Patched LQR Control for Robust Protection of Multi-mass Electrical Drives with Constraints Mario Vaˇsak Nedjeljko Peri´c
Krzysztof Szabat
Marcin Cychowski
Institute of Electrical Machines, Department of Electronic Engineering, Faculty of Electrical Drives and Measurements Cork Institute of Technology, Engineering and Computing, Wroclaw University of Technology, Rossa Avenue, Cork, Ireland University of Zagreb, Ul. Smoluchowskiego 19, Email:
[email protected] Unska 3, HR-10000 Zagreb, Croatia PL-50372 Wroclaw, Poland Emails:
[email protected], Email:
[email protected] [email protected]
Abstract—In this paper we successfully experimentally verify the recently introduced protective predictive controller patch for multi-mass drives subject to physical and safety constraints on their variables. The protective patch is obtained through set-theoretic computations that account for the drive model and constraints, as well as the state estimation uncertainties. It is used in combination with the classical LQR controller. The easy implementation of the patch online allows the correction of the LQR control signal in order to suppress the violation of drive constraints and thus avoid possibly dangerous and/or drive-harmful situations during the transients.
I. Introduction Various types of controllers for multi-mass electrical drives speed control emerged in past two decades, like state-controllers [1], [2], [3], polynomial controllers [4], sliding-mode [5] or neuro-fuzzy controllers [6], [7]. These controllers are usually coupled with a state-observer for hard-to-measure drive variables. The weakly damped modes of such drives are visible in the closed-loop response once the speed controller saturates. This way, the control system variables like shafts torsion torques or rotational masses speeds may exhibit oscillations of significant amplitudes which lower the drive life-time and also decrease safety of the drive operation. Only a very small fraction of appeared synthesis procedures focuses on this issue and actually takes into account the physical and safety constraints on variables in the drive, e.g. limitations of the available motor torque and the maximum allowable torsion of the elastic shaft. A systematic way to deal with constraints in the drive is offered by model predictive control (MPC) [8] algorithms where due to fast sampling optimization is pursued through offline pre-computations (explicit MPC) [9], [10]. Although many predictive control concepts arose for speed, torque, current or flux control in the electrical drives [11], [12], there is a lack of works that focus on MPC for multi-mass drives speed. In [13], an explicit MPC speed controller is proposed for a two-mass electrical drive which respects the
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inherent motor and shaft torque constraints during operation without significantly enlarging the transient time. However, for implementation reasons the control problem is not formulated in a way to guarantee constraints satisfaction. Recently, a protective algorithm for multi-mass drives that resides on predictive control, particularly the concept of invariant sets [14], is suggested [15]. It is used to patch any existing drive speed control scheme in a way to guarantee that the electrical drive system does not and will not violate the constraints imposed on its variables. In [15], the performance of the protective patch coupled with a Linear Quadratic Regulator (LQR) speed controller is presented, but also it is stressed that the patch may be used in combination with any other controller which is confirmed in [16]. A nice feature when LQR is used with the patch is the available procedure to check the stability of the closed-loop system [17]. In [16] it is also shown how the patch robust to a bounded state-estimation error can be computed. In this work, we use LQR control with the protective patch and verify the control system performance on a real laboratory two-mass drive set-up. The protective patch synthesis procedure from [16] is used to make the patch robust to a bounded state-estimation error. Prior to experimental verification, the control system performance is tested in simulations in extreme drive excitation conditions - sharp full loading/disloading load torque transients are combined with abrupt speed reversals. Moreover, for the chosen excitation it is determined how much certain parameters of the drive may differ from the ones used to compute the patch to still preserve the system from constraints violation. The paper is organized as follows. Section II outlines the state-space model of a two-mass electrical drive system. Section III outlines the robust protective patch design for the drive. Section IV focuses on a real experimental setup of a two-mass drive system. Through simulations and experiments it points out clear benefits regarding the use of the protective patch.
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II. Considered two-mass electrical drive system The considered two-mass electrical drive [18] is schematically shown in Fig. 1. It is assumed that the inner current T1
ms
ω1
me
Fig. 1.
T2
mL
Tc
ms
ω2
Considered two-mass electrical drive.
control loop (i.e. the motor torque control loop) can be modeled as a first-order lag system [18], such that the following relation holds between the reference torque signal mer and the actual motor torque me (all values are already presumed to be normalized): m ˙e=
1 (mer − me ), Tm
(1)
where Tm is the equivalent time constant of the torque control loop. This approximation makes it possible to abstract the actual nature of the drive (e.g., DC or AC). The (normalized) variables involved in the mechanical part of the drive, speed of the motor ω1 , speed of the load ω2 , the shaft torsion torque (or shortly: the shaft torque) ms and the load torque mL , are connected with the following relations: 1 (me − ms ) , (2) ω˙ 1 = T1 1 (ms − mL ) , (3) ω˙ 2 = T2 1 (ω1 − ω2 ) , (4) m˙ s = Tc where T1 , T2 and Tc are the mechanical time constants of the motor, load and shaft, respectively. Since the model is used in the predictive controller synthesis, the load torque mL , as a disturbance input, is assumed constant during predictions, i.e. m ˙ L = 0. (5) We presume that nonlinear effects of backlash or friction in the drive can be neglected for larger drive speed transients that are in focus of this research. Moreover, the static friction component is presumed incorporated in the load torque. The drive states vector x ∈ R5 and the drive input u ∈ R are defined as follows: ω 1 ω 2 ms me mL , (6) x = u = mer . (7) Based on equations (1)–(7) the following continuous-time state-space model is obtained: x˙ = Ac x + Bc u,
(8)
where Ac ∈ R5×5 and Bc ∈ R5 . For the controller synthesis actually the discrete time model is needed. The sampling time T is chosen in order to have 10-15 samples per period of the open-loop drive oscillatory response which roughly corresponds to [4] 1 , (9) T ≈ 2Ω0 where Ω0 is the angular frequency of the open-loop nondamped oscillations. Discretizing (8) with the sampling time T using zero-order hold method leads to the discretetime state-space model xk+1 = Axk + Buk ,
(10)
where xk , uk are shorter notations for x(kT ), u(kT ), respectively, and A ∈ R5×5 , B ∈ R5 . III. Design and implementation of the robustly patched LQR controller The block scheme of the closed-loop drive speed control system is given in Fig. 2. Designs of the protective patch and of the LQR controller can be considered separately – protective patch is universal and can be used with any control scheme [16]. load speed reference
patched LQR Speed mer' control (LQR) ωˆ1 , ωˆ 2 , mˆ s , mˆ e , mˆ L
ω 2r
Estimation
Fig. 2. patch.
P A T C H
mer
Torque control
M
ym measurements
Principle scheme of the control system with the protective
A. Constraints During its operation the control system should respect the physical and safety constraints on the drive variables. For the patch synthesis purpose, the constraints should be expressed as inequalities with linear combinations of states and inputs. In the following we present and explain the introduced constraints for the drive at hand. Since the generated motor torque and motor torque rate cannot be larger than their maximum available values, considering the drive at hand and its inverter, the following physical constraints are posed: |me | 1 |mer − me | Tm
≤ me,max ,
(11)
≤ μ·m ˙ e,max .
(12)
Constraint (12) models the available torque rate on the inverter. This limit is incorporated in constraints using (1) which relates the torque rate to the current torque and the reference torque. This is only an approximation of this constraint for the following reasons: (i) when the inverter voltage reaches its maximum value, the torque control
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(13)
input u (i.e. [x u] ∈ P xu ) and the successor state Ax+Bu is in I ∩ P x (controlled invariance property), and (ii) it is the largest of all sets that satisfy (i) (maximality property). Succeedingly through future time steps applying property (i) and taking into account property (ii), one may say that I contains all the states from which there exist drive control system transients in the future that do not violate given constraints. The computational scheme for obtaining I is given in Algorithm 1 (for more details and additional references see [19], [14]). Algorithm 1. The Maximum Controlled Invariant Set.
Issue (i) could be easily incorporated in constraints and for the case of a DC motor in the drive at hand it would replace the constraint (12) with
loop is broken and counter-electromotive force can also significantly affect the torque rate and (ii) mer is assumed constant between the sampling instants and the largest value of the torque rate, considering (1), appears at the beginning of the sample. Constraining mer in discrete time for the whole sampling period based on the torque rate at the beginning of the sample could be overconservative. Thus, right-hand side of (12) is enlarged by a factor μ. The factor μ relates the mean torque rate on the interval [0, T ] to the torque rate at the beginning of the sample: m ˙ e (0)
μ= 1 T
T 0
− Ttm
m ˙ e (0)e
= dt
T Tm T
1 − e− Tm
.
1 (mer − me ) ≤ μ · (k1 me + k2 ω1 + k3 ∗ Umax ) , (14) Tm
1 (mer − me ) ≥ μ · (k1 me + k2 ω1 + k3 ∗ (−Umax )) , (15) Tm
where the inverter voltage is presumed to span in the interval [−Umax , Umax ] and k1 , k2 , k3 depend on motor parameters like armature resistance, armature inductance, electromotive force constant and torque constant. Safety limits on the motor and load speeds are introduced: |ω1 | ≤ ω1,max , |ω2 | ≤ ω2,max .
(16) (17)
Very important for long-life of the mechanical parts of the drive is to prevent excessive shaft twist. Thus the constraint (18) |ms | ≤ ms,max
1) Initialize T := P x ; 2) Compute T + := {x| ∃u : [x u] ∈ PT } ∩ P x where PT = {[x u] ∈ P xu | Ax + Bu ∈ T }; 3) If T + = T set I := {x| ∃u : [x u] ∈ PT }, else T := T + , and goto Step 2.
The patch uses the information contained in the set shape of I in order to impose minimum-possible correction on the signal computed by the speed controller (mer in Fig. 2), such that the state is maintained in I. More in detail, the computation of I yields also the set PI = {[x u] | HI x + LI u ≤ KI }, see Step 2 of Algorithm 1. For certain x0 , the set (22) UI (x0 ) = {u|LI u ≤ KI − HI x0 } contains all the feasible inputs at the time-step k = 0 which are allowed to be applied to the system such that the next state is in I ∩ P x . Thus the constraints satisfaction is enforced for the successor state and remains feasible for all future times. The set UI (x) is computed on-line for given state x = x0 and the control input u := mer computed by the speed controller is passed through a simple saturation function to result in a safe control input u := mer to be applied to the drive, see Fig. 3.
is introduced. Finally it is assumed that the load torque is not greater than mL,max : |mL | ≤ mL,max .
u
(19)
UI (x)
Mentioned constraints on the drive variables are described with polytopes P x = {x|C x x ≤ C 1 } ⊂ R5 , P
xu
x
u
1
6
= {[x u] |D x + D u ≤ D } ⊂ R ,
u = u u
(20) Fig. 3.
(21)
where matrices C x and C 1 follow from (6), (11) and (16)(19), while Dx , Du and D1 follow from (6), (7) and (12). B. Patch design The off-line computation of the protective patch resides on the computation of the maximum controlled invariant set I for the system (10) subject to constraints described with polytopes P x and P xu [19], [14]. The set I is characterized by two important properties: (i) it contains only those states x for which there exists an admissible control
The protective patch action.
Due to imperfect measurements/estimates of the state and due to possible process-model parameters mismatch certain level of robustness needs to be included in the controlled invariant set. We consider here only robustness to a bounded state estimation error, but a similar procedure can be also used to take into account the effect of processmodel parameters mismatch. The robustness to the latter can however be considered partly included in the estimated load torque and partly in the robustness to the estimation error.
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The bounds on the state estimation error can be derived from the error probability distribution. We have (23)
IV. Application to an experimental two-mass drive system A. Description of the experimental set-up and the design parameters
where x ˆk1 |k2 is the estimate of x in instant k1 based on information until the instant k2 , δ ∈ D ⊂ R5 . D is a zerocentered ellipsoid which we here outer-approximate with a hyperrectangle. Using (23) we further obtain that the predicted state in the next instant is
The patched LQR controller described in this paper is tested experimentally on a pilot-scale two-mass drive system composed of two 500 W DC-motors connected by a long steel shaft (length 600 mm, diameter 7 mm), see Fig. 4. The driving motor and load machine are controlled
ˆk|k−1 + δ, xk = x
xk|k−1 + δ) + Buk = Aˆ xk|k−1 + Buk + w (24) xk+1 = A(ˆ
Power converter
DS 1104
Power converter
where w may be any from the set [16] W = proj{Aδ| δ ∈ D} × {0} . R4
In order to keep the state in P x regardless of the realization of the uncertainty w ∈ W, the computational scheme given in Algorithm 1 should be extended such that the maximum robust controlled invariant set, denoted with ˜ is computed, see Algorithm 2 [19], [20], [14]. I, Algorithm 2. The Maximum Robust Controlled Invariant Set. 1) Initialize T := P x ; 2) Compute T + := {x| ∃u : [x u] ∈ P˜T } ∩ P x where P˜T = {[x u] ∈ P xu | Ax+Bu ∈ T W}; 3) If T + = T set I˜ := {x| ∃u : [x u] ∈ P˜T }, else T := T + , and goto Step 2.
The set-operation denoted with in Step 2 of Algorithm 2 is the so-called Minkowski difference or Pontryagin difference [20]: T W = {x| x + w ∈ T ∀w ∈ W}. Using the set P˜I˜ in on-line patch implementation guarantees the feasibility of constraints satisfaction in the future even in the presence of any uncertainty from W. For uncertainty sets W 0 we have I˜ ⊆ I which facilitates the computation of I˜ once I is computed by replacing Step 1 in Algorithm 2 by T := I ∩ P x . Since the description of I˜ may become very complex, a satisfactory robustness may be achieved by stopping the iterations of Algorithm 2 prior to fulfilling the test in Step 3 and by outer-approximating I˜ using current set P˜T . This approach gives the designer an opportunity to weigh between the protection algorithm complexity and robustness level. In practice, the outer approximation of I˜ computed within m = 4 − 5 iterations of Algorithm 2 initialized with I ∩ P x results in a satisfactory constraints satisfaction robustness. The sole controller design can be considered apart from the protective patch design. However, since the protective patch requires to have estimates of all system states available on-line, the best way to use them also in the controller is by an optimal controller, simplest being the LQR controller. Details on the LQR controller implementation for this case are given in [17] and are thus omitted here.
Elastic shaft
Encoder
(25)
Driving motor
Fig. 4.
Encoder
Flywheels
Load Machine
Schematic diagram of the laboratory set-up.
by the dSpace 1104 control platform via two separate power converters. The inner torque control loop and the outer speed control loop are realized in software (Simulink + dSpace environment). The inner torque control loop consists of a PI torque controller, power converter (transistor H-bridge), armature winding and armature current sensor. The output signal of the controller shapes the PWM signal which is then passed to the transistors of the power converter which operates on a 10 kHz switching frequency. The parameters of the torque controller are tuned to ensure fast aperiodic step response of the torque control loop such that the first-order lag approximation in (1) is valid. In the considered experimental set-up both the motor and load speed measurements are realized with incremental encoders (36000 pulses per rotation). The torque and speed control loops are sampled at 0.2 ms and 5 ms, respectively. The nominal parameters of the two-mass drive system and the physical limitations of the motor torque amplitude and rate, the description of the estimation uncertainty set D as well as parameters of the safety constraints are listed in Table I. Observe a very large expected estimation error in mL ([−1, 1]). The patch is this way ”cautious” and ”prepared” for abrupt full-loading/disloading situations such that even then violation of constraints is avoided. B. Simulations The drive is in simulations tested in extreme excitation conditions. The speed reference experiences step changes between −1 and 1 (full reversals). The load torque experiences step changes asynchronously with the speed, between 0 and 1. The feedback states are obtained using Kalman estimator which is driven with the input mer and noised drive model signals of ω1 , ω2 and me . Noise levels
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ω1 , ω2 , ω2r
1 0.5 0
−0.5 −1 0
10
20
30
40
50
ms , me , mL
4
60
70
80
70
80
with patch
without patch me,max ms,max
ω1 ω2 ω2r
90
100
ms me mL
2
0
−2
−4 0
10 20 −me,max −ms,max
Fig. 5.
30
40
D ω1,max ω2,max ms,max mL,max
60
90
100
Performance of the LQR controller in simulation without and with the patch.
TABLE I Parameters of the considered two-mass electrical drive, estimation uncertainty set and safety constraints. Symbol T1 T2 Tc Tm me,max m ˙ e,max
50
time [s]
Value 203 ms 203 ms 1.2 ms 1.2 ms 3 670 s−1 [−0.001, 0.001] × [−0.004, 0.004]× ×[−0.1, 0.1] × [−0.15, 0.15] × [−1, 1] 1.2 1.2 1.5 1.45
on drive model signals correspond to those of measurement signals on the experimental drive set-up. Simulated response is shown in Fig. 5. From time 0 to 50 s the patch is turned off, and from 50 s to 100 s the patch is active. Clearly, all constraints violations that occur before 50 seconds are suppressed afterwards. The duration of speed transients from -1 to 1 with mL = 0 is about 23% enlarged with the patch turned on, e.g. the transient at t=16 s (patch off) lasts 300 ms and at t=92 s (patch on) lasts 370 ms. The simulation scenario shown in Fig. 5 was also used to test over which values can some of the drive parameters span such that still no constraints violation occurs with the designed patch during these demanding transients. The results are summarized in Table II which shows that the
protection algorithm possesses significant robustness also with respect to drive parameters variation, although this was not explicitly taken care of in the design. TABLE II Allowable parameter spans for the considered drive excitation pattern (one parameter changes, others are kept on the nominal value). Parameter T1 T2 Tc Tm
Allowable span 170 − 350 ms 170 − 620 ms 0.8 − 1.5 ms 0 − 6 ms
C. Experiments Experimental result obtained on the described experimental setup are given in Fig. 6. The critical constraint on the shaft torque is respected as long as the protection patch is kept active. Once the protection is switched off in cca. 5.5 s, large violations of the critical constraint on the shaft torque occur whenever the reference speed abruptly changes. V. Conclusion This paper presents the experimental verification of the recently introduced protection scheme for multi-mass electrical drives with constraints. The applied protective algorithm can be used to patch any drive control scheme in order to ensure the satisfaction of drive safety constraints like maximum speeds and shaft torques. The patch is designed to be robust to a state estimation error contained in
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ω1 , ω2 , ω2r
0.6 0.4 0.2 0
−0.2 −0.4
ms , me , mL
0
4
1
2
3
4
5
with patch
6
7
without patch
8
me,max ms,max
ω1 ω2 ω2r
9
10
ms me mL
2
0
−2
−4 0
1 2 −me,max −ms,max
Fig. 6.
3
4
5
time [s]
6
7
8
9
10
Performance of the LQR controller in experiments without and with the patch.
a bounded set, but it is shown that it possesses significant robustness also with respect to drive parameters variation. The robust protective patch is applied to a laboratory twomass drive system in combination with a classical LQR controller. References [1] J. Ji and S. Sul, “Kalman filter and LQ based speed controller for torsional vibration suppression in a 2-mass motor drive system,” IEEE Transactions on Industrial Electronics, vol. 42, no. 6, pp. 564–571, 1995. [2] J. Deur, T. Koledi´c, and N. Peri´ c, “Optimization of speed control system for electrical drives with elastic coupling,” in Proceedings of the 1998 IEEE International Conference on Control Applications, Trieste, Italy, September 1998, pp. 319–325. [3] K. Szabat and T. Orlowska-Kowalska, “Vibration suppression in a two-mass drive system using PI speed controller and additional feedbacks – comparative study,” IEEE Transactions on Industrial Electronics, vol. 54, no. 2, pp. 1193–1206, 2007. [4] J. Deur and N. Peri´c, “Design of polynomial speed controller for electrical drives with elastic transmission,” in Conference Record of the 8th European Conference on Power Electronics and Applications, Lausanne, Switzerland, September 1999, pp. 1–10. [5] P. Korondi, H. Hashimoto, and V. Utkin, “Direct torsion control of flexible shaft in an observer-based discrete-time sliding mode,” IEEE Transactions on Industrial Electronics, vol. 45, no. 2, pp. 291–296, 1998. [6] T. Orlowska-Kowalska and K. Szabat, “Control of the drive system with stiff and elastic couplings using adaptive neurofuzzy approach,” IEEE Transactions on Industrial Electronics, vol. 54, no. 1, pp. 228–240, 2007. [7] W. Li and Y. Hori, “Vibration suppression using single neuronbased PI fuzzy controller and fractional-order disturbance observer,” IEEE Transactions on Industrial Electronics, vol. 54, no. 1, pp. 117–126, 2007. [8] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained Model Predictive Control: Stability and Optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000.
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