Pole placement control: state space and polynomial approaches Lecture 1

Transcription

1 : state space and polynomial approaches Lecture 1 dynamical O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr o.sename November 7, 2017

2 Outline dynamical dynamical

3 References dynamical Some interesting books: K.J. Astrom and B. Wittenmark, Computer-Controlled Systems, Information and sciences series. Prentice Hall, New Jersey, 3rd edition, R.C. Dorf and R.H. Bishop, Modern Control Systems, Prentice Hall, USA, G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall, 2005

4 Course Objectives dynamical Main objective: design method : state space and polynomial approaches. Study of system properties in view of /observation Synthesis of state feedback ler for state space linear Design of state observers (estimators) Calculation of a 2 degrees-of-freedom ler using a polynomial approach for continuous-time and linear as well.

5 Why state space equations? dynamical dynamical where physical equations can be derived : electrical engineering, mechanical engineering, aerospace engineering, micro, process plants... include physical parameters: easy to use when parameters must be changed for the design State variables have physical meaning. Allow for including non linearities (state constraints ) Easy to extend to Multi-Input Multi-Output (MIMO) Advanced design methods are based on state space equations (reliable numerical optimisation tools)

6 Some physical examples dynamical

7 dynamical dynamical

8 General dynamical system dynamical Many dynamical can be represented by Ordinary Differential Equations (ODE) as { ẋ(t) = f ((x(t),u(t),t), x(0) = x0 (1) y(t) = g((x(t),u(t),t) where f and g are non linear functions.

9 Example: Lateral vehicle model The dynamical equations are as follows: dynamical

10 Vehicle model - synopsis [ ] ż s,z s ż us,z us (suspensions ) u ij Suspensions δ F szij F dx,y,z & M dx,y,z (external disturbances) xs y s z s dynamical (braking & steering ) [T bij,δ] Chassis (vehicle dynamics) ẍ s,ÿ s ψ,v, F sz,z us (road characteristics) Wheels [µ ij,z rij ] F tx,y,z λ ij β ij ω ij (tire, wheel dynamics) θ φ ψ

11 Vehicle model - dynamical equations ẍ s = ( ) (F txfr + F txfl )cos(δ) + (F txrr + F txrl ) (F tyfr + F tyfl )sin(δ) + m ψẏ s F dx /m ÿ s = ( ) (F tyfr + F tyfl )cos(δ) + (F tyrr + F tyrl ) + (F txfr + F txfl )sin(δ) m ψẋ s F dy /m z s = ( ) F szfl + F szfr + F szrl + F szrr +F dz /ms z usij = ( ) F szij F tzij /musij θ = ( ) (F szrl F szrr )t r + (F szfl F szfr )t f mhÿ s + (I y I z ) ψ φ+m dx /Ix φ = ( ) (F szrr + F szrl )l r (F szfr + F szfl )l f + mhẍ s +(I z I x ) ψ θ+m dy /Iy ψ = ( (F tyfr + F tyfl )l f cos(δ) (F tyrr + F tyrl )l r + (F txfr + F txfl )l f sin(δ) +(F txrr F txrl )t r + (F ) txfr F txfl )t f cos(δ) (F txfr F txfl )t f sin(δ) +(I x I y ) θ φ+m dz /Iz v ij R ij ω ij cosβ ij max(v ij,r ij ω ij cosβ ij ) λ ij = ω ij = ( RF txij (µ,λ,f n ) + T bij )/I w β ij = arctan ( ẋ ) ij ẏ ij dynamical

12 Definition of state space representations A continuous-time LINEAR state space system is given as : { ẋ(t) = Ax(t) + Bu(t), x(0) = x0 y(t) = Cx(t) + Du(t) x(t) R n is the system state (vector of state variables), u(t) R m the input y(t) R p the measured output A, B, C and D are real matrices of appropriate dimensions x 0 is the initial condition. n is the order of the state space representation. Matlab : ss(a,b,c,d) creates a SS object SYS representing a continuous-time state-space model (2) dynamical

13 Systems definition A state space system is as follows: { x((k + 1)h) = Ad x(kh) + B d u(kh), x(0) = x 0 y(kh) = C d x(kh) + D d u(kh) (3) dynamical where h is the sampling period. Matlab : ss(a d,b d,c d,d d,h) creates a SS object SYS representing a state-space model

14 Recall Laplace & Z-transform From Transfer Function to State Space dynamical H(s) to state space X U = den(s) Y X = num(s) H(z) to state space X U = den(z) Y X = num(z) Ẋ = AX + BU Y = CX + DU X k+1 = A d X k + B d U k Y k = C d X k + D d U k Y (s) = [ C[sI A] 1 B + D ] U(s) Y (z) = [ C d [zi A d ] 1 ] B d + D d U(z) }{{}}{{} H(s) H(z)

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16 Definition An equilibrium point x eq is stable if, for all ρ > 0, there exists a η > 0 such that: x(0) x eq < η = x(t) x eq < ρ, t 0 Definition An equilibrium point x eq is asymptotically stable if it is stable and, there exists η > 0 such that: dynamical x(0) x eq < η = x(t) x eq, when t These notions are equivalent for linear (not for non linear ones).

17 Analysis dynamical The stability of a linear state space system is analyzed through the characteristic equation det(si n A) = 0. The system poles are then the eigenvalues of the matrix A. It then follows: Proposition A system ẋ(t) = Ax(t), with initial condition x(0) = x 0, is stable if Re(λ i ) < 0, i, where λ i, i, are the eigenvalues of A. Using Matlab, if SYS is an SS object then pole(sys) computes the poles P of the LTI model SYS. It is equivalent to compute eig(a).

18 Analysis - Lyapunov The stability of a linear state space system can be analysed through the Lyapunov theory. It is the basis of all extension of stability for non linear, time-delay, time-varying... Theorem A system ẋ(t) = Ax(t), with initial condition x(0) = x 0, is asymoptotically stable at x = 0 if and only if there exist some matrices P = P T > 0 and Q > 0 such that: dynamical A T P + PA = Q (4) see lyap in MATLAB. Proof: The Lyapunov theory says that a linear system is stable if there exists a continuous function V (x) s.t.: V (x) > 0 with V (0) = 0 and V (x) = dv dx 0 A possible Lyapunov function for the above system is : V (x) = x T Px

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20 refers to the ability of ling a state-space model using state feedback. Definition Given two states x 0 and x 1, the system (2) is lable if there exist t 1 > 0 and a piecewise-continuous input u(t), t [0,t 1 ], such that x(t) takes the values x 0 for t = 0 and x 1 for t = t 1. Proposition The lability matrix is defined by C = [B,A.B,...,A n 1.B]. Then system (2) is lable if and only if rank(c ) = n. If the system is single-input single output (SISO), it is equivalent to det(c ) 0. Using Matlab, if SYS is an SS object then ctrb(sys) returns the lability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to ctrb(sys.a,sys.b) Exercices Test the lability of the previous examples: DC motor, suspension, inverted pendulum. dynamical

21 More on lability dynamical Other criteria Definition of stabilizability About the decomposition into the lable/unlable subspaces: use of ctrbf

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23 dynamical refers to the ability to estimate a state variable, from the knowledge of the system input and output variables Definition A linear system (2) is completely observable if, given the and the output over the interval t 0 t T, one can determine any initial state x(t 0 ). It is equivalent to characterize the non-observability as : A state x(t) is not observable if the corresponding output vanishes, i.e. if the following holds: y(t) = ẏ(t) = ÿ(t) =... = 0

24 Where does observability come from? dynamical Compare the transfer function of the two different and ẋ = ẋ = x + u y = 2x [ ] [ 1 x + 1 ] u y = [ 2 0 ] x

25 cont. Proposition The observability matrix is defined by O = C CA. CA n 1. Then system (2) is observable if and only if rank(o) = n. If the system is single-input single output (SISO), it is equivalent to det(o) 0. Using Matlab, if SYS is an SS object then obsv(sys) returns the observability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to OBSV(sys.a,sys.c). Exercices Test the observability of the previous examples: DC motor, suspension, inverted pendulum. Analysis of different cases, according to the considered number of sensors. dynamical

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27 State space analysis ( ) A system (state space representation) is stable iff all the eigenvalues of the matrix A d are inside the unit circle. definition Definition Given two states x 0 and x 1, the system (3) is lable if there exist K > 0 and a sequence of samples u 0,u 1,...,u K 1, such that x k takes the values x 0 for k = 0 and x 1 for k = K 1. dynamical definition Definition The system (3) is said to be completely observable if every initial state x(0) can be determined from the observation of y(k) over a finite number of sampling periods.

28 State space analysis (2) The system is lable iff C (Ad,B d ) = rg[b d The system is observable iff O (Ad,C d ) = rg[c d A d B d...a n 1 d B d ] = n C d A d...c d A n 1 d ] T = n dynamical Duality of (C d,a d ) of (A T d,ct d ). of (A d,b d ) of (B T d,at d ).

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30 Minimality dynamical Definition A state space representation of a linear system (2) of order n is said to be minimal if it is lable and observable. In this case, the corresponding transfer function G(s) is of minimal order n, i.e is irreducible (no cancellation of poles and zeros). When the transfer function is not of minimal order, there exists non lable or non observable modes.

31 Kalman decomposition When the linear system (2) is not completely lable or observable, it can be decomposed as shown. It is of course very important, in that case, to study the stability of the non lable and non observable modes, which refers to as the stabilizability and detectability, respectively Use ctrbf and obsvf in Matlab. dynamical