Particle accelerators require magnet power supplies with very high performance. Important power-supply considerations include reproducibility, stability and resolution—typical values are 100 PPM, 50 PPM, and 16 bits, respectively. Digital technology advances have made it possible to implement very complicated regulation schemes using DSPs or PLDs. These advances along with increased performance demands have resulted in designers using digital instead of analog regulators. This article presents a new digital-regulation scheme based on state estimation that may replace traditional analog PID regulators. The article presents a state-space model of the proposed system, including power supply, regulator, and estimator, along with Matlab and Simulink simulations.

Figure 1:  Generic accelerator-magnet power supply

Figure 2 shows the usual regulation scheme, which comprises one [outer] current loop and one [inner] voltage loop. The reference current, set by the local controller after being converted to an analog value, is compared to the actual current measured by a direct-current current transformer (DCCT). The resultant current error is then fed to the PI regulator whose output is the reference of the voltage loop. The voltage loop has a much higher bandwidth than the current loop—typical values are 1 kHz and 1 Hz, respectively).

Figure 3:  Simplified model of the power circuit

The numerical values in Figure 3 correspond to the correction magnet power supply, used in the simulations. In the figure, L₂ is the inductance of the output filter, C is the capacitance of the output filter, L₁ is the inductance of the super-conducting magnet, and R₁ is the internal resistance of the cables. You can write the state space equations of the system shown in Figure 3 as:

The three states are chosen to be the current through the inductance of the output filter, output voltage of the power supply and the output voltage of the DCCT. In the state-space equations, 1/g is the gain of the DCCT, A is the gain of the voltage amplifier, u is the steering signal of the power supply, and y is the actual current through the magnet. In the next step, we calculated the equivalent digital model of the system. The sample time, T, was 10^-4 seconds in the analog-to-digital conversion.

The numerical values of the digital state matrices after substituting the model parameters will be:

Figure 4:  The proposed regulation scheme, based on state estimation and pole placement

In the scheme shown in Figure 4, the pole-placement feedback is used to place the closed-loop poles of the system in our desired locations. Use of a digital integrator eliminates the steady-state error and improves system behavior.

where we have:

In these equations, X is the state vector and q the output of the integrator. The pole-placement feedback vector was calculated for the augmented system. The numerical value of the feedback vector is written as:

K_d = (K_d1  K_d2), K_d1 = (1.7056  0.0820  -41.9419), K_d2 = 1.7780

Where K_d1 is the feedback vector and K_d2 is the integrator gain.

where is the estimated states vector and C_d • (k) is the estimated output current at instant k. A block diagram of the system comprising the plant and estimator is shown in Figure 5.

Figure 5:  Block diagram of the plant and state estimator

You can show through some rather straightforward calculations that the states error vector, e(k), decays with the following dynamics:

From the last equation, it is clear that the dynamics of the error are determined from the eigenvalues of the (A_d - G • C_d) matrix. The three poles of the estimator should then be fast so that the estimated states reach the actual states after a very short time. You can find the estimator gain vector, G, by calculating a pole-placement feedback that locates the poles of the (A_d, TRANS(C_d)) system in our desired locations where TRANS(C_d) is the transposition of the output vector. The numerical value of the G vector in the simulations, is written as:

In these equations, X is the real states vector, is the estimates states vector, q is the output of the integrator, and U_ref is the reference input.

Figure 6:  Simulation model of the power supply and the proposed regulator

Figure 7:  The upper graph shows output voltage while the lower graph shows output current for step-input reference

Figure 8:  The prototype power supply and regulation board

Summary
This article proposes and shows the simulation results for a new digital-regulation technique for magnet power supplies. Design of the pole-placement feedback and the estimator was done using a state-space approach. Matlab simulations show the validity of the proposed scheme.