, analysis and of linear systems using state space representations Grenoble INP / GIPSA-lab Pole placement February 2018 -based optimal digital
of dynamical systems as state space representations of the state feedback Synthesis of the state feedback : the pole placement or how to ensure disturbance attenuation with a state feedback? and output feedback A preliminary property: -based optimal digital Pole placement -based optimal digital
Pole placement -based optimal digital
References Some interesting books: K.J. Astrom and B. Wittenmark, Computer-Controlled Systems, Information and systems sciences series. Prentice Hall, New Jersey, 3rd edition, 1997. R.C. Dorf and R.H. Bishop, Modern Control Systems, Prentice Hall, USA, 2005. G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, 2001. G. Franklin, J. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall, 2005 Pole placement -based optimal digital
The " design" process Plant study and modelling Determination of sensors and actuators (measured and led outputs, inputs) Performance specifications Control design (many methods) Simulation tests Implementation, tests and validation Pole placement -based optimal digital
The " design" process in CLEAR Pole placement -based optimal digital
About modelling... Identification based method System excitations using PRBS (Pseudo Random Binary Signal) or sinusoïdal signals Determination of a transfer function reproducing the input/ouput system behavior Knowledge-based method: Represent the system behavior using differential and/or algebraic equations, based on physical knowledge. Formulate a nonlinear state-space model, i.e. a matrix differential equation of order 1. Determine the steady-state operating point about which to linearize. Introduce deviation variables and linearize the model. Tools: Matlab/Simulink, LMS Imagine.Lab Amesim, Catia-Dymola, ADAMS, MapleSim... Pole placement -based optimal digital
Simulation of complex system (LMS Imagine.Lab AMESim) System Simulation for Controller Design What it means and what is required Simulation of the complete system using an assembly of components Components are described with analytical or tabulated models Multi-physics / Multi-level Control-oriented actuator models Description of physical phenomena based on few macroscopic parameters Models for static and dynamic responses, in time & frequency domains Pole placement -based optimal Restricted Siemens AG 2016 Page 9 Siemens PLM Software digital
of dynamical systems as state space representations Pole placement -based optimal digital
Why state space equations? dynamical systems where physical equations can be derived : electrical engineering, mechanical engineering, aerospace engineering, microsystems, process plants... include physical parameters: easy to use when parameters are changed for design. Need only for parameter identification or knowledge. State variables have physical meaning. Allow for including non linearities (state constraints, input saturation) Easy to extend to Multi-Input Multi-Output (MIMO) systems Advanced design methods are based on state space equations (reliable numerical optimisation tools) easy exportation from advanced modelling softwares Pole placement -based optimal digital
Towards state space representation What is a state space system? A "matrix-form" representation of the dynamics of an N- order differential equation system into a FIRST order differential equation in a vector form of size N, which is called the state. of a system state The state of a dynamical system is the set of variables, known as state variables, that fully describe the system and its response to any given set of inputs. Mathematically, the knowledge of the initial values of the state variables at t 0 (namely x i (t 0 ),i = 1,...,n), together with the knowledge of the system inputs for time t t 0, are sufficient to predict the behavior of the future system state and output variables (for t t 0 ). Pole placement -based optimal digital
of a NonLinear dynamical system Many dynamical systems can be represented by Ordinary Differential Equations (ODE). A nonlinear state space model consists in rewritting the physical equation into a first-order matrix form 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 and x(t) R n is referred to as the system state (vector of state variables), u(t) R m the vector of m inputs (actuators) y(t) R p the vector of p measured outputs (sensors) x 0 is the initial condition. Pole placement -based optimal digital
Example of a one-tank model Usually the hydraulic equation is non linear and of the form S dh dt = Q e Q s where H is the tank height, S the tank surface, Q e the input flow, and Q s the output flow defined by Q s = a H. the state space model The system is represented by an Ordinary Differential Equation whose solution depends on H(t 0 ) and Q e. Clearly H is the system state, Q e is the input, and the system can be represented as: { ẋ(t) = f (x(t),u(t)), x(0) = x0 (2) y(t) = x(t) with x = H, f (x,u) = a S x + 1 S u Pole placement -based optimal digital
Example: Underwater Autonomous Vehicle UAV Aster x Pole placement Actions: axial propeller to the velocity in Ox direction and 5 independent mobile fins : 2 horizontals fins in the front part of the vehicle (β 1, β 1 ). 1 vertical fin at the tail of the vehicle (δ). 2 fins at the tail of the vehicle (β 2, β 2 ). -based optimal digital
UAV modelling Physical model: M ν = G(ν)ν + D(ν)ν + Γ g + Γ u (3) η = J c (η 2 )ν (4) where: - M: mass matrix: real mass of the vehicle augmented by the "water-added-mass" part, - G(ν) : action of Coriolis and centrifugal forces, - D(ν): matrix of hydrodynamics damping coefficients, - Γ g : gravity effort and hydrostatic forces, - J c (η 2 ): referential transform matrix, - Γ u : forces and moments due to the vehicle s actuators. Pole placement -based optimal digital
UAV state definition A 12 dimensional state vector : X = [ η(6) ν(6) ] T. η(6): position in the inertial referential: η = [ η 1 η 2 ] T with η 1 = [ x y z ] T and η2 = [ φ θ ψ ] T. x, y and z are the positions of the vehicle, and φ, θ and ψ are respectively the roll, pitch and yaw angles. ν(6): velocity vector, in the local referential (linked to the vehicle) describing the linear and angular velocities (first derivative of the position, considering the referential transform: ν = [ ν 1 ν 2 ] T with ν 1 = [ u v w ] T and ν2 = [ p q r ] T Pole placement -based optimal digital
Exercise: a simple pendulum Let consider the following pendulum T l Pole placement θ M -based where θ is the angle (assumed to be measured), T the led torque, l the pendulum length, M its mass. Give the dynamical equations of motion for the pendulum angle (neglecting friction) and write the nonlinear state space model. optimal digital
of linear 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, e.g. A = [a ij ] i,j=1:n with n rows and n columns 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 (5) Pole placement -based optimal digital
A state space representation of a DC Motor Assumption: only the speed is measured. The dynamical equations are : Ri + L di dt + e = u e = K eω J dω = f ω + Γ m dt Γ m = K c i System of 2 equations( of order ) 1 = 2 state variables. ω A possible choice x = It gives: i ( ) ( ) f /J Kc /J 0 A = B = C = ( 0 1 ) K e /L R/L 1/L How to extend this definition when: measurement= motor angular position θ? Pole placement -based optimal digital
Examples: Suspension Let the following mass-spring-damper system. Pole placement where x 1 is the relative position (measured), M 1 the system mass, k 1 the spring coefficient, u the force generated by the active damper, and F 1 is an external disturbance. Applying the mechanical equations around the equilibrium leads to: M 1 ẍ 1 = k 1 x 1 + u + F 1 (6) -based optimal digital
Examples: Suspension cont. ( x1 The choice x = ẋ1 ) gives { ẋ(t) = Ax(t) + Bu(t) + Ed(t) y(t) = Cx(t) where d = F 1, y = x 1 with ( ) ( 0 1 0 A =, B = E = k 1 /M 1 0 1/M 1 ),and C = ( 1 0 ) Pole placement -based optimal digital
Example : Wind turbine modelling from CAD software Pole placement -based optimal digital
Some important issues A complete ADAMS or CATIA model can include 193 DOFs to represent fully flexible tower, drive-train, and blade components simulation model Different operating conditions according to the wind speed Control objectives: maximize power, enhance damping in the first drive train torsion mode, design a smooth transition different modes The model is obtained by a linearisation of a non linear electro-mechanical model (done by the software): { ẋ(t) = Ax(t) + Bu(t) + Ed(t) y(t) = Cx(t) where x 1 = rotor-speed x 2 = drive-train torsion spring force, x 3 = rotational generator speed u = generator torque, d : wind speed Pole placement -based optimal digital
Homework Let the following quarter car model with active suspension. z s and z us ) are the relative position of the chassis and of the wheel, m s (resp. m us ) the mass of the chassis (resp. of the wheel), k s (resp. k t ) the spring coefficient of the suspension (of the tire), u the active damper force, z r is the road profile. Choose some state variables and give a state space representation of this system Pole placement -based optimal digital
: how to get a linear model from a nonlinear one? The linearisation can be done around an equilibrium point or around a particular point defined by: { ẋeq (t) = f ((x eq (t),u eq (t),t), given x eq (0) (7) y eq (t) = g((x eq (t),u eq (t),t) Defining x = x x eq, ũ = u u eq, ỹ = y y eq this leads to a linear state space representation of the system, around the equilibrium point: { x(t) = A x(t) + Bũ(t), (8) ỹ(t) = C x(t) + Dũ(t) with A = f x x=x eq,u=u eq, B = f u x=x eq,u=u eq, C = g x x=x eq,u=u eq and D = g u x=x eq,u=u eq Usual case Usually an equilibrium point satisfies: 0 = f ((x eq (t),u eq (t),t) (9) Pole placement -based optimal digital For the pendulum, we can choose y = θ = f = 0.
Are state space representations equivalent to transfer? Equivalence transfer function - state space representation Consider a linear system given by: { ẋ(t) = Ax(t) + Bu(t), x(0) = x0 y(t) = Cx(t) + Du(t) Using the Laplace transform (and assuming zero initial condition x 0 = 0), (10) becomes: s.x(s) = Ax(s) + Bu(s) (s.i n A)x(s) = Bu(s) Then the transfer function matrix of system (10) is given by G(s) = C(sI n A) 1 B + D = N(s) D(s) Matlab: if SYS is an SS object, then tf(sys) gives the associated transfer matrix. Equivalent to tf(n,d) (10) (11) Pole placement -based optimal digital
Conversion TF to SS There mainly three cases to be considered Simple numerator y u = G(s) = 1 s 3 + a 1 s 2 + a 2 s + a 3 Numerator order less than denominator order Numerator equal to denominator order y u = G(s) = b 1s 2 + b 2 s + b 3 s 3 + a 1 s 2 = N(s) + a 2 s + a 3 D(s) y u = G(s) = b 0s 3 + b 1 s 2 + b 2 s + b 3 s 3 + a 1 s 2 = N(s) + a 2 s + a 3 D(s) Pole placement -based optimal digital
Canonical forms For the strictly proper transfer function: G(s) = c 0 + c 1 s +... + c n 1 s n 1 a 0 + a 1 s +... + a n 1 s n 1 + s n a very well-known specific state space representations, referred to as the lable canonical form is defined as: 0 1 0... 0 0 0 0 1 0... A =.........., B =. and 0. 0 1 0 a 0 a 1...... a n 1 1 C = [ ] c 0 c 1... c n 1. In Matlab, use canon Pole placement -based optimal digital
What is a canonical form for a physical system? It is worth noting that the following state space representation with 0 1 0... 0 0 0 0 1 0... A =.........., B =. 0. 0 1 0 a 0 a 1...... a n 1 1 C = [ 1 0...... 0 ] does correspond to the Nth-order differential equation d n y dt n + a d n 1 y n 1 dt n 1 +... + a 1ẏ + a 0y = u This indeed can be reformulated into N simultaneous first-order differential equations defining the state variables : x 1 = y,,x 2 = ẏ,,...x n = dn 1 y dt n 1, Pole placement -based optimal digital
How to compute the solution x(t) of a linear system? This theoretical problem is solved now using simulation tools (as Simulink) Case of the autonomous equation ẋ(t) = Ax(t) It is the generalization of the scalar case: if ẏ = αy then y(t) = exp(αt)y 0. The state x(t) with initial condition x(0) = x 0 is then given by x(t) = e At x(0) (12) To get an explicit analytical formula, this requires to compute the function e At, which can be done following one of the 3 methods to compute e At : 1. Inverse Laplace transform of (si n A) 1 2. Diagonalisation of A 3. Cayley-Hamilton method In Matlab : use expm(a*t) and not exp (if t is given). Pole placement -based optimal digital
How to compute the solution x(t) of a linear system? (cont..) General case of system (10) The state x(t), solution of system (10), is given by x(t) = e At x(0) }{{} free response t + 0 e A(t τ) Bu(τ)dτ } {{ } forced response Simulation of state space systems Use lsim. Example: t = 0:0.01:5; u = sin(t); lsim(sys,u,t) (13) Pole placement -based optimal digital
Properties Pole placement -based optimal digital
Non unicity Given a transfer function, there exists an infinity of state space representations (equivalent in terms of input-output behavior). Let { ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) (14) the transfer matrix being G(s) = C(sI n A) 1 B + D, and consider the change of variables x = Tz (T being an invertible matrix). Replacing x = Tz in the previous system gives: T ż(t) = ATz(t) + Bu(t) (15) y(t) = CTz(t) + Du(t) (16) Pole placement Hence ż(t) = T 1 ATz(t) + T 1 Bu(t) (17) y(t) = CTz(t) + Du(t) (18) -based optimal digital
Defining à = T 1 AT, B = T 1 B and C = CT, the transfer function of the previous system is: Using I n = T 1 T, we get G(s) = C(sI n Ã) 1 B + D (19) = C T (si n T 1 AT ) 1 T 1 B + D (20) (21) G(s) = C T T 1 (si n A) 1 T T 1 B + D = G(s) (22) Pole placement Exercise: For the quarter-car model, choose: x 1 = z s, x 2 = ż s, x 3 = z s z us, x 4 = ż s ż us and give the equivalent state space representation. -based optimal digital
Stability 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 An equilibrium point x eq is asymptotically stable if it is stable and, there exists η > 0 such that: x(0) x eq < η = x(t) x eq, when t These notions are equivalent for linear systems (not for non linear ones). Pole placement -based optimal digital
Stability Analysis 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). Pole placement -based optimal digital
The Phase Plane It consists in plotting the trajectory of the state variables (valid also for nonlinear systems). Trajectories that converge to zero are stable! { ẋ1 (t) = x 2 (t) given x ẋ 2 (t) = 5x 1 (t) 6x 2 (t) 1 (0) & x 2 (0) Pole placement -based optimal digital
Pole placement -based optimal digital
Objective of any system In one sentence: shape the response of the system to a given reference and get (or keep) a stable system in closed-loop, with desired performances, while minimising the effects of disturbances and measurement noises, and avoiding actuators saturation, this despite of modelling uncertainties, parameter changes or change of operating point. Steps to be achieved: Nominal stability (NS): The system is stable with the nominal model (no model uncertainty) Nominal Performance (NP): The system satisfies the performance specifications with the nominal model (no model uncertainty) Robust stability (RS): The system is stable for all perturbed plants about the nominal model, up to the worst-case model uncertainty (including the real plant) Robust performance (RP): The system satisfies the performance specifications for all perturbed plants about the nominal model, up to the worst-case model uncertainty (including the real plant). Pole placement -based optimal digital
About Feedback How to design a ler using a state space representation? Two classes of lers do exist (in red those studied in the course): Static lers (output or state feedback) Dynamic lers (output feedback or observer-based) What for? Closed-loop stability (of state or output variables) disturbance rejection Model tracking Input/Output decoupling Other performance criteria : H 2 optimal, H robust... Pole placement -based optimal digital
Why state feedback and not output feedback? Exercise: G(s) = y(s) u(s) = 1 s 2 s Follow the steps below: 1. Define x 1 = y, x 2 = ẏ. Write the differential equations that the state variables (x 1, x 2 ) do satisfy. Deduce the state space system representation, and check that this corresponds to the lable canonical form. 2. Case of output feedback= Proportional : Let us consider u = K p (y ref y) Compute the transfer function of the closed-loop system (with unitary feedback), and check that the closed-loop system poles are those given by the roots of the polynomial P BF (s) = s 2 s + K p. Can the closed-loop system be stabilized (chosen K p well)? 3. Case of state feedback : choose u = x 1 3x 2 + y ref From 1., compute the state space representation of the closed-loop system (replacing u by u = x 1 3x 2 + y ref ). What are the poles of the closed-loop system? Is the closed-loop system stable? Now, consider u = f 1 x 1 f 2 x 2 + y ref. How can we choose (f 1,f 2 ) such that the closed-loop system is stable? 4. To conclude, when the closed-loop system is stable, explain why the second law is efficient? Pole placement -based optimal digital
A preliminary property analysis: refers to the ability of ling a state-space model using state feedback. Given two states x 0 and x 1, the system (10) 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 (10) 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 crtb(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. Pole placement -based optimal digital
of the state feedback A state feedback ler for a continuous-time system is: u(t) = Fx(t) (23) where F is a m n real matrix. When the system is SISO, it corresponds to : u(t) = f 1 x 1 f 2 x 2... f n x n with F = [f 1, f 2,..., f n ]. When the system is MIMO we have u 1 u 2.. u m f 11... f 1n =.. f m1... f mn x 1 x 2.. x n Pole placement -based optimal digital
(2): stabilization Using state feedback lers (23), we get in closed-loop (for simplicity D = 0) { ẋ(t) = (A BF )x(t), y(t) = Cx(t) The stability (and dynamics) of the closed-loop system is then given by the eigenvalues of A BF. Indeed, in that case, the solution y(t) = C exp (A BF )t x 0 converges asymptotically to zero! (24) But what happens if the closed-loop system must also track a reference signal r? We might select u(t) = r(t) Fx(t). Therefore the closed-loop transfer matrix is : y(s) r(s) = C(sI n A + BF ) 1 B (25) for which the static gain is C( A + BF) 1 B and may differ from 1!! The law must be completed. Pole placement -based optimal digital
(3): complete solution for reference tracking When the objective is to track some reference signal r, i.e y(t) t r(t), the state feedback must be of the form: u(t) = Fx(t)+Gr(t) (26) where G is a m p real matrix to be determined. Then the closed-loop transfer matrix is defined as: G CL (s) = C(sI n A + BF) 1 BG (27) Therefore, the following choice for G ensures a unitary steady-state gain for the closed-loop system: Need to adapt when D 0 G = [C( A + BF ) 1 B] 1 (28) G CL (s) = [(C DF)(sI n A + BF) 1 B + D]G Pole placement -based optimal digital
Implementation in Simulink Pole placement -based optimal digital
How to synthetize the state feedback gain F? The pole placement Problem definition Given a linear system (5), does there exist a state feedback law (23) such that the closed-loop system poles are in predefined locations (denoted γ i, i = 1,...,n ) in the complex plane? Proposition Let a linear system given by A, B, and let γ i, i = 1,...,n, a set of complex elements (i.e. the desired poles of the closed-loop system). There exists a state feedback u = Fx such that the poles of the closed-loop system are γ i, i = 1,...,n if and only if the pair (A,B) is lable. In Matlab, use F=acker(A,B,P) or F=place(A,B,P) where P = [γ 1,...,γ n ] is the set of desired closed-loop poles. Remark predefined locations means that, according to the required closed-loop performances (settiling time, rise time, overshoot...), the designer has chosen a set of desired poles for the closed-loop system. Pole placement -based optimal digital
Illustration on the easy case of lable canonical forms Here we assume that the system state space model is of the form: 0 1 0... 0 0 0 0 1 0... A =.........., B =. and 0. 0 1 0 a 0 a 1...... a n 1 1 C = [ ] c 0 c 1... c n 1, corresponding to the transfer function: G(s) = c 0 + c 1 s +... + c n 1 s n 1 a 0 + a 1 s +... + a n 1 s n 1 + s n Pole placement Let F = [ f 1 f 2... f n ] Then 0 1 0... 0 0 0 1 0... A BF =......... 0. 0 1 (29) -based optimal digital a 0 f 1 a 1 f 2...... a n 1 f n
the case of lable canonical forms (cont..) From the specifications of the predefined closed-loop system poles locations, {γ i },i = 1,n., the desired closed-loop characteristic polynomial (denominator of the closed-loop transfer function) is given as: (s γ 1 )(s γ 2 )...(s γ n ) and can be developed as: (s γ 1 )(s γ 2 )...(s γ n ) = s n + α n 1 s n 1 +... + α 1 s + α 0 Therefore, from A BF given before, the chosen solution: f i = a i 1 + α i 1, i = 1,..,n ensures that the poles of A BF are {γ i },i = 1,n. Pole placement Remark the case of lable canonical forms is very important since, when we consider a general state space representation, it is first necessary to use a change of basis to make the system under canonical form, which will simplify a lot the computation of the state feedback gain F. -based optimal digital
How to specificy the desired closed-loop performances? The required closed-loop performances should be chosen in the following zone Pole placement -based optimal which ensures a damping greater than ξ = sinφ. γ implies that the real part of the CL poles are sufficiently negatives (so fast enough). digital
(2) Some useful rules for selection the desired pole/zero locations (for a second order system): Rise time : t r 1.8 ω n Seetling time : t s 4.6 ξ ω n Overshoot M p = exp( πξ /sqrt(1 ξ 2 )): ξ = 0.3 M p = 35%, ξ = 0.5 M p = 16%, ξ = 0.7 M p = 5%. Pole placement -based optimal digital
(3) Some rules do exist to shape the transient response. The ITAE (Integral of Time multiplying the Absolute value of the Error), defined as: ITAE = t e(t) dt 0 can be used to specify a dynamic response with relatively small overshoot and relatively little oscillation (there exist other methods to do so). The optimum coefficients for the ITAE criteria are given below (see Dorf & Bishop 2005). Order Characteristic polynomials d k (s) 1 d 1 = [s + ω n ] 2 d 2 = [s 2 + 1.4ω n s + ωn 2 ] 3 d 3 = [s 3 + 1.75ω n s 2 + 2.15ωn 2 s + ωn 3 ] 4 d 4 = [s 4 + 2.1ω n s 3 + 3.4ωn 2 s 2 + 2.7ωn 3 s + ωn 4 ] 5 d 5 = [s 5 + 2.8ω n s 4 + 5ωn 2 s 3 + 5.5ωn 3 s 2 + 3.4ωn 4 s + ωn 5 ] Pole placement 6 d 6 = [s 6 + 3.25ω n s 5 + 6.6ω 2 n s 4 + 8.6ω 3 n s 3 + 7.45ω 4 n s 2 + 3.95ω 5 n s + ω 6 n ] -based and the corresponding transfer function is of the form: H k (s) = ωk n, k = 1,...,6 d k (s) optimal digital
(4): responses of the optimum ITAE H k (s) k = 1,...,6 Pole placement -based optimal digital
or how to ensure disturbance attenuation with a state feedback? Let us consider the system: { ẋ(t) = Ax(t) + Bu(t)+Ed(t), x(0) = x0 y(t) = Cx(t) where d is the disturbance. Control objectives We wish to keep y following a reference signal r even in the presence of d, which means when d = 0 and r(t) 0 : when r = 0 and d(t) 0 : y(t) r(t), t y(t) 0, t BUT A state feedback ler may not allow to reject the effects of disturbances (particularly of input disturbances)!! (30) Pole placement -based optimal digital
Formulation of the Without integral Let consider the state feedback u(t) = Fx(t) + Gr(t) for the system { ẋ(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0 (31) y(t) = Cx(t) The tracking and disturbance rejection objectives can be formulated as y r 1? i.e. C( A + t BF) 1 BG = 1? y d 0? i.e. C( A + t BF) 1 BE = 0? However, there are few chances to find F and G such that both objectives, together with the pole placement one, are achieved! A solution to solve both problems: add and integral term A very useful method consists in adding an integral term (as usual on the tracking error) to ensure a unitary static closed-loop gain. Therefore the law is chosen as: t u(t) = Fx(t) H (r(τ) y(τ))dτ 0 Now the question is: how to find H? (and F too since a single design procedure is better in order to get a solution) Pole placement -based optimal digital
Synthesis of the The state space method It consists in first extending the system by introducing the new state variable: ż(t) = r(t) y(t) which [ ] leads, for the whole system, to define the extended state vector x. z Then the new open-loop state space representation is given as: [ ẋ(t) ż(t) ] = [ A 0 C 0 ][ x z ] y(t) = [ C 0 ][ x z Let us denote: A e = [ A 0 C 0 ] [ 0 + 1 ] [ B, B e = 0 ] [ B u(t) + 0 ] [ E r(t) + 0 ], C e = [ C 0 ] ] d(t) Pole placement -based optimal digital
The new state feedback is now defined as: [ ] x u(t) = [F H] = Fx(t) Hz(t) z Then the synthesis of the law u(t) (i.e of F e = [F H]) requires: the verification of the extended system lability, i.e of (A e,b e ) the specification of the desired closed-loop performances, i.e. a set P e of n + 1 desired closed-loop poles has to be chosen, the computation of the full state feedback F e using Fe=acker(Ae,Be,Pe) We then get the closed-loop system [ ] [ ][ ẋ(t) A BF BH x = ż(t) C 0 z y(t) = [ C 0 ][ ] x z ] [ 0 + 1 ] [ E r(t) + 0 ] d(t) Pole placement -based optimal digital
Integral scheme The complete structure has the following form: Pole placement When an observer is to be used (see next chapter), the action simply becomes: u(t) = F ˆx(t) Hz(t) -based optimal digital
and output feedback Pole placement -based optimal digital
A first insight To implement a state feedback, the measurement of all the state variables is necessary. If this is not available, we will use a state estimation through a so-called. or Estimation The estimation theory is based on the famous Kalman contribution to filtering problems (1960), and accounts for noise induced problems. The observation theory has been developed for Linear Systems by Luenberger (1971), and doe snot consider the noise effects. Other interest of observation/estimation In practice the use of sensors is often limited for several reasons: feasibility, cost, reliability, maintenance... An observer is a key issue to estimate unknown variables (then non measured variables) and to propose a so-called virtual sensor. Objective: Develop a dynamical system whose state ˆx(t) satisfies: (x(t) ˆx(t)) 0 t (x(t) ˆx(t)) 0 as fast as possible Pole placement -based optimal digital
How to simply (bad) compute x(t)? Let consider Knowing that: { ẋ(t) = Ax(t) + Bu(t), x(0) = x0 y(t) = Cx(t) y(t) = Cx(t) ẏ(t) = CAx(t) + CBu(t) ÿ(t) = CA 2 x(t) + CABu(t) + CB u(t)... =... y n 1 (t) = CA n 1 +... and given that we know the measurement, the inputs (and the system matrices), we can just perform some few computation to compute x(t) as: x(t) = C CA. CA n 1 1 y(t) ÿ(t). y n 1 (t) F (u(t), u(t),...,un 2 (t)) This requires the system to be observable (but still cannot work in practice when faced to measurement noises, modelling errors...) (32) Pole placement -based optimal digital
A preliminary property: refers to the ability to estimate a state variable (often not measured!!). A linear system (5) 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 Proposition The observability matrix is defined by O = C CA. CA n 1. Then system (10) 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. Pole placement -based optimal digital
cont. 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). Where does observability come from? Compare the transfer function of the two different systems* ẋ = x + u y = 2x and ẋ = [ 1 0 0 2 ] [ 1 x + 1 ] u Pole placement y = [ 2 0 ] x Exercices Test the observability of the previous examples: DC motor, suspension, inverted pendulum. Analysis of different cases, according to the considered number of sensors. -based optimal digital
Open loop (OL) observers: estimation from input data Such a method, consists in performing, in real-time (embedded computer), a simulation of the system model feeded by the known input variables. For a linear system, it means that we may define the OL observer as: { ˆx(t) = Aˆx(t) + Bu(t), given ˆx(0) (33) ŷ(t) = Cˆx(t) + Du(t) ˆx(t) R n is the estimated state of x(t). Now, IF ˆx(0) = x(0), then ˆx(t) = x(t), t 0. BUT x(0) is UNKNOWN so we cannot choose ˆx(0) = x(0), the estimation error (e = x ˆx) satisfies ė(t) = Ae(t) (could be unstable AND cannot be modified) the effects of disturbance and noise cannot be mitigated NEED FOR A FEEDBACK FROM MEASURED OUTPUTS TO CORRECT THE ESTIMATION ON LINE! Pole placement -based optimal digital
Closed-loop : estimation from input AND output data Objective: since y is KNOWN (measured) and is function of the state variables, use an on line comparison of the measured system output y and the estimated output ŷ. description: ˆx(t) = Aˆx(t) + Bu(t) + L(y(t) ŷ(t)) }{{} Correction ŷ(t) = Cˆx(t) + Du(t) (34) with ˆx 0 to be defined, and where ˆx(t) R n is the estimated state of x(t) and L is the n p constant observer gain matrix to be designed. Pole placement -based optimal digital
Analysis of the observer properties The estimated error, e(t) := x(t) ˆx(t), satisfies: ė(t) = (A LC)e(t) (35) If L is designed such that A LC is stable, then ˆx(t) converges asymptotically towards x(t). Proposition (34) is an observer for system (5) if and only if the pair (C,A) is observable, i.e. where O = C CA. CA n 1. rank(o) = n Pole placement -based optimal digital
The observer design is restricted to find L such that A LC is stable (so that (x(t) ˆx(t)) 0) and has some desired eigenvalues (so that t (x(t) ˆx(t)) 0 as fast as possible). This is still a pole placement problem. Usually the observer poles are chosen around 5 to 10 times higher than the closed-loop system, so that the state estimation is good as early as possible. This is quite important to avoid that the observer makes the closed-loop system slower. Design method In order to use the acker Matlab function, we will use the duality property between observability and lability, i.e. : (C,A) observable (A T,C T ) lable. Then there exists L T such that the eigenvalues of A T C T L T can be randomly chosen. As (A LC) T = A T C T L T then L exists such that A LC is stable. Matlab : use L=acker(A,C,Po) where Po is the set of desired observer poles. Pole placement -based optimal digital
Theoretical validation scheme using Simulink Written below for D = 0. Pole placement -based optimal digital
About the robustness of the observer Let assume that the systems is indeed given by { ẋ(t) = Ax(t) + Bu(t) + Edx (t), x(0) = x 0 y(t) = Cx(t) + Nν(t) where d x can represent input disturbance or modelling error, and ν stands for measurement noise. Then the estimated error satisfies: (36) ė(t) = (A LC)e(t) + Ed x LNν (37) Therefore the presence of d x or d y may lead to non zero estimation errors due to bias or variations. Then do not forget that you can: Provide an analysis of the observer performances/robustness due to d x or ν (see later) Design optimal observer when d x and ν represent noise effects (Kalman - lqe, see next course ) Pole placement -based optimal digital
Implementation Rules use a state-space block in Simulink enter formal matrices A =A-LC, B =[B L], C = eye(n), D = zeros(n,m)) Choose ˆx(0) x(0), Pole placement -based optimal alternative use of estim digital
-based Pole placement -based optimal digital
-based When an observer is built, we will use as law: u(t) = F ˆx(t) + Gr(t) (38) The closed-loop system is then { ẋ(t) = (A BF )x(t) + BF (x(t) ˆx(t)), y(t) = Cx(t) (39) Therefore the fact that ˆx(0) x(0) will have an impact on the closed-loop system behavior. The stability analysis of the closed-loop system with an observer-based state feedback needs to consider an extended state vector as: x e (t) = [ x(t) e(t) ] T Pole placement -based optimal digital
-based : stability analysis Defining x e (t) = [ x(t) e(t) ] T The closed-loop system with observer (34) and (38) is: [ ] [ ] A BF BF BG ẋ e (t) = x 0 A LC e (t) + r(t) (40) 0 The characteristic polynomial of the extended system is: det(si n A + BF) det(si n A + LC) If the observer and the are designed separately then the closed-loop system with the dynamic measurement feedback is stable, given that the and observer systems are stable and the eigenvalues of (40) can be obtained directly from them. This corresponds to the so-called separation principle. Remark: check pzmap of the extended closed-loop system. Pole placement -based optimal digital
Closed-loop analysis The closed-loop system from r to y is then computed from: y = [C 0] [ x(t) e(t) ] T which leads to y r = C(sI n A + BF )BG However if some disturbance acts as for: { ẋ(t) = Ax(t) + Bu(t) + Ed(t), x(0) = x0 y(t) = Cx(t) (41) Pole placement where d is the disturbance, then the extended system writes [ A BF BF ẋ e (t) = 0 A LC ] [ BG x e (t) + 0 ] [ E r(t) + E ] d(t) (42) which is a problem for the performances of closed-loop system and of the estimation (see later the Integral ). -based optimal digital
How to define the observer+state feedback as a "usual" ler? The observer-based ler is nothing else than a 2-DOF Dynamic Output Feedback ler. Indeed it comes from { ˆx(t) = Aˆx(t) + Bu(t) + L(y(t) ŷ(t)) (43) u(t) = F ˆx(t) + Gr(t) which can be written as (when D = 0) { ˆx(t) = (A BF LC)ˆx(t) + BGr(t) + Ly(t) We then can write: u(t) = F ˆx(t) + Gr(t) U(s) = K r (s)r(s) K y (s)y (s) with K r (s) = G F (si n A + BF + LC) 1 BG and K y (s) = F(sI n A + BF + LC) 1 L (44) Pole placement -based optimal digital
optimal Pole placement -based optimal digital
The objective of an optimal is to minimize a cost function which penalizes simultaneously the state and input behaviors, of the form 0 L(x,y)dt, i.e to reach a tradeoff between the transient response and the effort. This objective is defined through the following criteria always considered in the quadratic form: J = (x T Qx + u T Ru)dt 0 In that form: x T Qx is the state cost, u T Ru is the cost, Q and R are respectively the state and cost penalties. It can be proved that the state feedback that minimizes J in closed-loop (given Q and R) is obtained solving an Algebraic Riccati Equation (ARE) Pole placement -based optimal digital
Linear-Quadratic Regulator (LQR) design LQR problem solution Given a linear system ẋ(t) = Ax(t) + Bu(t), with (A,B) stabilizable, and given positive definite matrices Q = Q T > 0 and R = R T > 0, if there exists P = P T > 0 s.t: A T P + PA PBR 1 B T P + Q = 0 then the state feedback u = Kx such that: K = R 1 B T P minimizes the quadractic criteria J (for given Q and R). This problem is handled in Matlab through the lqr command. Pole placement -based optimal digital
digital Pole placement -based optimal digital
Toward digital Digital Usually lers are implemented in a digital computer as: Pole placement This requires the use of the discrete theory. (Sampling theory + Z-Transform) -based optimal digital
of the Z-Transform Mathematical definition Because the output of the ideal sampler, x (t), is a series of impulses with values x(kt e ), we have: by using the Laplace transform, x (t) = x(kt e )δ(t kt e ) k=0 L [x (t)] = x(kt e )e kst e k=0 Noting z = e st e, we can derive the so called Z-Transform X(z) = Z [x(k)] = x(k)z k k=0 Pole placement -based optimal digital
Properties Properties X(z) = Z [x(k)] = x(k)z k k=0 Z [αx(k) + βy(k)] = αx(z) + βy (z) Z [x(k n)] = z n Z [x(k)] Z [kx(k)] = z d dz Z [x(k)] Z [x(k) y(k)] = X(z).Y (z) lim x(k) = lim (z 1)X(z) k 1 z 1 The z 1 can be interpreted as a pure delay operator. Pole placement -based optimal digital
Zero order holder Sampler and Zero order holder A sampler is a switch that close every T e seconds. A Zero order holder holds the signal x for T e seconds to get h as: h(t + kt e ) = x(kt e ), 0 t < T e Pole placement -based optimal digital
Zero order holder (cont d) Model of the Zero order holder The transfer function of the zero-order holder is given by: G BOZ (s) = 1 s e st e s = 1 e st e s Influence of the D/A and A/D Note that the precision is also limited by the available precision of the converters (either A/D or D/A). This error is also called the amplitude quantization error. Pole placement -based optimal digital
Representation of the discrete linear systems The discrete output of a system can be expressed as: y(k) = h(k n)u(n) n=0 hence, applying the Z-transform leads to Y (z) = Z [h(k)]u(z) = H(z)U(z) H(z) = b 0 + b 1 z + + b m z m a 0 + a 1 z + + a n z n where n ( m) is the order of the system Corresponding difference equation: = Y U y(k) = 1 a n [ b0 u(k n) + b 1 u(k n + 1) + + b m u(k n + m) a 0 y(k n) a 2 y(k n + 1) a n 1 y(k 1) ] Pole placement -based optimal digital
Some useful transformations x(t) X(s) X(z) δ(t) 1 1 δ(t kt e ) e kste z k u(t) 1 s z 1 t s 2 e at s+a 1 1 e at 1 s(s+a) ω sin(ωt) cos(ωt) z 1 zt e (z 1) 2 z z e ate z(1 e ate ) (z 1)(z e ate ) zsin(ωt e) s 2 +ω 2 z 2 2zcos(ωT e)+1 s z(z cos(ωt e)) s 2 +ω 2 z 2 2zcos(ωT e)+1 Pole placement -based optimal digital
Poles, Zeros and Stability Equivalence {s} {z} The equivalence between the Laplace domain and the Z domain is obtained by the following transformation: z = e st e Two poles with a imaginary part witch differs of 2π/T e give the same pole in Z. Stability domain Pole placement -based optimal digital
Approximations for discretization Forward difference (Rectangle inferior) s = z 1 T e Backward difference (Rectangle superior) Pole placement s = z 1 zt e -based optimal digital
Approximations for discretization (cont d) Trapezoidal difference (Tustin) s = 2 T e z 1 z + 1 Pole placement -based optimal digital
Systems definition A discrete-time 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) where h is the sampling period. Matlab : ss(a d,b d,c d,d d,h) creates a SS object SYS representing a discrete-time state-space model From a discretization step (c2d) we have: h A d = exp(ah), B d = ( exp(aτ)dτ)b 0 For discrete-time systems, { 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) the discrete transfer function is given by (45) (46) G(z) = C d (zi n A d ) 1 B d + D d (47) where z is the shift operator, i.e. zx(kh) = x((k + 1)h) Pole placement -based optimal digital
Solution of state space equations - discrete case The state x k, solution of system x k+1 = A d x k with initial condition x 0, is given by The state x k, solution of system (45), is given by x 1 = A d x 0 (48) x 2 = A 2 d x 0 (49) x n = A n d x 0 (50) x 1 = A d x 0 + B d u 0 (51) x 2 = A 2 d x 0 + A d B d u 0 + B d u 1 (52) x n = A n n 1 d x 0 + A n 1 i d B d u i (53) i=0 Pole placement -based optimal digital
analysis (discrete-time systems) Stability A system (state space representation) is stable iff all the eigenvalues of the matrix F are inside the unit circle. definition Given two states x 0 and x 1, the system (45) is lable if there exist K 1 > 0 and a sequence of samples u 0,u 1,...,u K1, such that x k takes the values x 0 for k = 0 and x 1 for k = K 1. definition The system (45) 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. Pole placement -based optimal digital
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 Pole placement Duality of (C d,a d ) of (A T d,ct d ). of (A d,b d ) of (B T d,at d ). -based optimal digital
About sampling period Influence of the sampling period on the time response Pole placement Impose a maximal time response to a discrete system is equivalent to place the poles inside a circle defined by the upper bound of the bound given by this time response. The closer to zero the poles are, the faster the system is. -based optimal digital
Frequency analysis As in the continuous time, the Bode diagram can also be used. Example with sampling Time T e = 1s f e = 1Hz w e = 2π): Pole placement Note that, in our case, the Bode is cut at the pulse w = π. see SYSD = c2d(sysc,ts,method) in MATLAB. -based optimal digital
Frequency analysis As in the continuous time, the Bode diagram can also be used. Example with sampling Time T e = 1s f e = 1Hz w e = 2π): Pole placement Note that, in our case, the Bode is cut at the pulse w = π. see SYSD = c2d(sysc,ts,method) in MATLAB. Sampling Limitations Recall the Shannon theorem which imposes the sampling frequency at least 2 times higher than the system maximum frequency. Related to the anti-aliasing filter... -based optimal digital
About sampling period and robustness Influence of the sampling period on the poles In theory, smaller the sampling period T e is, closer the discrete system is from the continuous one. Pole placement But reducing the sampling time modify poles location... Poles and zeros become closer to the limit of the unit circle can introduce instability (decrease robustness). Sampling influences stability and robustness Over sampling increase noise sensitivity -based optimal digital
Stability Recall A linear continuous feedback system is stable if all poles of the closed-loop transfer function T (s) lie in the left half s-plane. The Z-plane is related to the S-plane by z = e st e = e (σ+jω)t e. Hence z = e σt e and z = ωt e Jury criteria The denominator polynomial (den(z) = a 0 z n + a 1 z n 1 + + a n = 0) has all its roots inside the unit circle if all the first coefficients of the odd row are positive. 1 a 0 a 1 a 2... a n k... a n 2 a n a n 1 a n 2... a k... a 0 3 b 0 b 1 b 2... b n 1 2 b n 1 b n 2 b n 3... b 0.. 2n + 1 s 0 b 0 = a 0 a n a n a 0 b 1 = a 1 a n 1 a n b k = a k a n k a n Pole placement -based a 0 optimal a 0 digital c k = b k b n 1 k b n 1 b 0
How to get a discrete ler First way Obtain a discrete-time plant model (by discretization) Design a discrete-time ler Derive the difference equation Second way Design a continuous-time ler Converse the continuous-time ler to discrete time (c2d) Derive the difference equation Now the question is how to implement the computed ler on a real-time (embedded) system, and what are the precautions to take before? Pole placement -based optimal digital
Discretisation The idea behind discretisation of a ler is to translate it from continuous-time to discrete-time, i.e. A/D + algorithm + D/A G(s) To obtain this, few methods exists that the Laplace operator (see lecture 1-2). Recall Pole placement s = z 1 T e s = z 1 zt e s = 2 T e z 1 z + 1 -based optimal digital
Implementation characteristics Anti-aliasing Practically it is smart to use a constant high sampling frequency with an analog filter matching this frequency. Then, after the A/D converter, the signal is down-sampled to the frequency used by the ler. Remember that the pre-filter introduce phase shift. Sampling frequency choice The sampling time for discrete-time are based on the desired speed of the closed loop system. A rule of thumb is that one should sample 4 10 times per rise time T r of the closed loop system. N sample = T r T e 4 10 where T e is the sampling period, and N sample the number of samples. Pole placement -based optimal digital
Delay Problematic Sampled theory assume presence of clock that synchronizes all measurements and signal. Hence in a computer based there always is delays ( delay, computational delay, I/O latency). Origins There are several reasons for delay apparition Execution time (code) Preemption from higher order process Interrupt Communication delay Data dependencies Hence the delay is not constant. The delay introduce a phase shift Instability! Pole placement -based optimal digital
Delay (cont d) Admissible delay (Bode) Measure the phase margin: PM = 180 + ϕ w0 [ř], where ϕ w0 is the phase at the crossover frequency w 0, i.e. G(jw 0 ) = 1 Then the delay margin is DM = 180w PMπ 0 [s] Exercise: compute delay margin for these 3 cases Pole placement -based optimal digital
A state space for continuous-time and discrete-time MIMO systems A first insight in optimal... that can be extended towards predictive (over a finite horizon) The state space is also considered in Robust, in order to design H lers provide a robustness analysis in the presence of parameter uncertainties prove the stability of a closed-loop system in the presence of non linearities (as state or input constraints) design non linear lers (feedback linearisation...) solve an optimisation problem using efficient numerical tools as Linear Matrix Inequalities Pole placement -based optimal digital