PID, LQR and LQR-PID on a Quadcopter Platform

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3 the parameters k, p, e were obtained applying a lqr function shown in equation 9, where Q is a square matrix of sixth order described in equation 10, adjusted to provide the most efficient values and R is a unitary vector. [k, p, e] = lqr(a, B, Q, R) (9) Q = (10) The output of the LQR function was the vector K shown in equation (11) containing the six elements that controls each state of the system individually, this makes a expansion of the state space function necessary. This expansion is obtained solving the equation 6 and the result is a system of six equations 12 to 17 that represent each one of the states. Fig. 3. Vertical position step response for LQR control Figure 3 shows the step response for the vertical position of the plant with no overshooting value and with an accommodating time of two seconds. It was also performed the LQR control for all other movements of the quadcopter from where analogous satisfactory results were obtained. Figure 4 shows the vertical-speed step response that was acquired when a step source was placed in the vertical position state K = (11) 1 = x 2 (12) Fig. 4. Vertical speed step response for LQR Control 2 = x x x x 6 (13) 3 = 10x 3 + 7u (14) 4 = 10x 4 + 7u (15) 5 = 10x 5 + 7u (16) 6 = 10x 6 + 7u (17) When a block diagram of the new expanded equations 12 to 17 is built, and the control law shown in equation 8 is applied, it becomes a closed loop system with six feedback gains and no inputs. As the regulators are known for leading the responses of the system to a null value, a disturbance (step) was performed at the state which the control was required and a result for the vertical position is presented in figure 3. As mentioned above, the speed response is null after a transition time because of the regulator controller. It does not mean that in a practical model the vertical speed goes to zero because there will always be a disturbance at the position state. B. ITAE - tuning PID 1) Vertical speed controller: The simplified transfer function that describes the vertical position is shown in equation 18: G w (s) = (18) s(s + 10) Once obtained the transfer function of the attitude, a controller function can be defined as: C(s) = (k3s2 + k1s + k2), (19) s where k 3, k 2, k 1 represents, respectively, the derivative, integral and proportional gains. Then, the transfer function of the closed-loop system can be dened as:

4 k3s 2 + k1s + k2 F w (s) = s 3 + (10 + k3)s 2 + k1s + k2. (20) Equalling the coefficients of the characteristic equation 20 with ITAE model optimum coefficients [12], values k 1, k 2 and k 3 were obtained. k1 = 860 (21) k2 = 8000 (22) k3 = 25 (23) After implementing the controller using the values found with the ITAE method, it was verified that the plant was not being controlled. That can be explained analysing the behaviour of the model, which tends to a negative response before achieving stability. This kind of process usually has its origin on two competing effects: a fast dynamic effect and a slow dynamic effect. These effects produce the negative start to the response before the step recovers to settle at a positive state value. To solve this problem, an additional negative unitary gain is placed at the output of the controller to maintain a negative feedback loop, which gives us a reverseacting controller. The phase delay compensators are also known for their properties of assisting for a better steady response with the consequence of shifting the poles to the right side of the root locus plane. This effect causes a destabilisation of the model and a longer transient time feedback [13]. Analysing the transfer function of the plant, it is possible to perceive the natural presence of one integrator, so the integral gain k 2, shown in equation 22 was set to null. Equations 21 and 23 (k 1 and k 3 gains) were applied in the ways mentioned above, producing the response shown at figure 5. simplified model shown in equation 18 and later applied to the space state matrices given by equations 3 through 5. 2) Vertical position controller: The transfer function of vertical position is described by equation 24, which will be controlled by new proportional, integral and differential gains: G z (s) = s 2 (s + 10). (24) The new vertical position controller C B can be described as C B (s) = (k3 B)s 2 + (k1 B )s + (k2 B ). (25) s Considering the same simplification shown in equation 18, and neglecting k 2B for the same reason as k 2 22, the vertical position transfer function was obtained, as shown in equation 26. F z (s) = C B (s)(k3s 2 + k1s) s(1 + C B (s)((k3 + 1)s 2 + (k1 + 10)s)) (26) Performing an analogous process to that described in Section IV-B1 for equation 26, the new gains k 1B, k 2B and k 3B are k 1B = (27) k 2B = (28) k 3B = 0.7 (29) Applying this new controller to the state space matrices given by equations 3, 4 and the modified equation 5 (to select the vertical positions as the system output), we obtained the response illustrated in figure 6. Once again, the model simplification was validated. Fig. 5. PID vertical speed step response. Figure 5 shows the step response for the vertical speed controlled by the gains calculated above. Equations 21 and 23 (respectively k1 and k3 gains) were obtained using the Fig. 6. PID Vertical position step response.

5 C. LQR - tuning PID 1) Vertical speed controller: As a comparative criterion, we decided to tune controllers for the same attitudes, but this time using the method proposed by [1], where the PID gains are calculated based on equations 30 trhough 34 [ Kp K d ] = Kp C 1 (30) K i = (I m + K d CB)K i (31) 2) Vertical position controller: Analogously, the same methodology can be applied for the vertical position controller, by using the state space matrices given by equation 24. Using matrix Q = diag[0, 0, 1000, 10000] as parameter for the software, the parameters k P z, k Iz and k Dz are as follows: k P z = (38) k Iz = 1 (39) Where: [ ] C C = CA CBK p (32) K p = (I m + K d CB) 1 (K p C + K d CA) (33) K i = (I m + K d CB) 1 K i (34) k Dz = (40) Figure 8 shows the step response for the vertical position controller. The methodology for obtaining the PID parameters is fully described in [1]. For better tuning results, it is necessary to execute the algorithm innumerous times until the desired response for the system is reached. In order to accomplish that, an algorithm was developed 2 that calculates automatically the gains based on the simplified space state matrices generated from equation 18 of the plant and the matrix Q (from the LQR control theory). Development time increased drastically and the controller could be tuned with reasonable parameters for the system. Using matrix Q = diag[0, 10000, 10000] as a parameter of our algorithm, the following gains k P w, k Iw and k Dw were obtained: k P w = (35) k Iw = 1 (36) k Dw = (37) Figure 7 shows the step response for the vertical speed controller. Fig. 7. PID - LQR vertical speed step response 2 Available at: https : //dl.dropbox.com/u/ /p ID LQ.m Fig. 8. PID - LQR vertical position step response. V. RESULT ANALYSIS Looking back at the step responses shown above, it is clear that each control system has a different response for the same attitudes. However, it is possible to choose the most suitable technique based on the characteristics that is needed for the quadcopter. A. Vertical speed responses Comparing figures 4, 5 and 7, it is possible to notice that the PID controller tuned by LQR theory is the fastest. The controller presented a fast response with no overshooting value. B. Vertical position responses The results for the vertical position controllers presented some differences between their performances. The PID controller tuned with ITAE performance indexes, presented in figure 6, showed a faster response compared to the other controllers. Even with less than ten percent overshooting, the settling time is around 0.25 seconds. When looking at the step response for the PID tuned with LQR theory (figure 8) and the classic LQR (figure 3), the settling time is about the same, but with no overshoot value. However, it is important to point out that the robustness of the controllers are not being analysed. Faster controllers does not necessarily means consistent responses under disturbances.

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