Servo Robust Control of Uncertain Mechanical Systems: Application in a Compressor/PMSM System

: High-speed Permanent Magnet Synchronous Motor (PMSM) systems have been widely used in industry and other ﬁelds for their advantages of having a simple structure, low processing cost and high efﬁciency. At present, the control precision of PMSM is required to be higher and higher, but it faces two major challenges. The ﬁrst is that the PMSM system possesses (possibly fast) time-varying uncertainty. The second is that there exist nonlinear portions in the PMSM system, such as nonlinear elasticity, etc. To resolve these challenges, a novel performance measure ˆ β is introduced as a dynamic depiction of the constraint-following error, and a new robust control design is proposed based on ˆ β . While this control renders guaranteed performance regardless of uncertainty, an optimal design of a control parameter is further pursued. This inquiry is summed up as a semi-inﬁnite constrained optimization problem. After the induction of the necessary condition, the candidate solutions can be identiﬁed. These are further screened by a sufﬁcient condition, which results in the actual solution. To verify the effectiveness of the control design, the compressor powered by a super high-speed PMSM system is simulated, and its performance is discussed.


Introduction
The modern AC motor actuators in the industrial field face the challenging requirements of high efficiency, high precision, and high performance with advances in digital electronics, industrial machine drives, and control theory. In AC motor actuators, the Permanent Magnet Synchronous Motor (PMSM) system is increasingly popular for its simple structure, low processing cost, high efficiency, and robustness. The PMSM has been widely used in high precision fields, such as robotics, aerospace, large telescopes and so on. However, PMSM is a typical nonlinear and multi-variable coupling system. Moreover, (possibly fast) time-varying uncertainties always affect its control performance and cause ripples in the actual operation [1]. Therefore, designing a high precision PMSM system control is difficult but rewarding.
The motion equation must be obtained first to achieve the control of the PMSM system. Pioneers proposed many forms of constrained motion equations which are based on D'Alembert's principle [2]. This principle is appropriate for many cases, but not in cases where the constraints are nonholonomic or non-ideal. To overcome the limitation, a simple and explicit decoupled motion equation, the Udwadia-Kalaba equation, is proposed. The theory is known as the Udwadia-Kalaba theory [3,4]. In contrast to the Lagrange method, the Udwadia-Kalaba theory does not need the hard-to-get multiplier. In addition, it provides an efficient way to handle any type of constraints, whether holonomic or nonholonomic constraints, ideal or non-ideal.
The analysis of PMSM system constraints is another central problem. Constraint following problems fall into two types: passive constraint problems and servo constraint problems [5]. The second type is the key issue for precise control design [6,7]. The main focus of servo constraints, which are model-based, is to figure out what the engineer should do, or to figure out the force/torque of the servo mechanism (such as the motor), in order to comply with the constraints. In this paper, the servo robust control is designed based on servo constraints.
At present, the traditional PID control, hysteresis control, and predictive control are commonly used for PMSM [8,9]. PID control has the characteristics of stability, reliability, and simple parameter setting. Nevertheless, it is difficult to meet the high precision application. Because it is easy to cause system overshoot and oscillation when the gain of PID control increases, therefore, scholars adopted many advanced approaches to control the PMSM system, such as sliding mode control [10], interval type-2 fuzzy dynamic high type control [11], predefined-time control [12] and fuzzy-set-based control [13], while the convergance speed is not fast enough when the system state is far from the equilibrium point and the calculation is complicated. Furthermore, the optimal design of the control parameter is not pursued. Robust control is another common tool to deal with uncertainty [14][15][16][17], which has been used in robot manipulators, quadrotors, aerospace and many other mechanical systems. A new Udwadia-Kalaba-based servo robust control approach is presented in this paper to meet the requirement of precise control.
Different from the previous robust control method, in which the control performance is directly measured by constraint error β [18,19], a generalized performance measureβ in the new proposed robust control is introduced as a dynamic depiction of the constraintfollowing error. The definition of β in the previous study is just a special case ofβ. Then, a more practical optimal design of control parameters is also considered [20,21] based on the general control performance.
In summary, this paper has made contributions to four aspects to realize the high precision control of the PMSM. First, a novel system performance measure (β) is taken as the control object, which is the "dynamic" generalization of the previous "static" β measure. Second, a new robust control method for constraint following is proposed, which can guarantee the uniform boundedness and uniform ultimate boundedness of the uncertain compressor/PMSM performance. Third, the relationship betweenβ and β is explored. Fourth, optimization of the control design parameter is pursued. This task is summed up as a constrained optimization problem of the compressor/PMSM and solved by invoking an necessary condition followed by a sufficient condition.

Servo Robust Control
In this section, a new servo robust control is proposed for the precision control of the compressor powered by a super high-speed PMSM system. First, the dynamic model of the uncertain compressor/PMSM system is introduced. Second, the proposed robust control is proved to guarantee the uniform boundedness and uniform ultimate boundedness of the uncertain system.

Dynamic Model of the Compressor/PMSM System
A compressor powered by a super high-speed PMSM is considered, which is shown in Figure 1. The system can be simplified as a two-mass system, where the first mass represents the super high-speed PMSM, the second mass represents the compressor, and the shaft is considered to be mass free, see Figure 2. Parameters of the system are given in Table 1.

Parameter
Description Units J m moment of inertia of the motor kg · m 2 J l moment of inertia of the load kg · m 2 c m friction of viscous motor N · m/(rad/s) c l friction of viscous load N · m/(rad/s) c s inner damping coefficient of the shaft N · m/(rad/s) k s elasticity coefficient of the shaft N · m/rad τ 1 control input N · m τ 2 load torque disturbance N · m Some parameters of this model are uncertain in the real situation. Based on [22], the uncertain dynamic model of the two-mass system without deadzone is: Here, x = [x 1 , x 2 ] T is the generalized coordinate vector,ẋ ∈ R 2 is velocity,ẍ ∈ R 2 is acceleration, ξ ∈ Ξ ⊂ R p , where ξ ∈ R p is the time-varying uncertain parameter, Ξ is the unknown but compact possible bound of ξ. In addition,

Remark 1.
The coordinate x can be chosen according to the specific situation, not necessarily a generalized coordinate. The dimension of ξ ∈ Ξ ⊂ R p is determined by the number of uncertain terms in the two-mass system.

Problem Formulation
Constraints can be classified as holonomic constraints and nonholonomic constraints. Bsed on [23], both holonomic constraints and nonholonomic constraints can be abbreviated to where is not restricted and m is the number of constraints.

Remark 2.
Mechanical systems may be subjected to both nonholonomic constraints and holonomic constraints. These constraint equations do not have to depend on each other.
Remark 3. Assumption 1 is vital. In the past, this was generally accepted as fact. However, the counter situation occurs when the coordinate x is not selected as the generalized coordinate [24]. (5) is consistent.

Definition 1. For a pair of given A and b, if there is at least one solutionẍ, then
The rank of A(x, t) is greater than 1. Furthermore, all constraints are consistent.
Problem. For the uncertain two-mass system (1), a control τ is designed so that the system finally meets the specified performance (5) under the action of control.

Servo Robust Control Design
Please note that functions M, C, and N all uncertain. They can be composed as: whereM(·),C(·) andN(·) are "nominal" portions, and ∆M(·), ∆C(·) and ∆G(·) are uncertain portions. In addition, above functions are all continuous. For the mechanical system Mẍ + Cẋ + N = τ, under the control τ, the system needs to follow the desired constraints Aẋ = c. In previous study, the control performance is directly measured by the constraint error In this paper, from a new angle, the control performance is defined aṡβ where g(0, t) = 0, andβ is the performance measure which is the general form to describe the control performance.

Remark 4.
The function Π(·) is the uncertainty bound. The servo robust control for the compressor/PMSM system is designed as where In (20), κ is a design parameter greater than or equal to 1 (that is then for given scalar constant > 0 Theorem 1. Under the conditions of Assumptions 1-5, the proposed control (18) renders the compressor/PMSM system satisfies following properties: (1) Uniform boundedness: For any r > 0 with β (t 0 ) ≤ r, there is a d(r) < ∞ such that β (t) ≤ d(r) for all t > t 0 ; (2) Uniform ultimate boundedness: For any r > 0 with β (t 0 ) ≤ r, there is a d > 0 such that β (t) ≤d for anyd > d as t ≥ t 0 + T(d, r), where T(d, r) < ∞.
Then, we analyze the third portion on the right side of (23). We decompose M −1 as We can obtain, According to (19), Next, according to (17) and µ =M −1 A T ∂V/∂βΠ According to τ 2 = −κημΠ 2 , we obtain By a direct algebra,

Relationship betweenβ and β
Previously, the control performance is defined as β, which is shown in (9). It presents a "static" measure of how constraints are followed. In this paper, the control performancê β provides a "dynamic" measure of the constraint-following. The novelty ofβ is the introduction of g(β, t), which allows the flexible design of the control.
The uniform boundedness and uniform ultimate boundedness of the novel system performance measureβ has been proved. There is a relationship betweenβ and β. Let According to Theorem 1, for all t > t 0 , β (t) ≤ d(r) ; and for all t > t 0 + T (T < ∞), β (t) ≤d. Then, there exists a supremumḡ such that for all t > t 0 , g(β, t) ≤ḡ, that is Furthermore, there exists a g such that for all t > t 0 + T (T < ∞), g(β, t) ≤ g, i.e.,

Design of the Performance Index
The proof of Theorem 1 shows that R decreases by increasing the design parameter of the control (κ). The decrease of R results in the decrease of the ultimate boundedness region sized. However, the increase of κ means an increase in the control cost. Therefore, we should consider the combined effect of changing κ on R and the control cost.
Let us consider the following performance index Here h(·) is a C 1 strictly increasing function h(·) : R + → R + , reflecting the control cost due to the use of κ. The choice of this function can be a part of the design, which should both reflect the control cost and facilitate the feasible solution. The problem is to seek κ ≥ 1, so that J(κ) renders a minimum value. This can be expressed as a semi-infinite constrained optimization problem: min κ≥1 J(κ). (44)

Solution of the Optimization Problem
To solve this optimization problem, we proceed to consider the necessary condition Due to the properties of γ 3 (·), as prescribed in Assumption 3, the derivative of γ − 1   3 exists. In addition, the derivative of h(·) also exists. Taking the derivatives results in Suppose the solution(s) to this equation exist, which are denoted by κ * . The sufficient condition for this optimization problem is then To sum up, necessary conditions provide candidate solutions for optimizing the problem, while sufficient conditions filter candidate solutions to obtain the (actual) solutions.

Lemma 1.
Suppose γ 3 (r) = l 1 r p and h(κ) = l 2 κ q , where l 1 , l 2 > 0, p, q > 0. Then, there always exists an unique solution to the necessary condition (45). The unique solution κ * is and Remark 5. For given l 1 , p, and q, κ * ≥ 1 can be assured by a proper choice of l 2 . This is the unique positive solution to the performance index. With (49), the minimum cost J is given by J min = J(κ)| κ=κ * (52)

Design Procedure
This is believed to be the most thorough study of an optimization problem, with all three fundamental issues completely solved: existence, uniqueness, and analytic expressions of both κ and J min . Since the assumptions are all very fundamental and the procedure only invokes standard numerical computations, the control scheme is quite easily applied. We shall further demonstrate this in the next section. The servo robust control proposed in this paper can be summarized in the following design step flow chart (Figure 3).

Constraints and Assumptions Verification
Firstly, according to the expression of M in (2), Assumption 1 is verified. Secondly, suppose that the load mass needs to track the desired trajectoryx 2 = sin t. The error between the coordinates of the load mass and the desired trajectory is defined as To obtain the desired trajectory quickly, the error can be modified aṡ Selecting k = 1, (54) can be written aṡ Differentiating (55) once to obtain Then, constraints (55) and (56) can be converted to the form of Aẋ = c and Hence, Assumption 2 is verified. Thirdly, to verify Assumption 3, the function g(β, t) and Lyapunov function V(·) should be selected. For the simulation of compressor/PMSM system, we select where H ∈ R and H > 0. Then, the performance measure˙β is represented bẏβ This in turn means in Assumption 3, the function V(·) can be selected as where Q > 0 is a scalar. Then, corresponding γ i ( β ), (i = 1, 2, 3) functions are, respectively, shown as Fourth, we consider the inertia moment of motor and load are uncertain and shown as J m =J m + ∆J m , J l =J l + ∆J l , whereJ m,l > 0 are constant nominal values and ∆J m,l are the uncertainties. We also consider the friction of viscous motor, the friction of viscous load, and the inner damping coefficient of the shaft are uncertain and they are, respectively, shown as c m =c m + ∆c m , c l =c l + ∆c l , and c s =c s + ∆c s , wherec m,l,s > 0 are constant nominal values and ∆c m,l,s are the uncertainty. The elasticity coefficient of the shaft is also considered to be uncertain and shown as k s =k s + ∆k s , wherek s > 0 is the constant nominal value and ∆k s is the uncertainty. Due toM(x, t)M −1 (x, ξ, t) − I > −1 thus Assumptions 4 can be verified.
Fifthly, we assume ∆J m ≤ ∆J m ≤ ∆J m , ∆J l ≤ ∆J l ≤ ∆J l , ∆c m ≤ ∆c m ≤ ∆c m , ∆c l ≤ ∆c l ≤ ∆c l , ∆c s ≤ ∆c s ≤ ∆c s , ∆k s ≤ ∆k s ≤ ∆k s ,. Then, Assumption 5 is met by choosing where

Simulations and Discussions
The region bounded by the norm of the system performance error ( e ) and t is expressed as where T t is the simulation time. Variable time step ODE15i integrator is used for simulation in MATLAB environment. The simulation results are shown in Figures 4-8. Figure 4 shows the comparison of system performance x 2 between the servo robust control and LQR control. The error between the system performance and the desired trajectory is shown in Figure 5. It is clearly shown that the servo robust control can reach the uniform boundedness more quickly than LQR control. In addition, the uniform ultimate bounded region under the servo robust control is much smaller than that under LQR control. The comparison of S is shown in Figure 6. The S under the servo robust control is more than five times smaller than that under LQR control. System performance x2 The servo robust control LQR Desired trajectory   Figure 7 shows the comparison of τ 1 under two controls (servo robust control and LQR control). At the beginning of the simulation, the LQR control has an overshoot, while the robust control is smooth. There is a surge in the control input of the proposed servo robust control. The reason for the surge in Figure 7 is the initial condition deviation. As shown in (52), the desired trajectory is x 2 = sint. The problem can be mitigated by adjusting the parameter κ.
The comparison of β (the "static" measure) andβ (the "dynamic" measure) under the servo robust control is shown in Figure 8. The "static" measure as well as the "dynamic" measure both can quickly reach a plateau.

Conclusions
To achieve high precision control of a nonlinear and uncertain compressor/PMSM system, the desired performance is translated into constraints for processing. Then, a general performance measure with dynamic characteristics is used to solve the constraint following the control problem. The introduction of g(β, t) produces the "dynamic" measureβ, which enables flexible control design. Regardless of uncertainty, the control renders the measurê β uniformly bounded and uniformly ultimately bounded. The next inquiry is to seek an optimal choice of the control design parameter κ. This process is separated into two steps. The first step is to explore the necessary condition, which generates candidate solutions. The second step is to use a sufficient condition to screen these candidate solutions to identify the actual optimal solution. The simulation results show the effectiveness of the new control design.
The paper addresses one performance index and one design parameter for the control design of the two-mass system. Future explorations may involve optimal design for multiple performance objectives and multiple design parameters.