# Pole placement control: state space and polynomial approaches Lecture 2

Size: px
Start display at page:

## Transcription

1 : state space and polynomial approaches Lecture 2 : a state O. Sename 1 1 Gipsa-lab, CNRS-INPG, 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 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

4 : a state -based

5 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

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 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

7 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 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 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 f 1n =.. f m1... f mn x 1 x 2.. x n -based

9 (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?

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 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)

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 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)

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 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

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 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 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

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 closed-loop poles and define the desired closed-loop 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 : continuous-time case -based

19 : 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.

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 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

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 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

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 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.

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 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.

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 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

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 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

35 -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.

36 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 ).

37 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.

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 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 ] [ ] [ 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 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 ] [ ] [ 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 observer-based ) - 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 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

44 : a state -based

45 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

46 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)

47 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) = 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 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

### Modelling, analysis and control of linear systems using state space representations

, 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

### Methods for analysis and control of dynamical systems Lecture 4: The root locus design method

Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.gipsa-lab.fr/ o.sename 5th February 2015 Outline

### Linear State Feedback Controller Design

Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University

### Control Systems Design

ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:

### Pole placement control: state space and polynomial approaches Lecture 1

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

### CONTROL DESIGN FOR SET POINT TRACKING

Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded

### Topic # Feedback Control Systems

Topic #15 16.31 Feedback Control Systems State-Space Systems Open-loop Estimators Closed-loop Estimators Observer Theory (no noise) Luenberger IEEE TAC Vol 16, No. 6, pp. 596 602, December 1971. Estimation

### Modelling, analysis and control of linear systems using state space representations

Modelling, analysis and of linear systems using state space O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr www.gipsa-lab.fr/ o.sename 12th January 2015 feedback optimal digital

### 5. Observer-based Controller Design

EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1

### Topic # Feedback Control

Topic #5 6.3 Feedback Control State-Space Systems Full-state Feedback Control How do we change the poles of the state-space system? Or,evenifwecanchangethepolelocations. Where do we put the poles? Linear

### State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state

### Methods for analysis and control of. Lecture 4: The root locus design method

Methods for analysis and control of Lecture 4: The root locus design method O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename Lead Lag 17th March

### Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.

### Pole placement control: state space and polynomial approaches Lecture 3

: state space and polynomial es Lecture 3 O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.fr www.gipsa-lab.fr/ o.sename 10th January 2014 Outline The classical structure Towards 2

### Topic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback

Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall

### Control Systems Design

ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output

### Module 08 Observability and State Estimator Design of Dynamical LTI Systems

Module 08 Observability and State Estimator Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha November

### Control System Design

ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science

### Topic # Feedback Control Systems

Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the

### Modern Control Systems

Modern Control Systems Matthew M. Peet Arizona State University Lecture 09: Observability Observability For Static Full-State Feedback, we assume knowledge of the Full-State. In reality, we only have measurements

### ME 132, Fall 2015, Quiz # 2

ME 132, Fall 2015, Quiz # 2 # 1 # 2 # 3 # 4 # 5 # 6 Total NAME 14 10 8 6 14 8 60 Rules: 1. 2 sheets of notes allowed, 8.5 11 inches. Both sides can be used. 2. Calculator is allowed. Keep it in plain view

### Intro. Computer Control Systems: F9

Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se

### Controls Problems for Qualifying Exam - Spring 2014

Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function

### Integral action in state feedback control

Automatic Control 1 in state feedback control Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 21-211 1 /

### D(s) G(s) A control system design definition

R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure

### Methods for analysis and control of. Lecture 6: Introduction to digital control

Methods for analysis and of Lecture 6: to digital O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename 6th May 2009 Outline Some interesting books:

### Module 03 Linear Systems Theory: Necessary Background

Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September

### Comparison of four state observer design algorithms for MIMO system

Archives of Control Sciences Volume 23(LIX), 2013 No. 2, pages 131 144 Comparison of four state observer design algorithms for MIMO system VINODH KUMAR. E, JOVITHA JEROME and S. AYYAPPAN A state observer

### Modeling and Analysis of Dynamic Systems

Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability

### Synthesis via State Space Methods

Chapter 18 Synthesis via State Space Methods Here, we will give a state space interpretation to many of the results described earlier. In a sense, this will duplicate the earlier work. Our reason for doing

### 6 OUTPUT FEEDBACK DESIGN

6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple

### ECE 388 Automatic Control

Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:

### Topic # Feedback Control Systems

Topic #1 16.31 Feedback Control Systems Motivation Basic Linear System Response Fall 2007 16.31 1 1 16.31: Introduction r(t) e(t) d(t) y(t) G c (s) G(s) u(t) Goal: Design a controller G c (s) so that the

### Richiami di Controlli Automatici

Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici

### Control Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC426 SC426 Fall 2, dr A Abate, DCSC, TU Delft Lecture 5 Controllable Canonical and Observable Canonical Forms Stabilization by State Feedback State Estimation, Observer Design

### Control Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich

Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:

### Computer Problem 1: SIE Guidance, Navigation, and Control

Computer Problem 1: SIE 39 - Guidance, Navigation, and Control Roger Skjetne March 12, 23 1 Problem 1 (DSRV) We have the model: m Zẇ Z q ẇ Mẇ I y M q q + ẋ U cos θ + w sin θ ż U sin θ + w cos θ θ q Zw

### sc Control Systems Design Q.1, Sem.1, Ac. Yr. 2010/11

sc46 - Control Systems Design Q Sem Ac Yr / Mock Exam originally given November 5 9 Notes: Please be reminded that only an A4 paper with formulas may be used during the exam no other material is to be

### Topic # Feedback Control Systems

Topic #20 16.31 Feedback Control Systems Closed-loop system analysis Bounded Gain Theorem Robust Stability Fall 2007 16.31 20 1 SISO Performance Objectives Basic setup: d i d o r u y G c (s) G(s) n control

### Dr Ian R. Manchester

Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign

### Time Response Analysis (Part II)

Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary

### Automatic Control II Computer exercise 3. LQG Design

Uppsala University Information Technology Systems and Control HN,FS,KN 2000-10 Last revised by HR August 16, 2017 Automatic Control II Computer exercise 3 LQG Design Preparations: Read Chapters 5 and 9

### Time Response of Systems

Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =

### Exam. 135 minutes, 15 minutes reading time

Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.

### Topic # Feedback Control Systems

Topic #16 16.31 Feedback Control Systems State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Fall 2007 16.31 16 1 Combined Estimators and Regulators

### Multivariable Control. Lecture 03. Description of Linear Time Invariant Systems. John T. Wen. September 7, 2006

Multivariable Control Lecture 3 Description of Linear Time Invariant Systems John T. Wen September 7, 26 Outline Mathematical description of LTI Systems Ref: 3.1-3.4 of text September 7, 26Copyrighted

### FRTN 15 Predictive Control

Department of AUTOMATIC CONTROL FRTN 5 Predictive Control Final Exam March 4, 27, 8am - 3pm General Instructions This is an open book exam. You may use any book you want, including the slides from the

### Lec 6: State Feedback, Controllability, Integral Action

Lec 6: State Feedback, Controllability, Integral Action November 22, 2017 Lund University, Department of Automatic Control Controllability and Observability Example of Kalman decomposition 1 s 1 x 10 x

### Classify a transfer function to see which order or ramp it can follow and with which expected error.

Dr. J. Tani, Prof. Dr. E. Frazzoli 5-059-00 Control Systems I (Autumn 208) Exercise Set 0 Topic: Specifications for Feedback Systems Discussion: 30.. 208 Learning objectives: The student can grizzi@ethz.ch,

### Lecture 9: Input Disturbance A Design Example Dr.-Ing. Sudchai Boonto

Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand d u g r e u K G y The sensitivity S is the transfer function

### EEE 184: Introduction to feedback systems

EEE 84: Introduction to feedback systems Summary 6 8 8 x 7 7 6 Level() 6 5 4 4 5 5 time(s) 4 6 8 Time (seconds) Fig.. Illustration of BIBO stability: stable system (the input is a unit step) Fig.. step)

### EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)

EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the

### Control Systems Lab - SC4070 Control techniques

Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from

### Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year

Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 21-211 1 / 39 Feedback

### SYSTEMTEORI - KALMAN FILTER VS LQ CONTROL

SYSTEMTEORI - KALMAN FILTER VS LQ CONTROL 1. Optimal regulator with noisy measurement Consider the following system: ẋ = Ax + Bu + w, x(0) = x 0 where w(t) is white noise with Ew(t) = 0, and x 0 is a stochastic

### Comparison of Feedback Controller for Link Stabilizing Units of the Laser Based Synchronization System used at the European XFEL

Comparison of Feedback Controller for Link Stabilizing Units of the Laser Based Synchronization System used at the European XFEL M. Heuer 1 G. Lichtenberg 2 S. Pfeiffer 1 H. Schlarb 1 1 Deutsches Elektronen

### Chapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control

Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design

### AN INTRODUCTION TO THE CONTROL THEORY

Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter

### TRACKING AND DISTURBANCE REJECTION

TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference

### State Feedback and State Estimators Linear System Theory and Design, Chapter 8.

1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 2 Homework 7: State Estimators (a) For the same system as discussed in previous slides, design another closed-loop state estimator,

### LMIs for Observability and Observer Design

LMIs for Observability and Observer Design Matthew M. Peet Arizona State University Lecture 06: LMIs for Observability and Observer Design Observability Consider a system with no input: ẋ(t) = Ax(t), x(0)

### EL2520 Control Theory and Practice

So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal

### Problem Set 4 Solution 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback

### Control Systems. State Estimation.

State Estimation chibum@seoultech.ac.kr Outline Dominant pole design Symmetric root locus State estimation We are able to place the CLPs arbitrarily by feeding back all the states: u = Kx. But these may

### DESIGN OF LINEAR STATE FEEDBACK CONTROL LAWS

7 DESIGN OF LINEAR STATE FEEDBACK CONTROL LAWS Previous chapters, by introducing fundamental state-space concepts and analysis tools, have now set the stage for our initial foray into statespace methods

### Lecture 7 (Weeks 13-14)

Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.

### 10/8/2015. Control Design. Pole-placement by state-space methods. Process to be controlled. State controller

Pole-placement by state-space methods Control Design To be considered in controller design * Compensate the effect of load disturbances * Reduce the effect of measurement noise * Setpoint following (target

### Outline. Classical Control. Lecture 1

Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction

### Intro. Computer Control Systems: F8

Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2

### EEE582 Homework Problems

EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use

### Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

### Singular Value Decomposition Analysis

Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control

### Control Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC4026 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) and Observability Algebraic Tests (Kalman rank condition & Hautus test) A few

### Raktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response

.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........

### State Feedback and State Estimators Linear System Theory and Design, Chapter 8.

1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 State Estimator In previous section, we have discussed the state feedback, based on the assumption that all state variables are

### Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10

Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

### Design Methods for Control Systems

Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9

### EL2520 Control Theory and Practice

EL2520 Control Theory and Practice Lecture 8: Linear quadratic control Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden Linear quadratic control Allows to compute the controller

### MODERN CONTROL DESIGN

CHAPTER 8 MODERN CONTROL DESIGN The classical design techniques of Chapters 6 and 7 are based on the root-locus and frequency response that utilize only the plant output for feedback with a dynamic controller

### Control System Design

ELEC ENG 4CL4: Control System Design Notes for Lecture #22 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Friday, March 5, 24 More General Effects of Open Loop Poles

### I. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching

I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline

### Chapter 20 Analysis of MIMO Control Loops

Chapter 20 Analysis of MIMO Control Loops Motivational Examples All real-world systems comprise multiple interacting variables. For example, one tries to increase the flow of water in a shower by turning

### Systems and Control Theory Lecture Notes. Laura Giarré

Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 17: Model-based Controller Feedback Stabilization Observers Ackerman Formula Model-based Controller L. Giarré- Systems and

### Final: Signal, Systems and Control (BME )

Final: Signal, Systems and Control (BME 580.) Instructor: René Vidal May 0th 007 HONOR SYSTEM: This examination is strictly individual. You are not allowed to talk, discuss, exchange solutions, etc., with

### ẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)

EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and

### 1 (30 pts) Dominant Pole

EECS C8/ME C34 Fall Problem Set 9 Solutions (3 pts) Dominant Pole For the following transfer function: Y (s) U(s) = (s + )(s + ) a) Give state space description of the system in parallel form (ẋ = Ax +

### Internal Model Principle

Internal Model Principle If the reference signal, or disturbance d(t) satisfy some differential equation: e.g. d n d dt n d(t)+γ n d d 1 dt n d 1 d(t)+...γ d 1 dt d(t)+γ 0d(t) =0 d n d 1 then, taking Laplace

### Topic # /31 Feedback Control Systems

Topic #16 16.30/31 Feedback Control Systems Add reference inputs for the DOFB case Reading: FPE 7.8, 7.9 Fall 2010 16.30/31 16 2 Reference Input - II On page 15-6, compensator implemented with reference

### ECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04-Nov-2008

ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 04-Nov-2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are

### Professor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley

Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the

### Control Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few

### Modelling and Control of Dynamic Systems. Stability of Linear Systems. Sven Laur University of Tartu

Modelling and Control of Dynamic Systems Stability of Linear Systems Sven Laur University of Tartu Motivating Example Naive open-loop control r[k] Controller Ĉ[z] u[k] ε 1 [k] System Ĝ[z] y[k] ε 2 [k]

### SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a

### Linear System Theory. Wonhee Kim Lecture 1. March 7, 2018

Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems

### FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control Lecture 8: Youla parametrization, LMIs, Model Reduction and Summary [Ch. 11-12] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 8: Youla, LMIs, Model Reduction

### to have roots with negative real parts, the necessary and sufficient conditions are that:

THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 543 LINEAR SYSTEMS AND CONTROL H O M E W O R K # 7 Sebastian A. Nugroho November 6, 7 Due date of the homework is: Sunday, November 6th @ :59pm.. The following

### Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

### Analysis and Design of Control Systems in the Time Domain

Chapter 6 Analysis and Design of Control Systems in the Time Domain 6. Concepts of feedback control Given a system, we can classify it as an open loop or a closed loop depends on the usage of the feedback.