: state space and polynomial approaches Lecture 2 : a state O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr www.gipsa-lab.fr/ o.sename -based November 21, 2017
Outline : a state : a state -based -based
About Feedback How to design a ler using a state space representation? Tow cases are possible : 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... : a state -based
: a state -based
State space representations Let consider continuous-time linear state space system given bt : { 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. (1) : a state -based
Why state feedback and not output feedback? Example: G(s) = y(s) u(s) = 1 s 2 s 1. Give the lable canonical form considering x 1 = y, x 2 = ẏ. 2. Case of output feedback= Proportional : u = K p y Compute the closed-loop transfer function and check that the closed-loop poles are given by the roots of the characteristic polynomial P BF (s) = s 2 s + K p. Can the closed-loop system be stabilized? 3. Case of state feedback : consider the law u = x 1 3x 2 + y ref Compute the state space representation of the closed-loop system. What are the poles of the closed-loop systems? Is it stable? If yes why this second solves the problem? : a state -based
Definition A state feedback ler for a continuous-time system is: where F is a m n real matrix. u(t) = Fx(t) (2) 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 : a state -based
Definition of the state feedback A state feedback ler for a continuous-time system is: u(t) = Fx(t) (3) : a state 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 -based
(2): stabilization Using state feedback lers (6), we get in closed-loop (for simplicity D = 0) { ẋ(t) = (A BF )x(t), (4) y(t) = Cx(t) The stability (and dynamics) of the closed-loop system is then given by the eigenvalues of A BF. Then the solution y(t) = C exp (A BF )t x 0 converges asymptotically to zero! Discrete-time systems Using u(k) = F d x(k) we get { x(k + 1) = (Ad B d F d )x(k), y(k) = C d x(k) (5) : a state -based Remark For both cases, the good choice of F (or F d ) may allow to stabilize the closed-loop system. But what happens if the closed-loop system must track a reference signal r?
(3): reference tracking Objective: y should track some reference signal r, i.e. y(t) r(t) t When the objective is to track some reference signal r, why not select: u(t) = r(t) Fx(t) (6) or u(k) = r(k) F d x(k) for discrete-time systems (7) : a state -based Can we ensure that y tracks the reference signal r, i.e y(t) t r(t)? No since, the closed-loop transfer matrix is : y(s) r(s) = C(sI n A + BF ) 1 B (8) for which the static gain is C( A + BF) 1 B and may differ from 1! (see examples)
(4): complete solution for reference tracking When the objective is to track some reference signal r, the state feedback can be selected as: u(t) = Fx(t)+Gr(t) (9) G is a m p real matrix. Then the closed-loop transfer matrix is : G CL (s) = C(sI n A + BF ) 1 BG (10) G is chosen to ensure a unitary steady-state gain as: G = [C( A + BF ) 1 B] 1 (11) : a state -based When D 0 G CL (s) = [(C DF)(sI n A + BF) 1 B + D]G Discrete-time systems (with D d =) u(k) = F d x(k) + G d r(k) (12) with G d = [C(I n A d + B d F d ) 1 B d ] 1 (13)
Implementation in Simulink : a state -based
: a state : a state -based
Problem definition Given a linear system (1), does there exist a state feedback law (6) such that the closed-loop 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. : a state -based
(2): case of the lable canonical form Let assume that the system G(s) = c 0+c 1 s+...+c n 1 s n 1 is given by a 0 +a 1 s+...+a n 1 s n 1 +s n 0 1 0... 0 0 0 0 1 0... A =.........., B =. and 0. 0 1 0 C = [ a 0 a 1...... a n 1 1 ] c 0 c 1... c n 1. Let F = [ f 1 f 2... f n ] Then 0 1 0... 0 0 0 1 0... A BF =......... 0. 0 1 (14) a 0 f 1 a 1 f 2...... a n 1 f n : a state -based
(3) From the specifications the desired closed-loop polynomial (s γ 1 )(s γ 2 )...(s γ n ) 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 Threfore 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. 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 (see next slide). Matlab: use F=place(A,B,P) where P is the set of desired closed-loop poles (old version F=acker(A,B,P) ) : a state -based
(4) Procedure for the general case: 1. Check lability of (A, B) 2. Calculate C = [B,AB,...,A n 1 B]. q 1 Note C 1 =... Define T = q n q n q n A. q n A n 1 1 3. Note Ā = T 1 AT and B = T 1 B (which are under the lable canonical form) 4. Choose the desired closed-loop poles and define the desired closed-loop characteristic polynomial: s n + α n 1 s n 1 +... + α 1 s + α 0 5. Calculate the state feedback u = F x with: : a state -based fi = a i 1 + α i 1,i = 1,..,n 6. Calculate (for the original system): u = Fx, with F = F T 1
: a state : continuous-time case -based
: what should be the closed-loop poles? The required closed-loop performances should be chosen in the following zone : a state -based which ensures a damping greater than ξ = sinφ. γ implies that the real part of the CL poles are sufficiently negatives.
(2) : a state 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%. -based
(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) : a state -based 1 d 1 = [s + ω n ] 2 d 2 = [s 2 + 1.4ω n s + ω 2 n ] 3 d 3 = [s 3 + 1.75ω n s 2 + 2.15ω 2 n s + ω 3 n ] 4 d 4 = [s 4 + 2.1ω n s 3 + 3.4ω 2 n s 2 + 2.7ω 3 n s + ω 4 n ] 5 d 5 = [s 5 + 2.8ω n s 4 + 5ω 2 n s 3 + 5.5ω 3 n s 2 + 3.4ω 4 n s + ω 5 n ] 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 ] and the corresponding transfer function is of the form: H k (s) = ωk n, k = 1,...,6 d k (s)
(4) 1.2 1 H1 H2 H3 H4 H5 H6 : a state 0.8 0.6 STEP RESPONSE OF TRANSFER FUNCTIONS WITH ITAE CHARACTERISTIC POLYNOMIALS -based 0.4 0.2 0 0 2 4 6 8 10 12 14 16 NORMALIZED TIME ω n t
: a state -based
Introduction 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. Observation 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. : a state -based 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
Open loop observers: a first approach to estimation from input data Let consider { ẋ(t) = Ax(t) + Bu(t), x(0) = x0 y(t) = Cx(t) + Du(t) Given that we know the plant matrices and the inputs, we can just perform a simulation that runs in parallel with the system { ˆx(t) = Ax(t) + Bu(t), given ˆx(0) ŷ(t) = Cˆx(t) + Du(t) (15) (16) : a state -based Therefore, if we would have 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) dynamics is determined by A, i.e satisfies ė(t) = Ae(t) (could be unstable AND cannot be modified) the effects of disturbance and noise cannot be attenuated (leads to estimation biais) NEED FOR A FEEDBACK FROM MEASURED OUTPUTS TO CORRECT THE ESTIMATION!
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)) }{{} ˆx 0 to be defined Correction 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. (17) : a state -based
Analysis of the observer properties The estimated error, e(t) := x(t) ˆx(t), satisfies: ė(t) = (A LC)e(t) (18) If L is designed such that A LC is stable, then ˆx(t) converges asymptotically towards x(t). Proposition (17) is an observer for system (1) if and only if the pair (C,A) is observable, i.e. : a state -based where O = C CA. CA n 1. rank(o) = n
design The observer design is restricted to find L such that A LC is stable (ensuring that (x(t) ˆx(t)) 0), and has some desired eigenvalues t (ensuring that (x(t) ˆx(t)) 0 as fast as possible). This is still a pole placement problem. Select the observer poles according to the systems closed-loop dynamical behavior (see later). Design method : a state -based In order to use the acker or place Matlab functions, 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.
Theoretical validation scheme using Simulink : a state -based
How to choose the observer poles? First This is quite important to avoid that the observer makes the closed-loop system slower. So the observer should be faster than you intend to make the regulator. Second Increasing the observe gain is actually possible since there is no saturation problem. However the measured outputs are often noisy. Trade-off between high bandwidth observers (very efficient for estimation but noise sensitive) and low bandwidth ones (less than noise sensitive but slower) : a state -based Rule of thumb 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.
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) + Dd y (t) where d x can represent input disturbance or modelling error, and d y stands for output disturbance or measurement noise. Then the estimated error satisfies: (19) ė(t) = (A LC)e(t) + Ed x LDd y (20) : a state -based 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 d y (see later) Design optimal observer when d x and d y represent noise effects (Kalman - lqe, see next course ) Design robust observer suing H approach (see next year)
Practical 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), : a state -based alternative use of estim
: a state -based -based
-based When an observer is built, we will use as law: u(t) = F ˆx(t) + Gr(t) (21) The closed-loop system is then (considering here D = 0) { ẋ(t) = (A BF )x(t) + BF (x(t) ˆx(t)), y(t) = Cx(t) (22) 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 : a state -based
-based : stability analysis Defining x e (t) = [ x(t) e(t) ] T The closed-loop system with observer (17) and (21) is: [ ] [ ] A BF BF BG ẋ e (t) = x 0 A LC e (t) + r(t) (23) 0 The characteristic polynomial of the extended system is: det(si n A + BF) det(si n A + LC) : a state -based 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 (23) can be obtained directly from them. This corresponds to the so-called separation principle. Remark: check pzmap of the extended closed-loop system.
Closed-loop analysis The closed-loop system from r to y is then computed from: y = [C 0] [ x(t) e(t) ] T : a state 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) where d is the disturbance, then the extended system writes (24) -based [ A BF BF ẋ e (t) = 0 A LC ] [ BG x e (t) + 0 ] [ E r(t) + E ] d(t) (25) which is a problem for the performances of closed-loop system and of the estimation (see later the Integral ).
The 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(Cˆx(t) y(t)) (26) u(t) = F ˆx(t) + Gr(t) which can be written as { ˆx(t) = (A BF LC)ˆx(t) + BGr(t) + Ly(t) u(t) = F ˆx(t) + Gr(t) (27) : a state -based We then can write: 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 K y (s) = F(sI n A + BF + LC) 1 L and the analysis can be done through the sensitivity functions.
: a state -based
or how to ensure disturbance attenuation with a state feedback? Preliminary remrak: a state feedback ler may not allow to reject the effects of disturbances (particularly of input disturbances). Let us consider the system: { ẋ(t) = Ax(t) + Bu(t)+Ed(t), x(0) = x0 (28) y(t) = Cx(t) where d is the disturbance. The objective is to keep y following a reference signal r even in the presence of d, i.e y r 1 t y d 0 t : a state -based
Formulation of the Introduction 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 to choose t u(t) = Fx(t) H (r(τ) y(τ))dτ 0 But the question is: how to find H? The state space method It consists in extending the system by adding a new state variable: : a state -based ż(t) = r(t) y(t) [ ] x which leads to define the extended state vector. 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 ] [ 0 + 1 ] [ B u(t) + 0 ] [ E r(t) + 0 ] d(t)
Synthesis of the Let us define: [ ] [ A 0 B A e =, B C 0 e = 0 The new state feedback is now of the form [ ] x u(t) = F e z ], C e = [ C 0 ] = Fx(t) Hz(t) denoting F e = [F H] Then the synthesis of the law u(t) requires: the verification of the extended system lability 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) : a state -based
Integral scheme The complete structure has the following form: r + 1 - s z H - u Plant F x (Or xˆ in case of an observer-based ) - y : a state -based When an observer is to be used, the action simply becomes: u(t) = F ˆx(t) Hz(t)
Integral scheme : a state Compute the closed-loop system representation and check that: the closed-loop system has a unitary gain the effect of the disturbance d in steady-state is nul -based
: a state -based
Some extension: reduced-order observers : a state In practice, since some output variables are measured it may be non necessary to get an estimation of all the state variables. For instance, if : [ ] x1 (t) x(t) = x 2 (t) and y(t) = x 1 (t) is measured, then it possible to estimate only x 2 (t), which is referred to as a reduced-order observer (order = n p). N.B: the use of a full-order observer also induces a filter effect on the measurements. -based
Performance analysis As seen in (27) the general form of a Dynamic Output Feedback ler is 2-DOF state space representation as: ] with ẋ K (t) = A K x K (t) + B K [ r(t) y(t) u(t) = C K x K (t) + D K [ r(t) y(t) [ BG A K = A BF LC, B K = L ] (29) ] [ G, C K = F, D K = 0 ] : a state -based This allows to compute the well known sensitivity functions since, from K (s) = D K + C K (SI n A K ) 1 B K, we can write: [ ] R(s) U(s) = K (s) := K Y (s) r (s)r(s) K y (s)y (s)
Performance analysis (SISO) The general scheme is of the form : a state -based The closed-loop system satisfies the equations y = 1 1+G(s)K y (s) (GK r r + d y GK y n + Gd i ) u = 1 1+K y (s)g(s) (K r r K y d y K y n K y Gd i ) Defining the Sensitivity function: S(s) = 1 1 + G(s)K y (s) The performance sensitivity functions are then y r = GK r y y y 1+GK y = SGK r d i = SG d y = S n = T u u u r = K r S di = T y dy = K y S u n = K y S
Performance analysis (MIMO) The closed-loop system satisfies the equations Defining (I p + G(s)K y (s))y(s) = (GK r r + d y GK r n + Gd i ) (I m + K y (s)g(s))u(s) = (K r r K y d y K y n K y Gd i ) Output and Output complementary sensitivity functions: : a state -based S y = (I p + GK y ) 1, T y = (I p + GK y ) 1 GK y, S y +T y = I p Input and Input complementary sensitivity functions: S u = (I m + K y G) 1, T u = K y G(I m + K y G) 1,S u +T u = I m and the performance sensitivity functions are then y r = S y y y y GK r d i = S y G d y = S y n = T y u u r = S u K r di = S u K y G d u u y = S u K y n = S u K y