In this thesis, the modeling, design and tuning of the GPC for the DC-DC buck converter is presented. List of Figures xvii 4.7 Closed-loop GPC response of a DC-DC buck converter for reference.
INTRODUCTION
- DC-DC buck converter
- Research motivation
- Contributions of the thesis
- Thesis organization
The derivation of GPC is also presented together with the stability analysis of the derived control law. In this thesis, a detailed derivation of the dynamics of the dual output converter device is presented.
PRELIMINARY
Introduction
The defined cost function is the weighted sum of squares of the predicted tracking and control error. An analogous version of the predictor in a state space model is also available in the literature.
Elements of GPC
- Plant model
- Cost function
- Optimized control law
Second, the quadratic cost function is a Lyapun function that will ensure the stability of the control law. When considering an infinite-time quadratic cost function, the stability of the resulting control input is inherently proven by .
Formulation of GPC for transfer function model
- Prediction and optimization
To have a unique solution of the Diophantine equation, polynomials E and F are chosen to be of degree equal to tok−1 and ennl, and polynomialsRanS are chosen to be of degree equal to k−1 and nm respectively. Current optimal control input is obtained by minimizing the cost function with respect to the control increment ∆op.
Closed-loop analysis of GPC
The pole and control law for GPC and GPCF (GPC with filter) are available in Table 2.1.
Effect of variation in tuning parameters
- Example 1
Based on the presented numerical simulation results, the weight can be set to λ = 100, which yields the desired output response in Figure 2.2. For a feasible solution in GPC, the control horizon is always less than or equal to the chosen forecast horizon. Figure 2.11 compares the closed loop output response (ycl) with the open loop response (yop).
Sensitivity functions of GPC
- Example 2
The control input sensitivity to noise (Sun) and disturbance (Sud) performance shown in Figures 2.12 and 2.13 is improved for GPCF. Similarly, the overall device sensitivity function improves with respect to uncertainty ( Su ) for the entire frequency range for the GPCF, as shown in Figure 2.16. As shown in Figures 2.14 and 2.15 for lower frequencies, the output responses Syn and Syd are less sensitive to GPC instead of GPCF.
Effect of noise and disturbance
Simulation results are shown in Figure 2.20 in the presence of an input step perturbation for GPC and GPCF. It can be observed from the output responses in Figure 2.20 that the effect of disturbance in GPCF is prominent than GPC. Similarly, the control input and change in control input response are shown in Figures 2.21 and 2.22 respectively.
Summary
From these simulation results, it can be observed that the variation of control input and consequently the change of control input is quite high in GPC, while it was significantly reduced in GPCF. Most of the tuning rules available in the literature are heuristic in nature or based on extensive simulations.
PROPOSED ANALYTICAL TUNING RULE OF GPC FOR FOPDT SYSTEM
Introduction
Lee and Yu [60] proposed an adaptation of the FOPDT model using regression analysis for DMC. In [63, 64], a convex optimization method is implemented to adjust the weighting matrices of the cost function, which drives the plant into the desired closed-loop system. A generalized mathematical expression for balancing the weights is proposed, which ensures the closed-loop stability of the controlled system.
Derivation of analytical expression
- Plant model mismatch
The first step of predictive control is to obtain the output prediction using the plant model. Once the prediction of the output is obtained, the optimized control input in (3.13) can be changed as . Therefore, the final closed-loop model of the system for the higher-order plant model is represented as.
Tuning rule for GPC
- Control horizon of one
- Control horizon of two
Similar to (3.36) and (3.37), inequality criteria can be obtained by choosing the cost function weights as positive value. For control horizon of two, the possible range to achieve the desired gain is obtained in the same way as in control horizon of one. The desired performance can be achieved with the proposed control horizon tuning algorithm of one and two for the selected plant model instead of deriving complicated mathematical expressions, which is not only difficult to derive, but also the performance is not significantly improved compared to lowering values of control horizon.
Stability analysis of cost function
Stability of the cost function can be proven if the cost function satisfies the Lyapunov stability criteria. The minimum of the finite horizon cost function (Jmin) is chosen as the Lyapunov function, Zo(ei). Proof: The Lyapunov value function Zo(f(ei)) = min∆u{Z(ei,∆ui)} is assumed to be the optimal value of the cost function.
Numerical simulation results
- Example 1
- Example 2
After obtaining the discrete closed loop transfer function, the range of feasible gain matrix is obtained. For this example, a closed-loop output response without overshoot is considered the desired specification with input constraint u(n) ≤ 2.
Summary
PRE-COMPUTED GPC FOR DC-DC BUCK CONVERTER
Introduction
72 4.1 Introduction the cost of an MPC or GPC accounts for about 70% of the total design cost of a system. Although the PID is not designed for optimal converter performance, a standard PID compensator is preferred due to its computational efficiency. The sensors used to measure the inductor current are more expensive than the voltage sensor.
Operating principle
Thus, the ripple in the induction current can be minimized using a high switching frequency and a higher induction value. The efficiency of the DC-DC buck converter mainly depends on the design parameters of the circuit. Circuit elements of the converter are selected to operate in CCM with the desired amount of capacitor voltage ripple and inductor current ripple.
Proposed tuning guidelines
For which forecast and control horizons are evaluated according to the tuning guide proposed in Chapter 3.
Simulation result
The overall closed-loop performance of the proposed GPC tuning algorithm is compared with PI control in Figure 4.8. A PI control with gain adjustment [71] has an offset error as the plant model response is observed in the presence of noise and disturbances. Furthermore, in the presence of noise, the performance of PI control deteriorates as the model behaves as a non-minimum phase system for a delayed plant model.
Experimental result and discussions
The waveform of the output voltage is observed in a digital oscilloscope and the waveform for the input voltage of 12 V is shown in Figure 4.11. Similarly, the input voltage also suddenly changes from 12 V to 10 V and the effect is shown in Figure 4.14. To remove the offset error shown in Figure 4.15, an integrator operator can be used before the GPC for reference tracking without offsets.
Modified GPC for reference tracking
A block diagram of a double-loop GPC for error-free reference tracking is shown in Figure 4.17. The PI outer-loop controller was tuned by performing extensive numerical simulation of the closed-loop model in MATLAB. To improve the reference tracking response during disturbances for the precomputed GPC, the outer loop PI control is designed as shown in Figure 4.20, which provides error-free reference tracking.
Summary
The controller design procedure for reference tracking is simplified by implementing the precomputed GPC control input using a DSP microcontroller. Tracking error during load and input changes is eliminated by using a PI control in the pre-computed closed-loop GPC external circuit.
DESIGN OF GPC FOR
MULTI-VARIABLE PLANT MDOEL
Introduction
As the number of inductors required in a SIMO topology is reduced, battery life and switching converter efficiency are improved. In this chapter, a dual-output DC-DC converter is discussed to demonstrate the GPC design process in the case of a SIMO converter. For this purpose, a single-inductor double-output (SIDO) DC-DC converter model is considered.
Design of GPC
Prediction of the augmented state matrix is obtained to determine the output prediction model, expressed as . When eigenvalues lie on the left half of the s-plane, the system is said to be stable one. If eigenvalues lie on the right half of the s-plane, the system is said to be unstable and if eigenvalues lie on the y-axis, the system is said to be marginally stable.
Observer design for the state estimation
106 5.4 Analysis of the stability of the control law can be written as It can be seen from (5.47) that the design of the observer and the design of the controller are independent of each other. It can be seen from (5.47) that the augmented eigenvalues of the system remain unchanged when the observer is designed for the system.
Stability analysis of the control law
The optimal of the chosen finite horizon cost function (J(∆u(k))) is considered as the Lyapunov function. To analyze the convergence of the obtained control law, the quadratic cost function (J(∆u)) is chosen as a candidate for the Lyapunov function,z(f(xn(k))). Therefore, it is proven that the first derivative of the chosen Lyapunov function yields a negative value.
Description of operation
- Transfer function matrix of SIDO DC-DC buck con- verterverter
In mode II operation shown in Figure 5.4, switches S1 and Sa are OFF, Sb, and the freewheeling diode D are ON, so the inductor current decreases to a value of IL2 with a slope of -Eob/L. Similarly, mode III operation is shown in Figure 5.5, switches S1 and Sb operate in the OFF state, while switch Sa and free-wheeling diode D are ON, which results in a decrease in the inductor current to a value of ILo with a slope of -Eob/ L. The transfer function matrix for the open-loop SIDO buck converter in Figure 5.1 is derived and expressed as
![Table 5.1: Design parameter for the SIDO buck converter [82]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10449001.0/136.892.113.732.313.1098/table-5-design-parameter-sido-buck-converter-82.webp)
Simulation result
- Steady state performance
- Performance during load variation
- Performance of the observer
In Figures 5.7 and 5.8, the closed loop model is simulated for a step disturbance signal of magnitude 0.5 and a noise signal of SN R = 48.2 dB for the resulting installation model. The cross-regulation effect for the designed GPC without an observer is shown in Figures 5.9 and 5.10. The performance of GPC for reference tracking of the set output voltage with and without an observer is shown in Figure 5.11.
Summary
CONCLUSION
Scope for future work
With this strategy it is possible to improve the performance of the controller without discarding the primary controller. This aspect is useful if it is not possible to change the primary controller and only the reference signal can be sent, or if one of the controllers is preferred for a lower frequency range and another for a higher frequency range. Although the stability of the controller has been established in this thesis, the uniqueness of the GPC control law is still an open topic that needs to be explored.
SUPPLEMENTARY FILES
Small signal modeling of a SIDO buck con- verter
Jury’s stability criteria [65]
The closed loop system is stable if all poles of the system lie inside the unit circle in az domain. Jurie's stability criteria [65] are similar to Routh-Hurwitz stability criteria in continuous time domain, where a Routh table is prepared and stability of the closed-loop system is determined from the elements of the table. For all the roots to lie inside the unit circle, necessary and sufficient conditions are given as.
Property of symmetric positive definite ma- trix using Schur’s compliment [84]
Rodr´ıguez, " Model Predictive Control - A Simple and Powerful Method to Control Power Converters," IEEE Transactions on Industrial Electronics, vol. Bemporad, "Model Predictive Control Tuning by Controller Matching,"IEEE Transactions on Automatic Control, vol. Jheng, "High Efficiency Single-Inductor Multiple-Output DC-DC Converter," IEEE Transactions on Power Electronics, vol.
