𝑡)−^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼. (3.3) The derivatives of 𝑉₁, 𝑉₂, and 𝑉₃ are computed as follows: 𝑉̇₁(𝑡)=2𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝐵𝐾𝑥𝑡−𝜂=2𝑥^𝑇(𝑡)𝑃(𝐴+𝐵𝐾)𝑥(𝑡)−2𝑥^𝑇(𝑡)𝑃𝐵𝐾^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼=2𝑥^𝑇(𝑡)𝑃(𝐴+𝐵𝐾)𝑥(𝑡)+2𝑥^𝑇(𝑡)(𝑌−𝑃𝐵𝐾)^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼+2𝑥^𝑇𝑡−𝜂𝑊^𝑡_𝑡−𝜂(𝛼)−𝑥̇𝑑𝛼2𝑥^𝑇(𝑡)𝑌^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼+2𝑥^𝑇𝑡−𝜂𝑊^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼=2𝑥^𝑇(𝑡)𝑃(𝐴+𝐵𝐾)𝑥(𝑡)+2𝑥^𝑇(𝑡)(𝑌−𝑃𝐵𝐾)𝑥(𝑡)−𝑥𝑡−𝜂+2𝑥^𝑇𝑡−𝜂𝑊𝑥(𝑡)−𝑥𝑡−𝜂−2𝑥^𝑇(𝑡)𝑌^𝑡_𝑡−𝜂𝑥̇(𝛼)𝑑𝛼+2𝑥^𝑇𝑡−𝜂𝑊^𝑡_𝑡−𝜂(𝛼)=1𝑥̇𝑑𝛼𝜂^𝑡_𝑡−𝜂2𝑥^𝑇(𝑡)(𝑃𝐴+𝑌)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵𝐾−𝑌+𝑊^𝑇𝑥𝑡−𝜂−2𝑥^𝑇𝑡−𝜂𝑊𝑥𝑡−𝜂−2𝑥^𝑇(𝑡)𝜂𝑌𝑥̇(𝛼)−2𝑥^𝑇𝑡−𝜂(𝛼𝑉̇𝜂𝑊𝑥̇)𝑑𝛼,₂(𝑡)=𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜂𝑄𝑥𝑡−𝜂=1𝜂^𝑡_𝑡−𝜂𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇𝑡−𝜂𝑄𝑥𝑡−𝜂𝑉̇𝑑𝛼,₃(𝑡)=⁰_−𝜂𝑥̇^𝑇(𝑡)𝑍𝑥̇(𝑡)−𝑥̇𝑡+𝛽𝑍𝑥̇𝑡+𝛽𝑑𝛽=^𝑡_𝑡−𝜂𝑥̇^𝑇(𝑡)(𝑡)(𝛼)(𝛼=1𝑍𝑥̇−𝑥̇𝑍𝑥̇)𝑑𝛼𝜂^𝑡_𝑡−𝜂𝑥^𝑇(𝑡)𝜂𝐴^𝑇𝑍𝐴𝑥(𝑡)+2𝑥^𝑇(𝑡)𝜂𝐴^𝑇𝑍𝐵𝐾𝑥𝑡−𝜂+𝑥^𝑇𝑡−𝜂𝜂𝐾^𝑇𝐵^𝑇𝑍𝐵𝐾𝑥𝑡−𝜂−𝑥̇^𝑇(𝛼)𝜂𝑍𝑥̇(𝛼)𝑑𝛼. (3.4)
Finally, we have 𝑉̇(𝑡)=1𝜂^𝑡_𝑡−𝜂𝜁^𝑇(𝑡,𝛼)Φ𝜁(𝑡,𝛼)𝑑𝛼, (3.5) where 𝜁()=𝑥(𝑡)𝑡,𝛼𝑥𝑡−𝜂(𝛼)𝑊𝑥̇,Φ=¹+𝜂𝐴^𝑇𝑍𝐴𝑊₂+𝜂𝐴^𝑇𝑍𝐵𝐾−𝜂𝑌∗𝑊₃+𝜂𝐾^𝑇𝐵^𝑇𝑍𝐵𝐾−𝜂𝑊∗∗−𝜂𝑍. (3.6) By the Schur complement theorem, we get from (3.1) that there must exist 𝜖>0 such that Φ<−𝜖𝐼, (3.7) where 𝐼 represents an identity matrix of an appropriate dimension. Substituting (3.7) into (3.5) yields 𝑉̇(𝑡)<−𝜖‖𝑥(𝑡)‖². (3.8) According to Lyapunov-Krasovskii theorem, if there exist 𝜖>0 such that 𝑉̇(𝑡)<−𝜀‖𝑥‖², the system (2.15) is asymptotic stable. So if the matrix inequality (3.1) holds, the system (2.15) is asymptotically stable. This completes the proof.

3.2. Robust Performance Analysis

Definition 3.2. A stable system in (2.15) is said to satisfy the 𝐻_∞ performance index 𝛾>0 if under the zero initial condition, ‖𝑧(𝑡)‖₂≤𝛾‖𝑑(𝑡)‖₂. (3.9) We can verify whether the performance requirement in (3.9) is satisfied through the following theorem.

Theorem 3.3. Under the given controller gain 𝐾 and the upper bound 𝜂>0 (in (2.13)), the system (2.15) satisfies the performance index 𝛾 in (3.9) if there exist matrices 𝑃>0, 𝑄>0, 𝑍>0, 𝑌 and 𝑊 such that the following matrix inequality holds: 𝑊₁𝑊₂−𝜂𝑌𝑃𝐵𝐴^𝑇𝐶^𝑇∗𝑊₃−𝜂𝑊0𝐾^𝑇𝐵^𝑇0∗∗−𝜂𝑍000∗∗∗−𝛾²𝐼𝐵^𝑇0∗∗∗∗−𝜂⁻¹𝑍⁻¹0∗∗∗∗∗−𝐼<0, (3.10) where 𝑊₁=𝑃𝐴+𝐴^𝑇𝑃+𝑌+𝑌^𝑇+𝑄, 𝑊₂=𝑃𝐵𝐾−𝑌+𝑊^𝑇, and 𝑊₃=−𝑄−𝑊−𝑊^𝑇.

Proof. Equation (3.10) implies (3.1). By Theorem 3.1, we know the system is stable.
Define 𝐽_𝑧𝑑=∫^∞₀[𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)]𝑑𝑡. 𝐽_𝑧𝑑 can be modified into 𝐽_𝑧𝑑=^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)+𝑉̇(𝑡)𝑑𝑡+𝑉(𝑡)|_𝑡=0−𝑉(𝑡)|_𝑡=∞. (3.11)
Under the zero initial condition, 𝑉(𝑡)|_𝑡=0=0. Because of 𝑉(𝑡)|_𝑡=∞≥0, we get 𝐽_𝑧𝑑≤^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)+𝑉̇(𝑡≤1)𝑑𝑡𝜂^∞₀^𝑡_𝑡−𝜂𝑥(𝑡)𝑥(𝑡−𝜂)𝑥̇(𝛼)𝑑(𝑡)^𝑇Ξ𝑥(𝑡)𝑥𝑡−𝜂(𝛼)𝑑(𝑡),𝑥̇ (3.12) where 𝑊Ξ=¹+𝐶^𝑇𝐶𝑊₂−𝜂𝑌𝑃𝐵∗𝑊₃−𝜂𝑊0∗∗−𝜂𝑍0∗∗∗−𝛾²𝐼+𝐴^𝑇𝐾^𝑇𝐵^𝑇0𝐵^𝑇𝐴𝜂𝑍^𝑇𝐾^𝑇𝐵^𝑇0𝐵^𝑇𝑇. (3.13)
By the Schur complement theorem, we know that (3.10) implies Ξ<0. Therefore, 𝐽_𝑧𝜔<0, that is, ^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)𝑑𝑡<0. (3.14) After simple manipulations, the above equation yields ‖𝑧(𝑡)‖²₂≤𝛾²‖𝑑(𝑡)‖²₂. (3.15) Equation (3.15) is equivalent to (3.9).

In (3.9), the left side variable 𝑧(𝑡) and the right one 𝑑(𝑡) have different units. So the ratio between them, 𝛾, does not have a clear physical meaning. In order to overcome this difficulty, we introduce the following relative sensitive functions.

Definition 3.4. The sensitive functions of the speed tracking error and the load torque disturbance are defined as 𝑆_𝑧=‖𝑍‖₂𝜔_ref,𝑆_𝑇=Δ𝑇_𝑙2T_𝑙0, (3.16) where 𝜔_ref is the reference tracking speed and 𝑇_𝑙0 is the nominal load torque.
Based on the above definition, (3.15) can be rewritten into 𝑆_𝑧≤𝛾𝑆_𝑇, (3.17) where 𝛾=(𝑇_𝑙0/𝜔_ref)𝛾.

Remark 3.5. In (3.17), both units of 𝑆_𝑧 and 𝑆_𝑇 are percentage. So (3.17) means how much percent of load torque disturbance yields how much percent of speed tracking error. 𝛾 is exactly the gain between two percentage variables (𝑆_𝑧 and 𝑆_𝑇) and can quantitatively reflect the capability to attenuate the load torque disturbance attenuation. 𝛾 is determined by the system's delay in (2.13) and the controller gain 𝐾 in (2.14). Although we cannot change the system's delay, we do have freedom to choose an appropriate 𝐾 to yield a better (smaller) 𝛾, which is the major task of the next section.

4. The Design of the Robust Controller

When we design the robust controller, the controller gain 𝐾 is unknown. So the matrix inequalities in Theorems 3.1 and 3.3 are bilinear matrix inequalities (BMIs). As a result, we cannot find a maximum 𝜂 or the minimum 𝛾 using convex optimization algorithms. In the subsequent part, we propose some methods to resolve this issue.

Define 𝑋=𝑃⁻¹ and Δ=diag{𝑋,𝑋,𝑋,𝐼,𝐼,𝐼}. Pre- and postmultiply (3.10) by Δ, we obtain𝑊₁𝑊₂−𝜂𝑌𝐵𝑋𝐴^𝑇𝑋𝐶^𝑇∗𝑊₃−𝜂𝑊0𝐹^𝑇𝐵^𝑇0∗∗−𝜂𝑋𝑆⁻¹𝑋000∗∗∗−𝛾²𝐼𝐵^𝑇0∗∗∗∗−𝜂⁻¹𝑆0∗∗∗∗∗−𝐼<0, (4.1)

where 𝑊₁=𝐴𝑋+𝑋𝐴^𝑇+𝑌+𝑌^𝑇+𝑄, 𝑊₂=𝐵𝐹−𝑌+𝑊^𝑇, 𝑊₃=−𝑄−𝑊−𝑊^𝑇, 𝑌=𝑋𝑌𝑋, 𝑄=𝑋𝑄𝑋, 𝑊=𝑋𝑊𝑋, 𝐹=𝐾𝑋, and 𝑆=𝑍⁻¹.

Define a matrix variable 𝑀<𝑋𝑆⁻¹𝑋, then matrix inequality (4.1) is equivalent to the combination of matrix inequalities (4.2), (4.3) and(4.4)𝑊₁𝑊₂−𝜂𝑌𝐵𝑋𝐴^𝑇𝑋𝐶^𝑇∗𝑊₃−𝜂𝑊0𝐹^𝑇𝐵^𝑇0∗∗−𝜂𝑀000∗∗∗−𝛾²𝐼𝐵^𝑇0∗∗∗∗−𝜂⁻¹𝑆0∗∗∗∗∗−𝐼<0,−𝑍𝑃𝑃−𝑁<0,𝑆=𝑍⁻¹,𝑁=𝑀⁻¹,𝑃=𝑋⁻¹. (4.2)(4.3)(4.4)

Based on the above transformation, a robust controller can be designed as follows.

Theorem 4.1. Under the given 𝜂>0 and 𝛾>0, the system (2.15) satisfies the performance index 𝛾 if there exist matrices 𝑋>0, 𝑃>0, 𝑄>0, 𝑍>0, 𝑀>0, 𝑆>0, 𝑁>0, 𝑌 and 𝑊 such that matrix inequalities (4.2), (4.3) and (4.4) hold. In this case, a robust 𝐻_∞ state feedback controller gain can be chosen as 𝐾=𝐹𝑋⁻¹.

It is noted that the conditions in Theorem 4.1 are not LMI because of the inverse matrix constraints in (4.3) and (4.4). Fortunately, there are some methods to efficiently solve these inequalities. In [16], a method is given to obtain the suboptimal delay 𝜂 or the suboptimal 𝛾 by setting 𝑆=𝑋. With more computational efforts, better results can be obtained by an iterative algorithm in [17]. That iterative algorithm is called cone complementary linearization (CCL) method. By adopting the CCL method, we get the following algorithm to cope with the nonlinear minimization problem subject to LMIs.

Algorithm 4.2. There are 4 steps.Step 1. Choose a sufficiently large initial variable 𝛾>0 such that there exists a feasible solution to matrix inequalities (4.2) and (4.3) and (4.4). Set 𝛾_min=𝛾.Step 2. Find a feasible set (𝑋₀,𝑃₀,𝑄₀,𝑍₀,𝑀₀,𝑆₀,𝑁₀,𝑌₀,𝑊₀) satisfying (4.2) and (4.3), and the matrix inequality in (4.5) (One way to get a feasible set is by setting 𝑆=𝑋 as the aforementioned suboptimal solution). Set 𝑘=0, 𝑆𝐼𝐼𝑍<0,𝑁𝐼𝐼𝑀<0,𝑃𝐼𝐼𝑋<0. (4.5)Step 3. Solve the LMI problem in (4.6), Min_{𝑋,𝑃,𝑄,𝑍,𝑀,𝑆,𝑁,𝑌,𝑊}tr𝑆_𝑘𝑍+𝑆𝑍_𝑘+𝑁𝑀_𝑘+𝑃_𝑘𝑋+𝑃𝑋_𝑘s.t.equations(4.2),(4.3),and(4.5). (4.6)Set 𝑋_𝑘+1=𝑋, 𝑃_𝑘+1=𝑃, 𝑄_𝑘+1=𝑄, 𝑍_𝑘+1=𝑍, 𝑀_𝑘+1=𝑀, 𝑆_𝑘+1=𝑆, 𝑁_𝑘+1=𝑁, 𝑌_𝑘+1=𝑌, and 𝑊_𝑘+1=𝑊.Step 4. If (3.10) holds, then set 𝛾_min=𝛾 and return to Step 2 after decreasing 𝛾 to some extent. If (3.10) is not satisfied within a specified number of iterations, to say 𝑘_max, then we stop the above iterations. Otherwise, set 𝑘=𝑘+1 and go back to Step 3.

5. Simulation Results

To verify the results in Theorems 3.1, 3.3, and 4.1, a MATLAB/SIMULINK simulation platform of networked PMSM servo systems is built up, which is shown in Figure 1. The sampling period ℎ is 10ms. The tracking reference speed is set to 1500 r/min. The nominal parameters of a PMSM is shown in Table 1.

Table 1 

The nominal parameters of a PMSM.

In the following simulations, the load torque disturbance is no larger than 30% of the nominal value. According to Theorem 4.1 and Definition 3.4, we obtain the relationship between the speed tracking error and 𝜂, which is demonstrated in Table 2.

Table 2 

The relationship between the speed tracking error and 𝜂.

Remark 5.1. The data in Table 2 shows the relationship between the speed tracking error and the maximum delay 𝜂. When the networked PMSM servo system needs higher tracking accuracy, it can tolerate less transmission delay and data packet dropouts. If the speed tacking error is less than 2.0%, 𝜂 is equal to 0.005s, which means that the system cannot tolerate even one data packet dropout under this circumstance.
In the sequel, we simulate 3 cases to demonstrate the effectiveness of the obtained results. In these simulations, the maximum relative speed tracking error is 5%, and the load torque disturbance is no larger than 30% of the nominal value. According to Definition 3.4, we get 𝛾≤𝛾_min=125. (5.1) By Theorem 4.1 (with 𝛾=𝛾_min), we try to maximize 𝜂 and reach a suboptimal solution with 𝜂=0.064s and the corresponding controller gain of 𝐾=[0,−0.0212].

5.1. Stability of System

Case 1. The first simulation is to verify the stability of the networked PMSM servo system. 𝜂 is set to 0.064s. Figure 5 shows the speed tracking error trajectory. After approximate 0.15s, the speed tracking error is almost zero. So this result demonstrates the correctness of the stability condition in Theorem 3.1.

Figure 5 

The speed tracking error.

5.2. Robustness against Transmission Delay and Data Dropouts

Case 2. In the second case, performance comparison is made between two approaches, the robust controller proposed in this paper and the PID controller.
The PID Controller
Its parameters are tuned to perform well under the network-delay-free situation. In the simulation, the network-induced delay is set to 0.1h (=0.001s).
Robust Controller
The robust controller is then applied as the central controller to regulate the speed loop. The network-induced delay 𝜂 is set to be 0.064s. 𝜂=0.064s=6.4h means that the system can tolerate 5 dropouts among any 6 consecutive data packets.
The results of this case are shown in Figure 6.
Although the PID controller suffers a much smaller delay than the robust controller, it generates much worse tracking performance than the robust controller. The reason lies in that the design of the robust controller takes the delay into account, while the design of the PID controller does not. So the effectiveness of Theorems 3.3 and 4.1 is confirmed.

Figure 6 

Speed response under the network delays and dropouts.

5.3. Robustness against Load Torque Disturbance

Case 3. The purpose of the third simulation is to verify that the proposed robust controller can effectively suppress the load torque disturbance. Under the worst condition of five consecutive data packet dropouts and network-induced delay up to 0.4h, a 30% load torque disturbance occurs at time 0.5s and disappears at time 0.55 s. Figure 7 shows the simulation results. As we see in Figure 7, when the load torque disturbance starts at time 0.5s, the speed deviates from the reference speed of 1500 r/min. However, after the disturbance disappears at time 0.55s, the speed of PMSM quickly returns back to the reference speed in about 0.03s. From the speed response curve, we can also see that the speed tracking error is less than 5% during the whole process. These simulation results confirm that the robust controller designed in this paper satisfies the accuracy demand and can effectively suppress the load torque disturbance.

Figure 7 

Speed response under the network delays and dropouts and the load torque disturbance.

6. Conclusion

The networked PMSM servo system has a promising future in industry applications. However, its performance may be degraded by the network delays and data packet dropouts. The load disturbance is also a detrimental factor for the control performance. In this paper, we propose a sufficient stability condition by the Lyapunov-Krasovskii method, quantitatively investigate the robustness of the system's performance against the load torque disturbance, and give a way to design a robust controller, which can either tolerate larger network delay or give better 𝐻_∞ performance. The simulations are done to verify the correctness of the stability result and demonstrate the superiority of the obtained controller in terms of performance robustness against the data packet dropouts and transmission delay as well as the load torque disturbance.

Acknowledgments

This paper was supported in part by the National Natural Science Foundation of China (60904012), the Program for New Century Excellent Talents in University (NCET0-10-0917), and the Doctoral Fund of Ministry of Education of China (20093402120017).