Introduction to Multicopter Design and Control

Transcription

1 Introduction to Multicopter Design and Control Lesson 10 Stability and Controllability Quan Quan, Associate Proessor BUAA Reliable Flight Control Group, Beihang University, China

2 Why is a multicopter dynamical system unstable and what is the DoC o a multicopter? 2016/12/25 2

3 Outline 1. Concepts o Stability 2. Stability Criteria 3. Basic Concepts o Controllability 4. Controllability o Multicopters 5. Conclusion 2016/12/25 3

4 1.Concepts o Stability An Example (a) Stable equilibrium (b) Unstable equilibrium Figure An example or stability Is the ball stable at position 1 and 3 ater it is perturbed? Stability is a concept associated to the equilibrium state o a dynamical system. Position 1 and 3 are equilibrium states while position 2 and 4 are not. he stability o position 1 and 3 are discussed in this lecture. 2016/12/25 4

5 1.Concepts o Stability Standard Form A nonlinear dynamical system with input is usually described by the ollowing nonlinear dierential equation x t, x, u + u g t, x c x t, x, g t, x t, x where x is the state, and u is the control input which is a unction o the state and time. For simplicity, the time-invariant dynamical system is considered x 0 x x x x * * c t, * 0 is the equilibrium state x t x c, t, t, * t, * x x x x c c c For dynamical system x x,0 is the equilibrium state 2016/12/25 5

6 1.Concepts o Stability hree Dierent Deinitions o Stability For dynamical system x x, x 0n 1 is the equilibrium state. he ollowing deinitions are derived rom [1]: Deinition I R, r, x 0 r x t R, t, the equilibrium state x 0n1 is said to be stable. (Lyapunov stability). Deinition I r, x0 rlimt xt 0, the equilibrium state x 0 is asymptotically stable. Deinition I,, r, such that he equilibrium state x 0 is exponentially stable. n1 t t, x t x 0 e, x 0 B 0, r, n1 Figure 10.2 Dierent stability Figures are rom [1] Slotine J-J, Li W. Applied Nonlinear Control. New Jersey, USA: Prentice Hall, /12/25 6

7 2. Stability Criteria Stability o Multicopters Consider the attitude control model in Lesson 6, and the eect o b is ignored: b ω J ω G a he state-space model is where: Θ b ω ω J τ b 1 A 0 I b ω he orces on a ixed-wing aircrat are thrust, gravity, drag and lit, where the lit is changed with the attitude. As a result, the attitude can be controlled by these eedback. hat is why multicopters rely on autopilots to hover, whereas ixed-wing aircrat do not need autopilots to keep orward light. he solution to the state-space equation is t zt e A z0 As the eigenvalues o A are zeros, the solution is unstable at 0 b z Az Θ ω /12/25 7 Θ , z b ω

8 2. Stability Criteria Invariant Set heorems[1] (1) Deinition A set S is an invariant set or a dynamical system i every system trajectory which starts rom the inside o S remains in S or all time thereater. (2) Local Invariant Set heorem Consider autonomous nonlinear system x x, where is continuous. Let V x be a scalar unction with continuous irst partial derivatives. Assume that: 1) or some l, the region l deined by Vx lis bounded; 2) V xxvx x0, xl Let R be the set o all points within where V x 0 l, and M be the largest invariant set in R. hen every solution originating in l tends to M as t x n l x V l x n R x V 0 Invariant M Set M l 2016/12/25 8

9 2. Stability Criteria Invariant Set heorems (3) Global Invariant Set heorem Consider autonomous nonlinear system x x, where is continuous. Let V xbe a scalar unction with continuous irst partial derivatives. Assume that: 1) x V x V is not required to be positive n 2) V x 0, x Let R be the set o all points where V x 0, and Mbe the largest invariant set in R. hen all states globally asymptotically converge to M as t What will happen i M contains only one equilibrium 0? 2016/12/25 9

10 2. Stability Criteria Stability Criteria or Simple System Consider a second-order dynamical system x v v u where uk is a PD controller 1 vx d k2 xxd z Az x xd 0n n In z, A d k2 n k v x I 1In heorem: I k1, k2 0, then limt z t 0. Furthermore z 02n 1is globally exponentially stable. 0 t Proo tips: z t e A z, the real parts o all the eigenvalues o A are negative, the theorem can be proved by decomposing the exponential unction based on Jordan canonical orm 2016/12/25 10

11 2. Stability Criteria Stability or Constrained System Consider the ollowing dynamical system gd a sat u,a u x v gd v sat u,a u where u k1 vxd k2 xxd is a PD controller and sat gd, a a u u, Figure Direction-guaranteed saturation unction u 1, u a u where u u u n, u max u,, u a, a, a u a 1 n 1 n Some open source autopilots, such as PX4, uses directionguaranteed saturation unction to keep the control direction the same as that o the original vector. Why? 2016/12/25 11

12 2. Stability Criteria Stability or Constrained System heorem: I k1, k2 0, then lim. Furthermore is locally exponentially t zt 0 z 02n 1 stable. Exponential stability requires Proo: x v V x 0 v sat gd u,a x v V xvx, x xxd v a uu he ollowing does not satisy v v x these conditions d A Lyapunov unction is designed and dierentiated k V vv uudu vv uuu uuu0 1 V 2 a a a k C 2 k2 k2 2 u Reer to Invariant Set heorem 2016/12/25 12

13 2. Stability Criteria Stability or constrained system (1) he largest invariant set k V k k 1 0 u 0n 1 1v 2x 0n 1 x v x k2 1v 2x 0 1 v 0n 1 Contains only one equilibrium: x 0n1, then n Namely the largest invariant set M contains only one point: x 0, v 0 n1 n1 k k (2) Stability. According to local invariant set theorem, z 02n 1is locally asymptotically stable. Furthermore, V z. hen z 02n 1 is globally asymptotically stable. As the control input u is not constrained in a neighborhood o the equilibrium z 02n 1. According to the last example, z 02n 1 is locally exponentially stable. 2016/12/25 13

14 3. Basic Concepts o Controllability he aircrat can still ly back! From In May 1983, two Israeli Air Force aircrat, an F-15 Eagle and an A-4 Skyhawk, collided in mid-air during a training exercise over the Negev region, in Israel. Notably, the F-15 with a crew o two managedtolandsaelyat a nearby airbase, despite having its right wing almost completely sheared o in the collision. he liting body properties o the F-15, together with its overabundant engine thrust, allowed the pilot to achieve this unique eat. Excellent pilot skill he attitude is Controllable 2016/12/25 14

15 3. Basic Concepts o Controllability Classical Controllability[2] Consider the ollowing linear time-invariant system (1) Deinition x AxBu, x, A, B, u n n n n m m I or any xt 0, there exists a bounded admissible control u t 0, t 1 deined on the inite time interval t0, t 1, which steers xt 0 to zero. hen the system is controllable. 2)Classical controllability criteria: the controllability matrix is o ull rank where, the controllability matrix is he system is controllable i and only i C n1, AB B AB A B rank C AB, [2] Chen C-. Linear System heory and Design (hird Edition). New York, USA: Oxord University Press, 1999 n 2016/12/25 15

16 3. Basic Concepts o Controllability A Simple Example Consider the ollowing linear time-invariant system where A =, C, = B 0 A B B AB 0 0 hen the pair (A,B) is uncontrollable. But the solution o the system is and it converges to zero. x t t e 0 = 0 t x 0 e It should be noticed that although any x0 approaches to zero, the convergence time is not inite. So the system is uncontrollable according to the deinition. However, both controllability and reachability are deined separately or discrete-time linear systems. 2016/12/25 16

17 3. Basic Concepts o Controllability he Limits o Classical Controllability Consider the ollowing linear time-invariant system n n, n nm x AxBu, x A, B, u Classical controllability theories oten require the origin to be an interior point o the control constraint set (a) Origin is the interior point Figure 10.4 Interior point (b) Origin is not the interior point An simple example (zero is not an interior point o the control constraint set) x xu, u0 It is controllable based on the classical controllability method. However, the state x i the initial value x0 0 no matter How close to zero it is. 2016/12/25 17

18 3. Basic Concepts o Controllability Positive Controllability Consider the ollowing linear time-invariant system n n, n nm x AxBu, x A, B, u (1) With control constraint, the controllability deinition is: I or any x t, there exists a bounded admissible control u 0 t0, t1 deined on the inite time interval t0, t 1, which steers x t 0 to zero. hen the system is controllable. Or else, the system is uncontrollable. (2) Controllability theorem (necessary and suicient conditions) [3] 1) he controllability matrix C AB, is o ull rank. 2) here is no real eigenvector v o satisying vbu0 or all A u Physical meaning? [3] Brammer R F. Controllability in linear Autonomous systems with positive controllers. SIAM Journal on Control, 1972, 10(2): /12/25 18

19 3. Basic Concepts o Controllability Positive Controllability (3) he necessity o condition 2): here is no real eigenvector v o A satisying vbu0 or all (4) Counter-example (suppose that condition 2) is not true) here is a real eigenvector o A satisying vbu or all hen vx vaxvbu v 1 y vx 1 u u I 0, vbu 1 0, then the state y, the system is uncontrollable! hereore, condition 2) is necessary. y y 1 0 vbu /12/25 19

20 4. Controllability o Multicopters [4] Multicopter System Modeling (1) Hovering model (2) State and system matrix x h vz x b y b zb 4 u x y z 4 g mg I4 88 A B 1 J J diag mj,, J, J xx yy zz 44 x AxB u g 8 (3) Control input he total thrust n 1, r 0, r U K i1 he system total thrust/moment vector Control input o the system Unidirectional [4] Du G-X, Quan Q, Yang B, Cai K-Y. Controllability analysis or multirotor helicopter rotor degradation and ailure. AIAA Journal o Guidance, Control, and Dynamics, 2015, 38(5): /12/25 20 u n i u B, u u B, U U u uu g, u

21 4. Controllability o Multicopters Classical Controllability First, the gravity is allocated to each propulsor u n r U, K u u i1 i i i At last, check i the controllability matrix C AB, is o ull rank, B BB Multicopter controllability based on classical controllability: Second, obtain the control input and control constraints 8n r zero is an interior point 2016/12/ gb, B B B g x AxB B g Ax B B B Ax B [5] Du G-X. Research on the Controllability Quantiication o Multirotor Systems. Doctoral hesis. Beihang University, China, [In Chinese] u But: (1)It depends on the control allocation method. (2) It needs to know gb,

22 4. Controllability o Multicopters Classical Controllability Available control authority index (Degree o Controllability, DoC): g, min g u, u G, G Figure 10.5 Degree o Controllability Hovering model x AxB u g I g, 0, then g Is an interior point o the constraint set u 2016/12/25 22

23 4. Controllability o Multicopters Positive Controllability he ollowing three statements are equivalent to each other: 1) g, 0 2) here is no real eigenvector v o A satisying vbu0 or all u 3) g is an interior point o Because o the simple AB, structure, it is easy to see that rank C AB he multicopter dynamical system is controllable i and only i g, 0 A Matlab oolbox or Calculating the Available Control Authority Index o Multicopters is available at /12/25 23

24 4. Controllability o Multicopters Controllability o PNPNPN and PPNNPN Hexacopter est the controllability o the hexacopter subjected to one propulsor ailure Figure 10.6 PNPNPN and PPNNPN hexacopter P denotes that a propulsor rotates clockwise while N anticlockwise able 10.1 Hexacopter parameters Parameter Value Units m kg g 9.80 m/s 2 d m K i, i = 1,, N J xx kg m 2 J yy kg m 2 J zz kg m 2 k µ /12/25 24

25 4. Controllability o Multicopters Controllability o PNPNPN and PPNNPN Hexacopter able 10.2 Hexacopter (PNPNPN and PPNNPN) controllability with one propulsor ailure Propulsor ailure Rank o C (A, B) PNPNPN PPNNPN ACAI Controllability ACAI Controllability No wear/ailure controllable controllable η 1 = uncontrollable controllable η 2 = uncontrollable controllable η 3 = uncontrollable controllable η 4 = uncontrollable controllable η 5 = uncontrollable 0 uncontrollable η 6 = uncontrollable 0 uncontrollable uncontrollable uncontrollable 2016/12/25 25

26 4. Controllability o Multicopters Controllability o PNPNPN and PPNNPN Hexacopter 1 N N o 2 o P P (a) o N Physical insights: the remaining propulsors o Fig 10.7(b) are still composed o a basic quadcopter coniguration that is symmetric about the mass center, whereas that o Fig 10.7(b) cannot. P 2016/12/25 26 P Figure 10.7 Physical insights o multicopter system controllability 4 (b) N

27 4. Controllability o Multicopters Controllability or a Degraded System (1) he degraded system x AxB u g (2) State and system matrix (3) Control constraints 6 x h vh p q, he total thrust u g mg x y 0 I A B J 3 0 0, ,, 3, u he yaw and yaw rate are removed he system total thrust/moment vector J diag mj, xx, Jyy [6] Du G.-X., Quan Q., and Cai K.-Y. Controllability Analysis and Degraded Control or a Class o Hexacopters Subject to Rotor Failures [J]. Journal o Intelligent & Robotic Systems, 2015, 78(1): U K r 1 n, r n 0, i i u B, u u B, U 2016/12/25 27

28 4. Controllability o Multicopters PNPNPN Hexacopter without Yaw States uncontrollable able 10.3 ACAI with dierent γ γ g, γ g, Assume that the maximum thrust o each propulsor is K ma g, 0,1 wo observations: 1) I 0.3, the degraded system is uncontrollable though the thrust provided by the let ive propulsors is still able to compensate or the weight o the hexacopter. his is consistent with [5]. 2) I 0.6, the ACAI o the multicopter system remains almost the same. 2016/12/25 28

29 4. Controllability o Multicopters Further Discussions (1) Control constraints, as well as the coniguration, determine the controllability o a multicopter system. In [6], the maximum thrust o the remaining ive propulsors o the PNPNPN hexacopter needs to satisy K 5mg/18( 0.3mg) to achieve a sae landing theoretically. As mentioned above, K 0.3mg (which is computed numerically) is also a guidance to design a hexacopter or the degraded system (ive propulsors work together). 2016/12/25 29

30 4. Controllability o Multicopters Further Discussions (2) Controllability analysis requires to clariy the equilibrium states irst, and the model aterwards. Some researchers in [7] presented periodic solutions to control a quadcopter ater losing one, two opposing or even three propellers. It seems a contradiction with the proposed results above. he quadcopter with propulsor ailures is uncontrollable at hover based on the controllability analysis method, namely the damaged quadcopter cannot hover. It should be noticed that this does not mean that the quadcopter cannot stay in the air anymore. he authors in [7] established new equilibrium states (not constant positions) or the quadcopter, which is controllable around the new equilibrium states. [7] Mark W. Mueller, and Raaello D Andrea. Stability and control o a quadrocopter despite the complete loss o one, two, or three propellers [C]. IEEE International Conerence on Robotics and Automation, 2014: /12/25 30

31 4. Controllability o Multicopters Application o the DoC (1) Health Evaluation (Online) (2) Multicopter design evaluation (Oline) x AxB FGd Propulsor has slow dynamics Basic idea: unknown dynamics, aults are lumped into d, and the DoC is computed[8] Is the design able to resist wind? [8] Bin Liu, Zhiyao Zhao, Binxian Yang, Quan Quan, Kai-Yuan Cai. A Real-ime Assessment Approach to Quadrotor Flight Control Capability, he 33rd Chinese control conerence, Nanjing, 2014, (in Chinese) 2016/12/25 31

32 5. Conclusion (1) Controllability is the basic o a dynamics system. (2) Classical controllability: 1) It depends on the control allocation method. 2) It needs to know gb, (3) Positive controllability connects the controllability o multicopters with the ACAI: 1) It does not depend on the control allocation method. 2) Can be used to show how controllable the system is (4) he ACAI can used to show the degree o controllability o multicopters, and it also can be used to show the saety o a multicopter. Please reer to [9] or more details degree o controllability. [9] Du G-X, Quan Q. Degree o controllability and its application in aircrat light control. Journal o Systems Science and Mathematical Sciences, 2014, 34(12): (in Chinese) 2016/12/25 32

33 Acknowledgement Deep thanks go to Guang-Xun Du or material preparation. Xunhua Dai 2016/12/25 33

34 hank you! All course PPs and resources can be downloaded at For more detailed content, please reer to the textbook: Quan, Quan. Introduction to Multicopter Design and Control. Springer, ISBN: It is available now, please visit /12/25 34