Pole placement control: state space and polynomial approaches Lecture 2


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1 : state space and polynomial approaches Lecture 2 : a state O. Sename 1 1 Gipsalab, CNRSINPG, FRANCE o.sename based November 21, 2017
2 Outline : a state : a state based based
3 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 observerbased) What for? Closedloop stability (of state or output variables) disturbance rejection Model tracking Input/Output decoupling Other performance criteria : H 2 optimal, H robust... : a state based
4 : a state based
5 State space representations Let consider continuoustime 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
6 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 closedloop transfer function and check that the closedloop poles are given by the roots of the characteristic polynomial P BF (s) = s 2 s + K p. Can the closedloop 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 closedloop system. What are the poles of the closedloop systems? Is it stable? If yes why this second solves the problem? : a state based
7 Definition A state feedback ler for a continuoustime 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 f 1n =.. f m1... f mn x 1 x 2.. x n : a state based
8 Definition of the state feedback A state feedback ler for a continuoustime 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 f 1n =.. f m1... f mn x 1 x 2.. x n based
9 (2): stabilization Using state feedback lers (6), we get in closedloop (for simplicity D = 0) { ẋ(t) = (A BF )x(t), (4) y(t) = Cx(t) The stability (and dynamics) of the closedloop 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! Discretetime 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 closedloop system. But what happens if the closedloop system must track a reference signal r?
10 (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 discretetime 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 closedloop 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)
11 (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 closedloop transfer matrix is : G CL (s) = C(sI n A + BF ) 1 BG (10) G is chosen to ensure a unitary steadystate 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 Discretetime 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)
12 Implementation in Simulink : a state based
13 : a state : a state based
14 Problem definition Given a linear system (1), does there exist a state feedback law (6) such that the closedloop 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 closedloop system). There exists a state feedback u = Fx such that the poles of the closedloop system are γ i, i = 1,...,n if and only if the pair (A,B) is lable. : a state based
15 (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 A = , B =. and C = [ a 0 a a n 1 1 ] c 0 c 1... c n 1. Let F = [ f 1 f 2... f n ] Then A BF = (14) a 0 f 1 a 1 f a n 1 f n : a state based
16 (3) From the specifications the desired closedloop 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 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 closedloop poles (old version F=acker(A,B,P) ) : a state based
17 (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 Note Ā = T 1 AT and B = T 1 B (which are under the lable canonical form) 4. Choose the desired closedloop poles and define the desired closedloop characteristic polynomial: s n + α n 1 s n 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
18 : a state : continuoustime case based
19 : what should be the closedloop poles? The required closedloop 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.
20 (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
21 (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 ω n s + ω 2 n ] 3 d 3 = [s ω n s ω 2 n s + ω 3 n ] 4 d 4 = [s ω n s ω 2 n s ω 3 n s + ω 4 n ] 5 d 5 = [s ω n s 4 + 5ω 2 n s ω 3 n s ω 4 n s + ω 5 n ] 6 d 6 = [s ω n s ω 2 n s ω 3 n s ω 4 n s ω 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)
22 (4) H1 H2 H3 H4 H5 H6 : a state STEP RESPONSE OF TRANSFER FUNCTIONS WITH ITAE CHARACTERISTIC POLYNOMIALS based NORMALIZED TIME ω n t
23 : a state based
24 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 socalled. 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 socalled 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
25 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!
26 Closedloop : 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
27 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
28 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 closedloop 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.
29 Theoretical validation scheme using Simulink : a state based
30 How to choose the observer poles? First This is quite important to avoid that the observer makes the closedloop 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. Tradeoff 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 closedloop system, so that the state estimation is good as early as possible.
31 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)
32 Practical implementation Rules use a statespace block in Simulink enter formal matrices A =ALC, B =[B L], C = eye(n), D = zeros(n,m)) Choose ˆx(0) x(0), : a state based alternative use of estim
33 : a state based based
34 based When an observer is built, we will use as law: u(t) = F ˆx(t) + Gr(t) (21) The closedloop 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 closedloop system behavior. The stability analysis of the closedloop system with an observerbased state feedback needs to consider an extended state vector as: x e (t) = [ x(t) e(t) ] T : a state based
35 based : stability analysis Defining x e (t) = [ x(t) e(t) ] T The closedloop 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 closedloop 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 socalled separation principle. Remark: check pzmap of the extended closedloop system.
36 Closedloop analysis The closedloop 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 closedloop system and of the estimation (see later the Integral ).
37 The ler The observerbased ler is nothing else than a 2DOF 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.
38 : a state based
39 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
40 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 closedloop 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 openloop state space representation is given as: [ ẋ(t) ż(t) ] = [ A 0 C 0 ][ x z ] y(t) = [ C 0 ][ x z ] [ ] [ B u(t) + 0 ] [ E r(t) + 0 ] d(t)
41 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 closedloop performances, i.e. a set P e of n + 1 desired closedloop poles has to be chosen, the computation of the full state feedback F e using Fe=acker(Ae,Be,Pe) We then get the closedloop system [ ] [ ][ ẋ(t) A BF BH x = ż(t) C 0 z y(t) = [ C 0 ][ ] x z ] [ ] [ E r(t) + 0 ] d(t) : a state based
42 Integral scheme The complete structure has the following form: r s z H  u Plant F x (Or xˆ in case of an observerbased )  y : a state based When an observer is to be used, the action simply becomes: u(t) = F ˆx(t) Hz(t)
43 Integral scheme : a state Compute the closedloop system representation and check that: the closedloop system has a unitary gain the effect of the disturbance d in steadystate is nul based
44 : a state based
45 Some extension: reducedorder 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 reducedorder observer (order = n p). N.B: the use of a fullorder observer also induces a filter effect on the measurements. based
46 Performance analysis As seen in (27) the general form of a Dynamic Output Feedback ler is 2DOF 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)
47 Performance analysis (SISO) The general scheme is of the form : a state based The closedloop 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) = 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
48 Performance analysis (MIMO) The closedloop 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
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