Linear quadratic regulator

Transcription

1 Chapter 5 Linear quadratic regulator 5.1 Introduction The linear quadratic regulator is one of the most simple optimal controller that can be developed. It is a stable and explicit controller. Its key points are that it is ease to calculate and that it is the best optimal controller that can be done. Its main disadvantage is that can t work with constraints. Can be developed a MPC that work in a similar way than the LQR. These controllers are designed as Constrained LQR. However these controllers have a performance worse than conventional LQR. Its main advantage is the possibility to use constraints. 5.2 Problem design Considering the following discrete linear system x k+1 = Ax k + Bu k y = Cx k (5.1) The linear quadratic regulator propose the following cost function to minimize J = x Q + u R (5.2) where x is the state vector, u is the control action vector, Q is the weighting matrix to evaluate the state and R is the weighting matrix to evaluate the control action. The solution of this controller can be obtained resolving a Ricatti equation. In order to change the equilibrium point from origin to x D it s used the following modification where u = Kx + M y D, M is a matrix that resolve x = M y and K is the discrete control law 26

2 of the LQR. Supposing that the steady state can be reached, this matrix can be obtained from the following way M = (C (I 2 2 (A + BK)) 1 B) 1 (5.3) 5.3 Matlab Simulation. In order to emulate this controller it is made a scheme in Simulink/Matlab. Figure 5.1: LQR scheme (Simulink/Matlab) It is simulate two situations: a sunny day in which can be obtained a lot of energy from the solar panels and a cloudy day in which most of the energy is obtained from the storage systems. The controller LQR has a explicit control law u = K LQR x obtained resolving the Riccati 27

3 equation. In this case, the control law is obtained using the command dlqr from M atlab. It necessary to use the weighting matrixes in order to calculate the matrix K LQR. The weighting matrixes are Q y = 5 8 ; R = 1 1 K LQR =, 12, 6, 1, 5 In figure 5.1 in color cyan can be watched a full state observer. It is used to obtain the state vector x. Due to the matrix C of the linearized system is not singular, the output and the state of the system can be calculated directly from: x = C 1 y (5.4) However, the observer provide a certain degree of smoothness and it is relatively simple to make thus has been decided to calculate it and introduce it in the scheme. The observer gain is, 12 L =, 5 The economic cost function to optimize was the equation 3.1 that can be rewritten as equation 3.3. Can be watched that the real cost function of the LQR is J = v 2 R + y y D 2 Q f eco (5.5) where f eco is the economic cost function,u and v are related through equation u = v + L w (5.6) where w are the system disturbances,u are control action that attack the linearized system and v are the control action before the feedforward. Therefore this controller it is not an economic controller due to the economic cost function is not minimized. This controller will move the system to the desired operational point minimizing the equation 5.5. The problem is that it is not the economic cost function A sunny day. In this case it is supposed a sunny day and therefore there are a excess of generated energy that in a economic system must be stored in the storage system according to the minimization of a cost function, in this case the equation 3.1. This following profile represent the best condition of working and if the system is good designed the controller must have its best performance in 28

4 this case. This profile is a typical design case and was provided by L. valverde Disturbances (Sunny day): Generation and Solar Panels hours 12 Disturbances: Sunny day hours Figure 5.2: Disturbances for LQR simulation (Sunny day) 29

5 In figure 5.2(a) can be watched that the panels produce a lot of energy in a range of time. Thus the controller should has to manage this energy excess so that it is necessary to buy or waste a minimal amount of energy. It will be the main feature of a good economic controller. In order to obtain good results, the chosen weighting matrixes are: Q y = 1 1 ; R = 5 8 The weighting values in the matrix Q are increase in order to compensate the effect of the disturbances. SOC(%) Battery SOC SOC real SOC operation point MHL(%) Metal Hidrure Level (MHL) MHL real MHL operation point Figure 5.3: Curves (Sunny day):(a)soc LQR,(b)HML LQR In figure 5.3(a),(b) is shown the evolution in 24h of the storage system when the desired operational point are SOC = 6%, HML = 45% and the initial state of the storage system are SOC = 5%, HML = 5%. In this figures is exposed that the hydrogen system is only used at the beginning (5.4) in order to reach the desired operational point. This graphics show how the system converges to this steady state. It s clear that it is not a good economical behaviour due to in economically optimal-controlled system the plant never reaches a steady state. Figure 5.7 show the energy that is wasted or purchased to the GP N. It is a indicator of the 3

6 15 Hydrogen Power hours Figure 5.4: Hydrogen power P LQR H2 (Sunny day). Power grid Figure 5.5: power sypply P LQR grid (Sunny day). 31

7 performance of the controller. The cost function take into account two economic aspect, one is the minimization of the control action applied to the system. In particular P grid that show the purchased power. A economically well-controller plant must to reduce this power near to zero. In the cost function it is exposed providing a high weighting factor about this power. In this case the signal P grid is not near to zero thus it s not a good economical controller ALL Signals Hydrogen Grid batteries Pnet Panels Figure 5.6: All LQR curves (Sunny day). In figure 5.8 is shown all powers. In this graphics can be observed easily that the demand is satisfied by general power supply (P grid ) and when there are excess of energy it is wasted. It is a very bad economic performance due to the hydrogen system and batteries are not used to store the excess of energy. In figure 5.7 (a) can be watched that the demand is satisfied so this control objective is satisfied. In figure 5.7(b) is shown the evolution of the economic cost function with a average value of The cost function is mainly composed by the term J(u) which indicates the evolution of the powers of the plant. The term J(y) of the cost function is near to zero so this is another indicator that this controller is not economical A cloudy day. In a cloudy day the energy provide by solar panels will be poor. Therefore in this case the demand should be supported by the storage systems but examining the other test the more likely it will be that all the energy is provided by the GP N. The storage system will be moved to the operational point exposed above. 32

8 11 Provided Power supply Cost(no Units) Cost Function Cost function values Cost J(u) setpoint desviation J(y) Average Cost function Figure 5.7: LQR Curves:(a)Satisfaction of the (Sunny day),(b)economic cost function (Sunny day) 33

9 25 2 Disturbances (Cloudy day): Generation and Solar Panels hours 15 Disturbances: Cloudy day hours Figure 5.8: Disturbances for LQR simulation (Cloudy day) 34

10 ALL Signals Hydrogen Grid batteries Pnet Panels Figure 5.9: All LQR curves (Cloudy day). In figure 5.9 can be observed that in this case happens the same that in the previous case, the energy faults are satisfied using the energy purchased from GP N. In figure 5.1(b) can be observed that the average cost is lightly low in this case. In this case the value of the cost function is about Can be confuse that this controller has a better performance in a cloudy day than in a sunny day but this result is due to that the disturbances are smaller in this case than in the previous case. 35

11 11 Provided Power supply Cost Function Cost function values Cost J(u) setpoint desviation J(y) Average Cost function Cost(no Units) Figure 5.1: Curves: (a)satisfaction of the (Cloudy day),(b)economic cost function (Cloudy day) 36