(1)Source: Materials of the international scientific conference EPE-PEMC 2004 Riga, Latvia (11th International Power Electronics and Motion Control Conference).
Abstract - The contribution presents the method, one of several possible ones, to be applied in the torque load observers design. The method employs the quadratic discrete-time quality factors theory of linear stationary discrete-time systems.
In many methods for optimal control of electrical drive, like: the minimum-time control, optimal stabilization, position control, the load torque must be determined in order to define the control vector. This problem can be solved by direct measurement methods or by the use of a load torque observer. The first approach requires the use of 8 tensometers located on the motor shaft, whose purpose is determination of the shaft torsion value. Thus, knowing the shaft diameter and the material it is made of, the load torque can be computed. This method is much more precise in comparison with observer application, but it needs high precision in tensometers mounting and the measuring system is particularly sensitive to any mechanical damages. For that reason the method of direct load torque measuring is not employed for industrial solutions.
The estimation of the load torque by means of the observer requires measuring of typical values only, like: the speed and torque generated by a motor. Since these quantities are also measured in a typical cascade control system, the problem solving is allocated to the computer system.
A. Discrete - time full order observer
The following system of equations for dynamic discrete-time control system is taken into consideration:
(1)
Additionally we assume, that the model of the process is non ideal [x^[k] = x(k)] and due to difficulties with measurements of all vector state variables we use the output vector for constructing the error vector described as [1]:
e(k) = y(k) - Cx^(k) = C(x(k) - x^(k)) (2)
In result we obtain the equation of the discrete-time full order observer
x^(k + 1) = Ax^(k) + Bu(k) + LC(x(k) - x^(k)) (3)
where L is gain matrix of error estimation. Diagram of this observer is shown on Fig.l.
Now the equation of discrete-time error vector is:
e(k + 1) = Ae(k) - LCe(k) = (A-LC)e(k) (4)
and its solution is
e(k) = (A-LC)ke(0). (5)
In order to implement the observer algorithm into a computer system it is convenient to convert of the observer equation (3)
x^(k +1) = (A- LC)x^(k) + Bu(k) +Ly(k). (6)
New diagram of observer is shown on Fig.2
B. Mathematic models of electrical drives
We take into consideration well known models of electrical drives. A DC drive supplied by a controlled rectifier is described by the following system of equations: [2],[3],[4]:
I '(t) = RI(t) + φeNω(t) + KpU(t) (7a)
Jω '(t) = Me(t) - Mm(t) (7b)
Me(t) = φeNI(t) (7c)
where:
I, ω - armature current, angular speed,
Mm - load torque,
Kp - amplifier gain,
φeN - nominal exciting flux,
J - moment of inertia,
Tm, T - starting and electromagnetic time constant,
R - generalized resistance (the sum of all resistances in the armature circuit),
L - total inductance (the sum of all inductances in the armature circuit).
The mechanical part of a drive can be described by following system of state equations:
(8a)
where:
(8b)
(8c)
Discrete form of the system (8) is following (Ts is sampling time):
(9a)
(9b)
Ñ. LQ problem solution for the load torque observer
The LQ problem was designed for synthesis a control systems. Its practical application in a observation problems is possible only after applying Duality Theory [5],[6]. Then the discrete-time dual system is
(10)
where 
(see Eq.(1)).
For some reasons the continuous-time quality factor cannot be applied to the discrete-time load torque observer. Therefore we assume a discrete-time quality factor in the form [7],[8],[9]:
(11)
After solving the discrete Algebraic Riccati Equation (ARE) [10],[11] we obtain matrix K. In load torque observer problem Ê is a horizontal vector, so in the basic problem observer matrix is
(12)
Then we obtain the system of equations for the load torque observer in the form:
(13)
where ω^ is estimation of angular speed (auxiliary quantity), M^m is estimation of load torque.
Diagram of the discrete-time torque load observer for dc drive is shown in Fig.3. Matrixes QT = Q >= 0 i RT = R > 0 of index
(11) are in form:
(14)
where q1 > 0, q2 > 0, r > 0. If q2 = 0 then pair (Q, A) is not detectable and LQ problem is not possible to solution.
Presently, only several selected results of the simulation tests are presented. These tests were carried out using MATLAB-SlMULINK software. Typical cascade control system of a dc drive with the P-type speed controller and Pi-type current controller was tested. Two different pairs of balance coefficients of the quality factor were assumed (q1 = 1, q2 = 100 and q1 = 1, q2 = 1000). Sampling time was Ts = 0.001.
The waveforms of a current, speed and estimated load torque start and steady state are presented in Figures 4 and 5. Step change in the load torque appears at time t = 1 [s].
The waveforms of the same quantities during the drive start under active load torque are shown in Fig.6.
The contribution presents the method, one of several possible ones, to be applied in the load torque observers design. This method is based on complying with quadratic discrete - time quality factors and theory of linear stationary discrete - time systems. The method employs the quadratic discrete-time quality factors theory of linear stationary discrete-time systems.
DC drive data using in the simulations:
PN = 18 [kW], UN = 440 [V], IN = 47 [A],
nN = 1800 [rpm], ωN = 188 [rad/s], ω0 = 200,3 [rad/s],
R = 1,8 [Ω], L = 99 [mH], T = L/R = 55 [ms],
φeN = 2,197 [Vs/rad], λN = 2 [Imax/IN], J = 0,69 [kgm2],
Kp = 75 [V/V], p = 50IN [A/s].