Robust control of MIMO systems

Transcription

1 Robust control of MIMO systems Olivier Sename Grenoble INP / GIPSA-lab September 2017 Olivier Sename (Grenoble INP / GIPSA-lab) Robust control September / 162

2 O. Sename [GIPSA-lab] 2/ Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

3 O. Sename [GIPSA-lab] 3/162 About the course Organization 20H lessons in class 10H Matlab sessions (E3) and 16H (MISCIT) Homework see given document + course complements : tools for norm definitions, and Introduction to Linear Matrix Inequalities Exercices Matlab project Evaluation One final exam Homework Ability and report on one Matlab project

4 Reference books To be studied during the course S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: analysis and design, John Wiley and Sons, chap 1 to 3 available K. Zhou, Essentials of Robust Control, Prentice Hall, New Jersey, book slides available J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory, Macmillan Publishing Company, New York, book available Carsten Scherer s courses cscherer/., Lecture slides available (MSc Course "Robust Control", MSc Course "Linear Matrix Inequalities in Control") + all the MATLAB demo, examples and documentation on the Robust Control toolbox (mathworks.com/products/robust) Other references (some in french) G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design, Prentice Hall, New Jersey, csd.newcastle.edu.au G. Duc et S. Font, Commande Hinf et -analyse: des outils pour la robustesse, Hermès, France, D. Alazard, C. Cumer, P. Apkarian, M. Gauvrit, et G. Ferreres, Robustesse et commande optimale, Cépadues Editions, O. Sename [GIPSA-lab] 4/162

5 O. Sename [GIPSA-lab] 5/162 0 Robust control in 1 slide? Sensitivity function S(s) = 1 1+L(s) Complementary Sensitivity function : y = = G(s)K(s) 1 + G(s)K(s) r L(s) 1 + L(s) r = T (s).r

6 Robust control in 1 slide? 0 Sensitivity function S(s) = 1 1+L(s) Complementary Sensitivity function : y = = G(s)K(s) 1 + G(s)K(s) r L(s) 1 + L(s) r = T (s).r Why S is the key function in control? S allows to characterize many things: S = 1 T S = r y r. For performance analysis: (S(ω = 0) = steady-state error, bandwidth ) y if output disturbance d y then, = S. d y S to be minimized! distance from -1 to Nyquist plot = inf ω 1 L(jω) = [ sup ω ] L(jω) Robustness w.r.t model uncertainties Robust control: Find K s.t S satisfies all requirements O. Sename [GIPSA-lab] 5/162

7 O. Sename [GIPSA-lab] 5/162 0 Robust control in 1 slide? Sensitivity function S(s) = 1 1+L(s) Complementary Sensitivity function : y = = G(s)K(s) 1 + G(s)K(s) r L(s) 1 + L(s) r = T (s).r Example: an uncertain mass-spring-damper system controlled by a proportional gain. Robust stability condition : S(jω) < 1, ω W (jω)

8 O. Sename [GIPSA-lab] 6/162 1 Some definitions 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

9 O. Sename [GIPSA-lab] 7/162 Definition of LTI systems 1 Some definitions Definition (LTI dynamical system) Given matrices A R n n, B R n nw, C R nz n and D R nz nw, a Linear Time Invariant (LTI) dynamical system (Σ LT I ) can be described as: Σ LT I : { ẋ(t) = Ax(t) + Bw(t) z(t) = Cx(t) + Dw(t) (1) where x(t) is the state which takes values in a state space X R n, w(t) is the input taking values in the input space W R nw and z(t) is the output that belongs to the output space Z R nz. The LTI system locally describes the real system under consideration and the linearization procedure allows to treat a linear problem instead of a nonlinear one. For this class of problem, many mathematical and control theory tools can be applied like closed loop stability, controllability, observability, performance, robust analysis, etc. for both SISO and MIMO systems. However, the main restriction is that LTI models only describe the system locally, then, compared to nonlinear models, they lack of information and, as a consequence, are incomplete and may not provide global stabilization.

10 O. Sename [GIPSA-lab] 8/162 1 Some definitions Signal norms Reader is also invited to refer to the famous book of Zhou et al., 1996, where all the following definitions and additional information are given. All the following definitions are given assuming signals x(t) C, then they will involve the conjugate (denoted as x (t)). When signals are real (i.e. x(t) R), x (t) = x T (t). Definition (Norm and Normed vector space) Let V be a finite dimension space. Then p 1, the application. p is a norm, defined as, v p = ( v i p) 1/p Let V be a vector space over C (or R) and let. be a norm defined on V. Then V is a normed space. i (2)

11 O. Sename [GIPSA-lab] 9/162 L signal norms 1 Some definitions Definition (L 1, L 2, L norms) The 1-Norm of a function x(t) is given by, + x(t) 1 = x(t) dt (3) 0 The 2-Norm (that introduces the energy norm) is given by, x(t) 2 = = + 0 x (t)x(t)dt 1 + 2π X (jω)x(jω)dω (4) The second equality is obtained by using the Parseval identity. The -Norm is given by, x(t) = sup x(t) (5) t X = sup X(s) = sup X(jω) (6) Re(s) 0 ω if the signals that admit the Laplace transform, analytic in Re(s) 0 (i.e. H ).

12 O. Sename [GIPSA-lab] 10/162 L and H spaces 1 Some definitions Definition (L space) L is the space of piecewise continuous bounded functions. It is a Banach space of matrix-valued (or scalar-valued) functions on C and consists of all complex bounded matrix functions f(jω), ω R, such that, sup σ[f(jω)] < (7) ω R Definition (H and RH spaces) H is a (closed) subspace in L with matrix functions f(jω), ω R, analytic in Re(s) > 0 (open right-half plane). The real rational subspace of H which consists of all proper and real rational stable transfer matrices, is denoted by RH. Example In control theory s+1 (s+10)(s+6) RH s+1 (s 10)(s+6) RH s+1 (s+10) RH (8)

13 O. Sename [GIPSA-lab] 11/162 1 Some definitions The H norm definition H norm Definition (H norm) The H norm of a proper LTI system defined by the state space representation (A, B, C, D) from input w(t) to output z(t) and which belongs to RH, is the induced energy-to-energy gain (induced L 2 norm) defined as, G(jω) = sup ω R σ (G(jω)) = z(s) sup 2 w(s) H2 w(s) 2 (9) = z max 2 w(t) L2 w 2 Physical interpretations of the H norm This norm represents the maximal gain of the frequency response of the system. It is also called the worst case attenuation level in the sense that it measures the maximum amplification that the system can deliver on the whole frequency set. For SISO (resp. MIMO) systems, it represents the maximal peak value on the Bode magnitude (resp. singular value) plot of G(jω), in other words, it is the largest gain if the system is fed by harmonic input signal. Unlike H 2, the H norm cannot be computed analytically. Only numerical solutions can be obtained (e.g. Bisection algorithm, or LMI resolution).

14 O. Sename [GIPSA-lab] 12/162 Well-posedness 1 Some definitions Stability issues Consider G = s 1 s + 2, K = 1 Therefore the control input is non proper: u = s + 2 s 1 (r n dy) d i DEF: A closed-loop system is well-posed if all the transfer functions are proper I + K( )G( ) is invertible In the example ( 1) = 0 Note that if G is strictly proper, this always holds.

15 O. Sename [GIPSA-lab] 13/162 Internal stability 1 Some definitions Stability issues DEF: A system is internally stable if all the transfer functions of the closed-loop system are stable Figure: Control scheme For instance : ( y u ) ( (I + GK) 1 GK (I + GK) 1 G = K(I + GK) 1 K(I + GK) 1 G G = 1 s 1, K = s 1 s + 1, ( y u ) ( 1 s+2 = s+1 (s 1)(s+2) s 1 s+2 1 s+2 ) ( r di ) ) ( r d i ) There is one RHP pole (1), which means that this system is not internally stable. This is due here to the pole/zero cancellation (forbidden!!).

16 O. Sename [GIPSA-lab] 14/162 Input-Output Stability 1 Some definitions Stability issues Definition (BIBO stability) A system G (ẋ = Ax + Bu; y = Cx) is BIBO stable if a bounded input u(.) ( u < ) maps a bounded output y(.) ( y < ). Now, the quantification (for BIBO stable systems) of the signal amplification (gain) is evaluated as: and is referred to as the PEAK TO PEAK Gain. Definition (L 2 stability) y γ peak = sup 0< u < u A system G (ẋ = Ax + Bu; y = Cx) is L 2 stable if u 2 < implies y 2 <. Now, the quantification of the signal amplification (gain) is evaluated as: y 2 γ energy = sup 0< u 2 < u 2 and is referred to as the ENERGY Gain, and is such that: γ energy = sup ω G(jω) := G For a linear system, these stability definitions are equivalent (but not the quantification criteria).

17 O. Sename [GIPSA-lab] 15/162 Small Gain theorem 1 Some definitions Stability issues Consider the so called M loop. Figure: M form Theorem Suppose M(s) in RH and γ a positive scalar. Then the system is well-posed and internally stable for all (s) in RH such that 1/γ if and only if M < γ

18 O. Sename [GIPSA-lab] 16/162 2 Introduction to LMIs 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

19 O. Sename [GIPSA-lab] 17/162 Brief on optimisation 2 Introduction to LMIs Background in Optimisation Definition (Convex function) A function f : R m R is convex if and only if for all x, y R m and λ [0 1], f(λx + (1 λ)y) λf(x) + (1 λ)f(y) (10) Equivalently, f is convex if and only if its epigraph, epi(f) = {(x, λ) f(x) λ} (11) is convex. Definition ((Strict) LMI constraint) A Linear Matrix Inequality constraint on a vector x R m is defined as, F (x) = F 0 + m F i x i 0( 0) (12) where F 0 = F T 0 and F i = F T i R n n are given, and symbol F 0( 0) means that F is symmetric and positive semi-definite ( 0) or positive definite ( 0), i.e. { u u T F u(>) 0}. i=1

20 O. Sename [GIPSA-lab] 18/162 Convex to LMIs 2 Introduction to LMIs Background in Optimisation Example Lyapunov equation. A very famous LMI constraint is the Lyapunov inequality of an autonomous system ẋ = Ax. Then the stability LMI associated is given by, x T P x > 0 x T (A T P + P A)x < 0 (13) which is equivalent to, [ P 0 F (P ) = 0 A T P + P A ] 0 (14) where P = P T is the decision variable. Then, the inequality F (P ) 0 is linear in P. LMI constraints F (x) 0 are convex in x, i.e. the set {x F (x) 0} is convex. Then LMI based optimization falls in the convex optimization. This property is fundamental because it guarantees that the global (or optimal) solution x of the the minimization problem under LMI constraints can be found efficiently, in a polynomial time (by optimization algorithms like e.g. Ellipsoid, Interior Point methods).

21 O. Sename [GIPSA-lab] 19/162 2 Introduction to LMIs Background in Optimisation LMI problem Two kind of problems can be handled Feasibility: The question whether or not there exist elements x X such that F (x) < 0 is called a feasibility problem. The LMI F (x) < 0 is called feasible if such x exists, otherwise it is said to be infeasible. Optimization: Let an objective function f : S R where S = {x F (x) < 0}. The problem to determine V opt = inf x S f(x) is called an optimization problem with an LMI constraint. This problem involves the determination of V opt, the calculation of an almost optimal solution x (i.e., for arbitrary ɛ > 0 the calculation of an x S such that V opt f(x) V opt + ɛ, or the calculation of a optimal solutions x opt (elements x opt S such that V opt = f(x opt)).

22 O. Sename [GIPSA-lab] 20/162 2 Introduction to LMIs Background in Optimisation Semi-Definite Programming (SDP) Problem LMI programming is a generalization of the Linear Programming (LP) to cone positive semi-definite matrices, which is defined as the set of all symmetric positive semi-definite matrices of particular dimension. Definition (SDP problem) A SDP problem is defined as, min c T x under constraint F (x) 0 (15) where F (x) is an affine symmetric matrix function of x R m (e.g. LMI) and c R m is a given real vector, that defines the problem objective. SDP problems are theoretically tractable and practically: They have a polynomial complexity, i.e. there exists an algorithm able to find the global minimum (for a given a priori fixed precision) in a time polynomial in the size of the problem (given by m, the number of variables and n, the size of the LMI). SDP can be practically and efficiently solved for LMIs of size up to and m 1000 see ElGhaoui, 97. Note that today, due to extensive developments in this area, it may be even larger.

23 O. Sename [GIPSA-lab] 21/162 2 Introduction to LMIs LMI in control The state feedback design problem Stabilisation Let us consider a controllable system ẋ = Ax + Bu. The problem is to find a state feedback u(t) = Kx(t) s.t the closed-loop system is stable. Using the Lyapunov theorem, this amounts at finding P = P T > 0 s.t: which is obviously not linear... Solution; use of change of variables (A BK) T P + P(A BK) < 0 A T P + PA K T B T P P BK < 0 First, left and right multiplication by P 1 leads to with Q = P 1 and Y = KP 1. P 1 A T + AP 1 P 1 K T B T BKP 1 < 0 QA T + AQ + Y T B T + BY < 0 The problem to be solved is therefore formulated as an LMI! and without any conservatism!

24 O. Sename [GIPSA-lab] 22/162 The Bounded Real Lemma 2 Introduction to LMIs LMI in control The L 2 -norm of the output z of a system Σ LT I is uniformly bounded by γ times the L 2 -norm of the input w (initial condition x(0) = 0). A dynamical system G = (A, B, C, D) is internally stable and with an G < γ if and only if there exists a positive definite symmetric matrix P (i.e P = P T > 0 s.t AT P + P A P B C T B T P γ I D T C D γ I < 0, P > 0. (16) The Bounded Real Lemma (BRL), can also be written as follows (see Scherer) I 0 A B 0 I C D T 0 P 0 0 P γ 2 I I I 0 A B 0 I C D 0 (17) Note that the BRL is an LMI if the only unknown (decision variables) are P and γ (or γ 2 ).

25 O. Sename [GIPSA-lab] 23/162 Quadratic stability 2 Introduction to LMIs LMI in control This concept is very useful for the stability analysis of uncertain systems. Let us consider an uncertain system ẋ = A(δ)x where δ is an parameter vector that belongs to an uncertainty set. Theorem The considered system is said to be quadratically stable for all uncertainties δ if there exists a (single) "Lyapunov function" P = P T > 0 s.t A(δ) T P + PA(δ) < 0, for all δ (18) This is a sufficient condition for ROBUST Stability which is obtained when A(δ) is stable for all δ.

26 O. Sename [GIPSA-lab] 24/162 Some useful lemmas 2 Introduction to LMIs Some useful lemmas Schur lemma: allows to a convert a quadratic constraint (ellipsoidal constraint) into an LMI constraint. Lemma Let Q = Q T and R = R T be affine matrices of compatible size, then the condition [ ] Q S S T 0 R is equivalent to R 0 Q SR 1 S T 0 Kalman-Yakubovich-Popov lemma: convert frequency inequalities into Linear Matrix Inequalities (used in robust control for dissipative systems) Projection Lemma: allows to eliminate variable by a change of basis (projection in the kernel basis). It is involved in one of the H solutions, see (Doyle,Scherer). Completion Lemma: allows to simplify the number of variables when a matrix and its inverse enter in a LMI. Finsler s lemma: This Lemma allows the elimination of matrix variables.

27 O. Sename [GIPSA-lab] 25/162 2 Introduction to LMIs Some useful lemmas Interests of LMIs LMIs allow to formulate complex optimization problems into "Linear" ones, allowing the use of convex optimization tools. Usually it requires the use of different transformations, changes of variables... in order to linearize the considered problmems: Congruence, Schur complement, projection lemma, Elimination lemma, S-procedure, Finsler s lemme... Examples of handled criteria stability H, H 2, H 2 /H performances robustness analysis: Small gain theorem, Polytopic uncertianties, LFT representations... Robust control and/or observer design pole placement stability, stabilization with input constraints Passivity constraints Time-delay systems

28 O. Sename [GIPSA-lab] 26/162 3 What is the H performance? 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

29 O. Sename [GIPSA-lab] 27/162 3 What is the H performance? H norm How to define the system gain? SISO systems z = Gd, the gain at a given frequency is simply z(ω) d(ω) = G(jω)d(ω) = G(jω) d(ω) The gain depends on the frequency, but since the system is linear it is independent of the input magnitude How to generalize to MIMO systems? we may select: z(ω) 2 d(ω) 2 = G(jω)d(ω) 2 d(ω) 2 = G(jω) 2? Which seems to be "independent" of the input magnitude. But this is not a correct definition. Indeed the input direction is of great importance

30 O. Sename [GIPSA-lab] 28/162 3 What is the H performance? H norm The gain of a MIMO system as induced L 2 norm? Let consider [ 5 4 G = 3 2 How to define and evaluate its gain? Consider ( five ) different inputs: ( ) ( ) d 1 = d 0 2 = d 1 3 = ] ( d 4 = ) ( 0.6 d 5 = 0.8 ) The input magnitudes are: d 1 2 = d 2 2 = d 3 2 = d 4 2 = d 5 2 = 1 But the ( corresponding ) outputs ( ) are 5 4 z 1 = z 3 2 = 2 ( z 3 = ) ( z 4 = ) ( 0.2 z 5 = 0.2 ) and the ratio are z 2 / d 2 z 1 2 z = = 4.47 d 1 2 d 2 2 z 3 2 d 3 2 = 7.27 z 4 2 d 4 2 = 0.99 z 5 2 d 5 2 = 0.28 So the gain value differs function of the input vector direction.

31 O. Sename [GIPSA-lab] 29/162 3 What is the H performance? H norm How the Singular Value Decomposition can provide such a MIMO gain definition? Below is represented z 2 / d 2 as a function of d20/d10 (where d = [d10, d20] T ) We can see that, depending on the ratio d20/d10, the gain varies between 0.27 and 7.34., where σ(g) = 7.34 and σ(g) = We then have these mathematical definitions: MAXIMUM SINGULAR VALUE max d 0 Gd 2 d 2 = σ(g) MINIMUM SINGULAR VALUE min d 0 Gd 2 d 2 = σ(g)

32 O. Sename [GIPSA-lab] 30/ Frequency (rad/sec) 3 What is the H performance? H norm Characterization of the H norm as induced L 2 norm Finally, in the case of a transfer matrix G(s) : (m inputs, p outputs) u vector of inputs, y vector of outputs. σ(g(jω)) z(ω) 2 d(ω) 2 σ(g(jω)) Singular Values Example of A two-mass/spring/damper system largest singular 2 inputs: F 1 and F 2 2 outputs: x 1 and x 2 value Hinf norm = db Singular Values (db) smallest singular value G=ss(A,B,C,D): LTI system [ninf,fpeak] = hinfnorm(g): Compute H norm and freq norm(g,inf): Compute H norm normhinf(g): Compute H norm sigma(g): plot max and min SV

33 O. Sename [GIPSA-lab] 31/162 3 What is the H performance? H norm computation 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

34 O. Sename [GIPSA-lab] 32/162 How to compute the H norm? 3 What is the H performance? H norm computation As said before, H norm cannot be computed analytically. Only numerical solutions can be obtained (e.g. Bisection algorithm, or LMI resolution). Method 1: Since G(jω) = sup ω R σ (G(jω)), the intuitive computation is to get the peak on the Bode magnitude plot, which can be estimated using a thin grid of frequency points, {ω 1,..., ω N }, and then: G(jω) max σ{g(jω k)} 1 k N Method 2: Let the dynamical system G = (A, B, C, D) RH : G < γ if and only if σ (D) < γ and the Hamiltonian H has no eigenvalues on the imaginary axis, where ( A + BR H = 1 D T C BR 1 B T ) C T (I n + DR 1 D T )C (A + BR 1 D T and R = γ 2 D T D C) Use norm(sys,inf)or hinfnorm(sys,tol)in Matlab. Method 3 (Bounded Real Lemma): A dynamical system G = (A, B, C, D) is internally stable and with an G < γ if and only is there exists a positive definite symmetric matrix P (i.e P = P T > 0 s.t AT P + P A P B C T B T P γ I D T C D γ I < 0, P > 0. (19)

35 O. Sename [GIPSA-lab] 33/162 3 What is the H performance? What is H control? 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

36 O. Sename [GIPSA-lab] 34/162 3 What is the H performance? What is H control? Towards H control: the General Control Configuration This approach has been introduced by Doyle (1983). The formulation makes use of the general control configuration. P is the generalized plant (contains the plant, the weights, the uncertainties if any) ; K is the controller. The closed-loop transfer matrix from w to z is given by: T zw(s) = F l (P, K) = P 11 + P 12 K(I P 22 K) 1 P 21 where F l (P, K) is referred to as a lower Linear Fractional Transformation.

37 O. Sename [GIPSA-lab] 35/162 3 What is the H performance? What is H control? Problem definition The overall control objective is to minimize some norm of the transfer function from w to z, for example, the H norm. Definition (H optimal control problem) H control problem: Find a controller K(s) which based on the information in y, generates a control signal u which counteracts the influence of w on z, thereby minimizing the closed-loop norm from w to z. Definition (H suboptimal control problem) Given γ a pre-specified attenuation level, a H sub-optimal control problem is to design a stabilizing controller that ensures : T zw(s) = max ω σ(tzw(jω)) γ The optimal problem aims at finding γ min (done using hinfsyn in MATLAB). Remarks It is worth noting that the H control problem is a disturbance attenuation, formulated in the worst-case performance analysis. z is then often defined as an "error signal" (to be minimized)

38 O. Sename [GIPSA-lab] 36/162 4 Why H control is adapted to control engineering? 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

39 O. Sename [GIPSA-lab] 37/162 Introduction 4 Why H control is adapted to control engineering? Objectives of any control system 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. This is formulated as: 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).

40 O. Sename [GIPSA-lab] 38/162 4 Why H control is adapted to control engineering? Sensitivity functions The control structure - SISO case Figure: Complete control scheme In the SISO case, it leads to: 1 y = 1+G(s)K(s) (GKr + dy GKn + Gd i) 1 u = 1+K(s)G(s) (Kr Kdy Kn KGd i) Loop transfer function L = G(s)K(s) 1 Sensitivity function S(s) = 1+L(s) Complementary Sensitivity function T (s) = L(s) 1+L(s) N.B. S is often referred to as the Output Sensitivity.

41 O. Sename [GIPSA-lab] 39/162 4 Why H control is adapted to control engineering? Sensitivity functions Stability and robustness margins... Classical definitions: Stability L(jω π) < 1, where ω π is the phase crossover frequency defined by φ(l(jω π)) = π. Gain Margin: indicates the additional gain that would take the closed loop to the critical stability condition (G M (db) = [ L(jω π ] db ) Phase margin: quantifies the pure phase delay that should be added to achieve the same critical stability condition (Φ M = 180 o + arg[l(jω c)], where L(jω c ) = 1(0dB)) Delay margin: quantifies the maximal delay that should be added in the loop to achieve the same critical stability condition, P M/ω c

42 O. Sename [GIPSA-lab] 40/162 4 Why H control is adapted to control engineering? Sensitivity functions Robustness margins... It is important to consider the module margin that quantifies the minimal distance between the curve and the critical point (-1,0j): this is a robustness margin. The MODULE MARGIN is a robustness margin. Indeed, the influence of plant modelling errors on the CL transfer function: is given by: T = K(s)G(s) 1 + K(s)G(s) T T = 1 G 1 + K(s)G(s) G = S. G G A good module margin implies good gain and phase margins: M = 1 / MS M S = max S(jω) = S ω Good value M S < 2 (6dB) GM M S M S 1 andp M 1 M S For M S = 2, then GM > 2 and P M > 30 Last: M T = max ω T (jω) A good value : M T < 1.5(3.5dB)

43 O. Sename [GIPSA-lab] 41/162 4 Why H control is adapted to control engineering? Sensitivity functions Input/Output performances Defining two new sensitivity functions : Plant Sensitivity: SG = S(s).G(s) (often referred to as the Input Sensitivity, e.g in Matlab) Controller Sensitivity: KS = K(s).S(s)

44 O. Sename [GIPSA-lab] 42/162 Using Matlab 4 Why H control is adapted to control engineering? Sensitivity functions % Determination of the sensitivity fucntions L=series(G,Khinf) % Loop transfer function L=GK S=inv(1+L); % S= 1/(1+L) poles=pole(s) T= feedback(l,1) polet=pole(t) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SG=S*G; polesg=pole(sg) KS=Khinf*S; poleks=pole(ks) %%%% w=logspace(-3,3,500); %% to be adjusted subplot(2,2,1), sigma(s,w), title('sensitivity function') subplot(2,2,2), sigma(t,w), title('complementary sensitivity function') subplot(2,2,3), sigma(sg,w), title('sensitivity*plant') subplot(2,2,4), sigma(ks,w), title('controller*sensitivity') Remark: for SISO systems, use bodemag instead of sigma but not for MIMO ones!

45 O. Sename [GIPSA-lab] 43/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis and specification using the sensitivity functions: the SISO case- Dynamical behavior As mentioned in Skogestad & Postlethwaite s book: The concept of bandwidth is very important in understanding the benefits and trade-offs involved when applying feedback control. Above we considered peaks of closed-loop transfer functions, which are related to the quality of the response. However, for performance we must also consider the speed of the response, and this leads to considering the bandwidth frequency of the system. In general, a large bandwidth corresponds to a faster rise time, since high frequency signals are more easily passed on to the outputs. A high bandwidth also indicates a system which is sensitive to noise and to parameter variations. Conversely, if the bandwidth is small, the time response will generally be slow, and the system will usually be more robust. Definition Loosely speaking, bandwidth may be defined as the frequency range [ω 1, ω 2 ] over which control is effective. In most cases we require tight control at steady-state so ω 1 = 0, and we then simply call ω 2 the bandwidth. The word "effective" may be interpreted in different ways : globally it means benefit in terms of performance.

46 O. Sename [GIPSA-lab] 44/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis and specification using the sensitivity functions: the SISO case- Bandwidth definitions Definition (ω S ) The (closed-loop) bandwidth, ω S, is the frequency where S(jω) crosses 3dB (1/sqrt2) from below. Remark: S < 0.707, frequency zone, where e/r = S is reasonably small Definition (ω T ) The bandwidth (in term of T ), ω T, is the frequency where T (jω) crosses 3dB (1/sqrt2) from above. Definition (ω c) The bandwidth (crossover frequency), ω c, is the frequency where L(jω) crosses 1 (0dB), for the first time, from above.

47 O. Sename [GIPSA-lab] 45/162 Some remarks 4 Why H control is adapted to control engineering? Sensitivity functions Remark Usually ω S < ω c < ω T Remark In most cases, the two definitions in terms of S and T yield similar values for the bandwidth. In other cases, the situation is generally as follows. Up to the frequency ω S, S is less than 0.7, and control is effective in terms of improving performance. In the frequency range [ω S, ω T ] control still affects the response, but does not improve performance. Finally, at frequencies higher than ω T, we have S 1 and control has no significant effect on the response. Remark Usually ω S < ω c < ω T Finally the following relation is very useful to evaluate the rise time: t r 2.3 ω T

48 O. Sename [GIPSA-lab] 46/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis: answer to... Analysis of S: Steady state error in tracking and output disturbance rejection : S(ω = 0) = 0? Maximum peak criterion (Module Margin): S < 2? bandwidth of S Analysis of T : Steady state error in tracking : T (ω = 0) = 1?. Attenuation of measurement noise: T (jω) small when ω? Maximum of T, T < 1.5? ω T bandwidth of T + rise time evaluation t r Analysis of KS: Input saturation: u(t) < u max? (where u max < KS r max ). Attenuation of measurement noise: KS(jω) small when ω? Analysis of SG: Steady state error in input disturbance rejection : SG(ω = 0) = 0? Attenuation of input disturbance effet: SG(jω) small in the frequency range of interest?.

49 O. Sename [GIPSA-lab] 47/162 4 Why H control is adapted to control engineering? Sensitivity functions From analysis to specification... templates Objective : good performance specifications are important to ensure better control system Mean : give some templates on the sensitivity functions For simplicity, presentation for SISO systems first. Sketch of the method: 1 Robustness and performances in regulation can be specified by imposing frequential templates on the sensitivity functions. 2 If the sensitivity functions stay within these templates, the control objectives are met. 3 These templates can be used for analysis and/or design. In the latter they are considered as weights on the sensitivity functions 4 The shapes of typical templates on the sensitivity functions are given in the following slides Mathematically, these specifications may be captured by an upper bound, on the magnitude of a sensitivity function, given by another transfer function, as for S: S(jω) where W e(s) is a WEIGHT selected by the designer. 1 W e(jω), ω WeS 1

50 O. Sename [GIPSA-lab] 48/162 4 Why H control is adapted to control engineering? Sensitivity functions Template on the sensitivity function S - Weighted sensitivity Typical specifications in terms of S include: 1 Minimum bandwidth frequency ω S 2 Maximum tracking error at selected frequencies. 3 System type, or the maximum steady-state tracking error ɛ 0 4 Shape of S over selected frequency ranges. 5 Maximum peak magnitude of S, S < M S. The peak specification prevents amplification of noise at high frequencies, and also introduces a margin of robustness; typically we select M S = 2. How to select the template function It should: be close to the control objectives avoid too much under -or over- estimation be simple enough to be used later in the control design step

51 O. Sename [GIPSA-lab] 49/162 4 Why H control is adapted to control engineering? Sensitivity functions Template on the sensitivity function S S(s) = 1 W e(s) = K(s)G(s) s + ω bε s/m S + ωb Generally ε 0 is considered, M S < 2 (6dB) or (3dB - cautious) to ensure sufficient module margin. ω b influences the CL bandwidth : ω b faster rejection of the disturbance faster CL tracking response better robustness w.r.t. parametric uncertainties Singular Values (db) Template on the sensitivity function S Template on the Sensitivity function S M S =2(6dB) ω b s.t. 1/W e =0dB ε =1e Frequency 10(rad/sec)

52 4 Why H control is adapted to control engineering? Sensitivity functions Template on the function KS KS(s) = 1 W u(s) = K(s) 1 + K(s)G(s) ε 1s + ω bc s + ω bc /M u M u chosen according to LF behavior of the process (actuator constraints: saturations) ω bc influences the CL bandwidth : ω bc better limitation of measurement noises roll-off starting from ω bc to reduce modeling errors effects Singular Values (db) Template on the Controller*Sensitivity KS M u = 2 ω bc for 1 /W u = 0dB ɛ c =1e Frequency (rad/sec) O. Sename [GIPSA-lab] 50/162

53 O. Sename [GIPSA-lab] 51/162 4 Why H control is adapted to control engineering? Sensitivity functions Template on the function T T (s) = 1 W T (s) = K(s)G(s) 1 + K(s)G(s) ε T s + ω bt s + ω bt /M T Generally ε T 0 is considered, M T < 1.5 (3dB) to limit the overshoot. ω bt influences the bandwidth hence the transient behavior of the disturbance rejection properties: ω bt better noise effects rejection better filtering of HF modelling errors Singular Values (db) Template on the Complementary sensitivity function T MT =1.5 ωbt s.t. 1/WT =0dB ɛt =1e Frequency 10(rad/sec)

54 O. Sename [GIPSA-lab] 52/162 4 Why H control is adapted to control engineering? Sensitivity functions Template on the function SG SG(s) = 1 W SG (s) = G(s) 1 + K(s)G(s) s + ω SGε SG s/m SG + ω SG M SG allows to limit the overshoot in the response to input disturbances. Generally ε SG 0 is considered, ω SG influences the CL bandwidth : ω SG = faster rejection of the disturbance. 20 Template on the Sensitivity*Plant SG 10 Singular Values (db) M SG =10 ω sg s.t. 1/W SG =0dB ɛ SG=1e Frequency 10(rad/sec)

55 O. Sename [GIPSA-lab] 53/162 4 Why H control is adapted to control engineering? Sensitivity functions 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

56 O. Sename [GIPSA-lab] 54/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis and specification using the sensitivity functions: the MIMO case Figure: Complete control scheme The output & the control input satisfy the following equations : BUT : K(s)G(s) G(s)K(s)!! (I p + G(s)K(s))y(s) = (GKr + d y GKn + Gd i ) (I m + K(s)G(s))u(s) = (Kr Kd y Kn KGd i )

57 O. Sename [GIPSA-lab] 55/162 4 Why H control is adapted to control engineering? Sensitivity functions Sensitivity functions- The MIMO case Definitions Output and Output complementary sensitivity functions: S y = (I p + GK) 1, T y = (I p + GK) 1 GK, S y + T y = I p Input and Input complementary sensitivity functions: S u = (I m + KG) 1, T u = KG(I m + KG) 1, S u + T u = I m Properties T y = GK(I p + GK) 1 T u = (I m + KG) 1 KG S uk = KS y

58 O. Sename [GIPSA-lab] 56/162 4 Why H control is adapted to control engineering? Sensitivity functions MIMO Input/Output performances Defining two new sensitivity functions : Plant Sensitivity: S yg = S y(s).g(s) Controller Sensitivity: KS y = K(s).S y(s)

59 O. Sename [GIPSA-lab] 57/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis and specification using the sensitivity functions: the MIMO case- Some classical analysis criteria (1) The transfer function KS y(s) should be upper bounded so that u(t) does not reach the physical constraints, even for a large reference r(t) The effect of the measurement noise n(t) on the plant input u(t) can be made «small» by making the sensitivity function KS u(s) small (in High Frequencies) The effect of the input disturbance d i (t) on the plant input u(t) + d i (t) (actuator) can be made «small» by making the sensitivity function S u(s) small (take care to not trying to minimize T u which is not possible)

60 O. Sename [GIPSA-lab] 58/162 4 Why H control is adapted to control engineering? Sensitivity functions Performance analysis and specification using the sensitivity functions: the MIMO case- Some classical analysis criteria (2) The plant output y(t) can track the reference r(t) by making the complementary sensitivity function T y(s) equal to 1. (servo pb) The effect of the output disturbance d y(t) (resp. input disturbance d i (t) ) on the plant output y(t) can be made «small» by making the sensitivity function S y(s) (resp. S yg(s) ) «small» The effect of the measurement noise n(t) on the plant output y(t) can be made «small» by making the complementary sensitivity function T y(s) «small»

61 O. Sename [GIPSA-lab] 59/162 Trade-offs 4 Why H control is adapted to control engineering? Sensitivity functions But Some trade-offs are to be looked for... These trade-offs can be reached if one aims : S + T = I to reject the disturbance effects in low frequencies to minimize the noise effects in high frequencies It remains to require: S y and S yg to be small in low frequencies to reduce the load (output and input) disturbance effects on the controlled output T y and KS y to be small in high frequencies to reduce the effects of measurement noises on the controlled output and on the control input (actuator efforts)

62 O. Sename [GIPSA-lab] 60/162 Synthesis 4 Why H control is adapted to control engineering? Sensitivity functions Provide a clear and detailed frequency-domain performance analysis using the sensitivity functions in order to explain the trade-off performance/robustness/actuator constraints. Qualitative analysis Use of S y, T y, KS y, S yg to: predict the behavior of the output w.r.t different external inputs (reference, disturbance, noise) predict the behavior of the control input w.r.t different external inputs (reference, disturbance, noise) Quantitative analysis Use of S y, T y, KS y, S yg to: Stability analysis and margins. Compute the steady-state errors in tracking, output and input disturbance attenuations. Give the maximum of the input/output gains to analyze the transient behaviors of the output and control input (incl. saturation). Give all the bandwidths of the sensitivity functions Evaluate the rise time in tracking Evaluate the robustness margins

63 O. Sename [GIPSA-lab] 61/162 4 Why H control is adapted to control engineering? Sensitivity functions A first insight into the performance specifications for the MIMO case The direct extension of the performances objectives to MIMO systems could be formulated as follows: 1 Disturbance attenuation/closed-loop performances: σ(s y(jω)) < 1 W 1 (jω) with W 1 (jω) > 1 2 Actuator constraints: for ω < ω b σ(ks y(jω)) < 1 W 2 (jω) with W 2 (jω) > 1 for ω > ω h 3 Robustness to multiplicative uncertainties: with W 3 (jω) > 1 for ω > ω t σ(t y(jω)) < 1 W 3 (jω) However these objectives do not consider the specific MIMO structure of the system, i.e. the input-output relationship between actuators and sensors. It is then better to define the objectives accordingly with the system inputs and outputs.

64 O. Sename [GIPSA-lab] 62/162 4 Why H control is adapted to control engineering? Sensitivity functions Towards MIMO systems Let us consider a system with 2 inputs and 1 output and define: Therefore and the sensitivity functions are: G = ( G 1 G 2 ), K = ( K1 K 2 ) ( ) K1 G GK = G 1 K 1 + G 2 K 2, KG = 1 K 1 G 2 K 2 G 1 K 2 G 2 S y = G 1 K 1 + G 2 K 2, KS y = ( K 1 1+G 1 K 1 +G 2 K 2 K 2 1+G 1 K 1 +G 2 K 2 While a single template W e is convenient for S y it is straightforward that the following diagonal template should be used for KS y: ( ) W 1 W u(s) = u (s) 0 0 Wu 2(s) where W 1 u and W 2 u are chosen in order to account for each actuator specificity (constraint). )

65 O. Sename [GIPSA-lab] 63/162 4 Why H control is adapted to control engineering? Sensitivity functions The MIMO general case Let us consider G with m inputs and p outputs. In the MIMO case the simplest way is to defined the templates as diagonal transfer matrices, i.e. using (M Si, ω bi, ε i ) In that case, a weighting function should be dedicated for each input, and for each output. These weighting functions may of course be different if the specifications on each actuator (e.g. saturation), and on each sensor (e.g. noise), are different. In addition, during the performance analysis step, take care to plot, in addition to the MIMO sensitivity functions, the individual ones related to each input/output to check if the individual specification is met. Hence, in the simplest case: 1 If the specifications are identical then it is sufficient to plot: 1 σ(s y(jω)) and, for all ω We(jω) 1 σ(ks y(jω)) and, for all ω Wu(jω) 2 If the specifications are different, one should plot 1 σ(s y(i, :)) and W e i, for all ω, i = 1,..., p 1 σ(ks y(k, :)) and Wu k, for all ω, k = 1,..., m. i.e. p plots for all output behaviors and m plots for the input ones. 3 In a very general case, plot σ(s y) with σ(1/w e)

66 O. Sename [GIPSA-lab] 64/162 4 Why H control is adapted to control engineering? Sensitivity functions More on weighting functions When tighter (harder) objectives are to be met... the templates can be defined more accurately by transfer functions of order greater than 1, as W e(s) = ( ) s/ms + ωb k, s + ω b ε if a roll-off of 20 k db per decade is required. Take care to the choice of the parameters (M S, ω b, ε) to avoid incoherent objectives! Wu and Wu square without parameter modification 20 $1/W_u$ and $1/Wu_^2$ with parameter modification 20 Original weight (specifications) Mu=2, ɛ1 = 0.01, ωbc=100 for Wu 10 Mu=2, wbc=100, epsi1= Singular Values (db) Singular Values (db) Mu = 2, ε1 = 0.01, ωbc=200 for 1/Wu 2 case 2 Mu = 2, ε1 = 0.01, ωbc=400 for 1/Wu 2 case Mu = 2, ε1 = 0.01, ωbc=100 for 1/Wu 2 case Frequency (rad/sec) Frequency (rad/sec)

67 O. Sename [GIPSA-lab] 65/162 Final objectives 4 Why H control is adapted to control engineering? Sensitivity functions In terms of control synthesis, all these specifications can be tackled in the following problem: find K(s) s.t. W es y W uks y W W T T y 1 es y 1 W SG S yg 1 W uks y 1 W T T y 1 W SG S yg Often, the simpler following one (referred to as the mixed sensitivity problem) is studied: Find K s.t. WeSy W uks y 1 since the latter allows to consider the closed-loop output performance as well as the actuator constraints.

68 O. Sename [GIPSA-lab] 66/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

69 O. Sename [GIPSA-lab] 67/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design How to consider performance specification in H control? Illustration on the H SISO problem: T ew(s) = WeS W uks γ In that case the closed-loop system T ew(s) must have 1 input and 2 outputs. Since S = r y and r KS = u, the control scheme needs only one external input r. r Control objectives: y = Gu = GK(r y) tracking error : ε = Sr u = K(r y) = K(r Gu) actuator force : u = KSr To cope with that control objectives the following control scheme is considered: Objective w.r.t sensitivity functions: W es 1, W uks 1. Idea: define 2 new virtual controlled outputs: e 1 = W esr e 2 = W uksr

70 O. Sename [GIPSA-lab] 68/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design The mixed sensitivity H control design - Problem definition The performance specifications on the tracking error & on the actuator, given as some weights on the controlled output, then leads to the new control scheme: The associated general control configuration is :

71 O. Sename [GIPSA-lab] 69/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design The mixed sensitivity H control design - Problem definition The corresponding H suboptimal control problem is therefore to find a controller K(s) such that T ew(s) = WeS W uks γ with in Matlab T ew(s) [ = F l (P, ] K) [ = P 11 + P ] 12 K(I P 22 K) 1 P 21 We WeG = + K(I + GK) 0 W 1 I [ u ] W = es W uks

72 O. Sename [GIPSA-lab] 70/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design What about disturbance attenuation? To account for input disturbance rejection, the control scheme must include d i : When W d = 1 this corresponds to the closed-loop system: [ ] W T ew = es y W es yg W uks y W ut u The new H control problem therefore includes the input disturbance rejection objective, thanks to S yg that should satisfy the same template as S, i.e an high-pass filter! Remarks: Note that W ut u is an additional constraint that may lead to an increase of the attenuation level γ since it is not part of the objectives. Hopefully T u is low pass, and W u as well. The input weight has to be on u not u + d i which would lead to an unsolvable problem.

73 4 Why H control is adapted to control engineering? The mixed sensitivity H control design Improve the disturbance attenuation The previous problem, allows to ensure the input disturbance rejection, but does not provide any additional d-o-f to improve it (without impacting the tracking performance). In order to decouple both performance objectives, the idea is to add a disturbance model that indeed changes the disturbance rejection properties. Let then consider : d i (t) = W d.d. In that case the closed-loop system is This corresponds to the closed-loop system. [ ] W T ew = es y W es ygw d W uks y W ut uw d and the template expected for S yg is now 1 W d.w e. First interest: improve the disturbance weight as W d = but this has a price (see Fig. below for an example) Sensitivity*Plant SG Controller*Sensitivity KS with W d =1 30 with W d = Magnitude (db) with W d =100 Singular Values (db) with W d = Frequency (rad/sec) O. Sename [GIPSA-lab] 71/162 Frequency (rad/sec)

74 O. Sename [GIPSA-lab] 72/162 More generally... 4 Why H control is adapted to control engineering? The mixed sensitivity H control design To include multiple objectives in a SINGLE H control problem, there are 2 ways: 1 add some external inputs (reference, noise, disturbance, uncertainties...) 2 add new controlled outputs Of course both ways increase the dimension of the problem to be solved...thus the complexity as well. Moreover additional constraints appear that are not part of the objectives... General rule: first think simple!!

75 O. Sename [GIPSA-lab] 73/162 4 Why H control is adapted to control engineering? The mixed sensitivity H control design Some extensions: the 2-DOF case In some cases it is intresting to decouple the transient response in tracking fropm the stabilization loop (as in RST controllers). This is the case of 2 dof control structure. Pay attention when building P since: External inputs: r, d i, d y and n Control Input: u Controlled ouputs z 1 and z 2 Measurements: r and y + n The controller solution will the be such as u = [ ] [ ] r K r K y y n

76 O. Sename [GIPSA-lab] 74/162 4 Why H control is adapted to control engineering? Some performance limitations 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

77 O. Sename [GIPSA-lab] 75/162 Introduction 4 Why H control is adapted to control engineering? Some performance limitations Framework Main extracts of this part: see Goodwin et al "Performance limitations in control are not only inherently interesting, but also have a major impact on real world problems." Objective : take into account the limitations inherent to the system or due to actuators constraints, before designing the controller... Understanding what is not possible is as important as understanding what is possible! Example of structural constraints S + T = 1, ω We then cannot have, for any frequency ω 0, S(jω 0 ) < 1 and T (jω 0 ) < 1. This implies that, disturbance and noise rejection cannot be achieved in the same frequency range. Bode s Sensitivity Integral for open-loop stable systems It is known that, for an open loop stable plant: log S(jω) dω = 0 0 Then the frequency range where S(jω) < 1 is balanced by the frenquencies where S(jω) > 1

78 O. Sename [GIPSA-lab] 76/162 Bode sensitivity 4 Why H control is adapted to control engineering? Some performance limitations Nice interpretation of the balance between reduction and magnification of the sensitivity. For open-loop unstable systems we have a stronger constraint: N p log det(s(jω)) dω = π Re(p i ), 0 i=1 where p i design the N p RHP poles. Therefore, in the presence of RHP poles, the control effort necessary to stabilize the system is paid in terms of amplification of the sensitivity magnitude.

79 O. Sename [GIPSA-lab] 77/162 4 Why H control is adapted to control engineering? Some performance limitations The interesting case of systems with RHP zeros Theorem Let G(s) a MIMO plant with one RHP zero at s = z, and W p(s) be a scalar weight. Then, closed-loop stability is ensured only if: W p(s)s(s) W p(s = z) To illustrate the use of that theorem, if W p is chosen as: ( ) s/ms + ωb W p(s) =, s + ω b ε and, if the controller meets the requirements, then Therefore a necessary condition is: W p(s)s(s) 1 z/m S + ωb z + ω b ε To conclude, if z is real, and if the performance specifications are such that : M S = 2 and ε = 0, then a necessary condition to meet the performance requirements is : 1 ω b z 2

80 O. Sename [GIPSA-lab] 78/162 5 How to solve an H control problem? 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

81 O. Sename [GIPSA-lab] 79/162 5 How to solve an H control problem? The Static State feedback case A first case: the state feedback control problem Let consider the system: ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) (20) z(t) = Cx(t) + D 11 w(t) + D 12 u(t) The objective is to find a state feedback control law u = Kx s.t: T zw(s) γ The method consists in applying the Bounded Real Lemma to the closed-loop system, and then try to obtain some convex solutions (LMI formulation). This is achieved if and only is there exists a positive definite symmetric matrix P (i.e P = P T > 0 s.t (A B 2K) T P + P (A B 2 K) P B 1 (C D 12 K) T γ I D T 11 γ I < 0, P > 0. (21)

82 O. Sename [GIPSA-lab] 80/162 5 How to solve an H control problem? The Static State feedback case Solution of the state feedback control problem Use of change of variables First, left and right multiplication by diag(p 1, I n, I n), and use Q = P 1 and Y = KP 1. It leads to AQ + B 2Y + QA T + Y T B2 T B 1 QC T + Y T D12 T γ I D11 T < 0, Q > 0. (22) γ I The state feedback controller is then: K = YQ 1

83 O. Sename [GIPSA-lab] 81/162 5 How to solve an H control problem? The Dynamic Output feedback case 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

84 O. Sename [GIPSA-lab] 82/162 5 How to solve an H control problem? The Dynamic Output feedback case The Dynamic Output feedback case It will be shown how to formulate such a control problem using "classical" control tools. The procedure will be 2-steps: Get P : Build the General Control Configuration scheme s.t. the closed-loop system matrix does correspond to the tackled H problem (for instance the mixed sensitivity problem). Use of Matlab, sysic tool. A state space representation of P, the generalized plant, is needed. Compute K: Use an optimisation algorithm that finds the controller K solution of the considered problem. The calculation of the controller, solution of the H control problem, can then be done using the Riccati approach or the LMI approach of the H control problem [Zhou et al.(1996)zhou, Doyle, and Glover] [Skogestad and Postlethwaite(1996)]. Notations: ẋ = Ax + B 1 w + B 2 u P z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u P = A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 with x R n : { plant state variables state variables of weights} w R nw : external inputs u R nu control inputs z R nz : controlled outputs y R ny measured outputs (inputs of the controller)

85 O. Sename [GIPSA-lab] 83/162 Problem formulation 5 How to solve an H control problem? The Dynamic Output feedback case Let K(s) be a dynamic output feedback LTI controller defined as { ẋk (t) = A K(s) : K x K (t) + B K y(t), u(t) = C K x K (t) + D K y(t). where x K R n, and A K, B K, C K and D K are matrices of appropriate dimensions. Remark. The controller will be considered here of the same order (same number of state variables) n than the generalized plant, which here, in the H framework, the order of the optimal controller. With P (s) and K(s), the closed-loop system N(s) is: N(s) : { ẋcl (t) = A CL x cl (t) + B CL w(t), z(t) = C CL x cl (t) + D CL w(t), (23) where x T cl (t) = [ x T (t) x T K (t)] and ( ) A + B2 D A CL = K C 2 B 2 C K, B K C 2 A K ( ) B1 + B B CL = 2 D K D 21, B K D 21 C CL = ( ) C 1 + D 12 D K C 2, D 12 C K, D CL = B 1 + B 2 D K D 21. The aim is of course to find matrices A K, B K, C K and D K s.t. the H norm of the closed-loop system (23) is as small as possible, i.e. γ opt = min γ s.t. N(s) < γ.

86 O. Sename [GIPSA-lab] 84/162 5 How to solve an H control problem? The Riccati approach 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

87 O. Sename [GIPSA-lab] 85/162 5 How to solve an H control problem? The Riccati approach Asuumptions for the Riccati method A1: (A, B 2 ) stabilizable and (C 2, A) detectable: necessary for the existence of stabilizing controllers A2: rank(d 12 ) = n u and rank(d 21 ) = n y: Sufficient to ensure the controllers are proper, hence realizable ( ) A jωin B A3: ω R, rank 2 = n + n C 1 D u 12 ( A jωin B A4: ω R, rank 1 A5: C 2 D 21 ) = n + n y Both ensure that the optimal controller does not try to cancel poles or zeros on the imaginary axis which would result in CL instability D 11 = 0, D 22 = 0, D 12 T [ C 1 D 12 ] = [ 0 I nu ], [ ] [ B1 D D T 21 = 21 ] 0 I ny not necessary but simplify the solution (does correspond to the given theorem next but can be easily relaxed)

88 O. Sename [GIPSA-lab] 86/162 The problem solvability 5 How to solve an H control problem? The Riccati approach The first step is to check whether a solution does exist of not, to the optimal control problem. Theorem (1) Under the assumptions A1 to A5, there exists a dynamic output feedback controller u(t) = K(.) y(t) such that the closed-loop system is internally stable and the H norm of the closed-loop system from the exogenous inputs w(t) to the controlled outputs z(t) is less than γ, if and only if ( i the Hamiltonian H = imaginary axis. A γ 2 B 1 B1 T B 2 B2 T C1 T C 1 A T ) has no eigenvalues on the ii there exists X 0 t.q. A T X + X A + X (γ 2 B 1 B1 T B 2 B2 T ) X + CT 1 C 1 = 0, ( A iii the Hamiltonian J = T γ 2 C1 T C 1 C2 T C ) 2 B 1 B1 T has no eigenvalues on the A imaginary axis. iv there exists Y 0 t.q. A Y + Y A T + Y (γ 2 C T 1 C 1 C T 2 C 2) Y + B 1 B T 1 = 0, v the spectral radius ρ(x Y ) γ 2.

89 Controller reconstruction 5 How to solve an H control problem? The Riccati approach Theorem (2) If the necessary and sufficient conditions of the Theorem 1 are satisfied, then the so-called central controller is given by the state space representation [ ]  Z K sub (s) = L F 0 with  = A + γ 2 B 1 B T 1 X + B 2F + Z L C 2 F = B T 2 X, L = Y CT 2 Z = ( I γ 2 Y X ) 1 The Controller structure is indeed an observer-based state feedback control law, with u 2 (t) = B2 T X ˆx(t), where ˆx(t) is the observer state vector ) ˆx(t) = A ˆx(t) + B 1 ŵ(t) + B 2 u(t) + Z L (C 2 ˆx(t) y(t). (24) and ŵ(t) is defined as ŵ(t) = γ 2 B1 T X ˆx(t). Remark. ŵ(t) is an estimation of the worst case disturbance. Z L is the filter gain for the OE problem of estimating O. Sename [GIPSA-lab] ˆx(t) in the presence of the worst case disturbance 87/162

90 O. Sename [GIPSA-lab] 88/162 5 How to solve an H control problem? The LMI approach for H control design 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

91 O. Sename [GIPSA-lab] 89/162 5 How to solve an H control problem? The LMI approach for H control design The LMI approach for H control design- Solvability In this case only A1 is necessary. The solution is base on the use of the Bounded Real Lemma, and some relaxations that leads to an LMI problem to be solved [Scherer(1990)]. when we refer to the H control problem, we mean: Find a controller K for system P such that, given γ, F l (P, K) < γ (25) The minimum of this norm is denoted as γ and is called the optimal H gain. Hence, it comes, γ = min Tzw(s) (A K,B K,C K,D K )s.t.σa CL C (26) As presented in the previous sections, this condition is fulfilled thanks to the BRL. As a matter of fact, the system is internally stable and meets the quadratic H performances iff. P = P T 0 such that, AT CL P + PA CL PB CL CCL T BCL T P γ infty 2 I DCL T < 0 (27) C CL D CL I where A CL, B CL, C CL, D CL are given in (23). Since this inequality is not an LMI and not tractable for SDP solver, relaxations have to be performed (indeed it is a BMI), as proposed in [Scherer et al.(1997)scherer, Gahinet, and Chilali].

92 O. Sename [GIPSA-lab] 90/162 5 How to solve an H control problem? The LMI approach for H control design The LMI approach for H control design- Problem solution Theorem (LTI/H solution [Scherer et al.(1997)scherer, Gahinet, and Chilali]) [ ] AK B A dynamical output feedback controller of the form K(s) = K that solves the H C K D K control problem, is obtained by solving the following LMIs in (X, Y, Ã, B, C and D), while minimizing γ, M 11 ( ) T ( ) T ( ) T M 21 M 22 ( ) T ( ) T M 31 M 32 M 33 ( ) T 0 (28) [ M 41 M ] 42 M 43 M 44 X In 0 I n Y where, M 11 = AX + XA T + B 2 C + CT B T 2 M 21 = Ã + AT + C T 2 D T B T 2 M 22 = YA + A T Y + BC 2 + C2 T B T M 31 = B1 T + DT D T 21 B2 T M 32 = B1 T Y + D21 T B T M 33 = γ I nu M 41 = C 1 X + D 12 C M42 = C 1 + D 12 DC2 M 43 = D 11 + D 12 DD21 M 44 = γ I ny (29)

93 O. Sename [GIPSA-lab] 91/162 Controller reconstruction 5 How to solve an H control problem? The LMI approach for H control design Once A, B, C, D, X and Y have been obtained, the reconstruction procedure consists in finding non singular matrices M and N s.t. M N T = I X Y and the controller K is obtained as follows D K = D C K = ( C D cc 2 X)M T B K = N 1 (30) ( B YB 2 D c) A K = N 1 (Ã YAX YB 2D cc 2 X NB cc 2 X YB 2 C cm T )M T where M and N are defined such that MN T = I n XY (that can be solved through a singular value decomposition plus a Cholesky factorization). Remark. Note that other relaxation methods can be used to solve this problem, as suggested by [Gahinet(1994)].

94 O. Sename [GIPSA-lab] 92/162 6 Robust analysis 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

95 O. Sename [GIPSA-lab] 93/162 6 Robust analysis Introduction Introduction A control system is robust if it is insensitive to differences between the actual system and the model of the system which was used to design the controller How to take into account the difference between the actual system and the model? A solution: using a model set BUT : very large problem and not exact yet A method: these differences are referred as model uncertainty. The approach 1 determine the uncertainty set: mathematical representation 2 check Robust Stability 3 check Robust Performance Lots of forms can be derived according to both our knowledge of the physical mechanism that cause the uncertainties and our ability to represent these mechanisms in a way that facilitates convenient manipulation. Several origins : Approximate knowledge and variations of some parameters Measurement imperfections (due to sensor) At high frequencies, even the structure and the model order is unknown (100 Choice of simpler models for control synthesis Controller implementation Two classes: parametric uncertainties / neglected or unmodelled dynamics

96 O. Sename [GIPSA-lab] 94/162 6 Robust analysis Representation of uncertainties 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

97 O. Sename [GIPSA-lab] 95/162 Example 1: uncertainties 6 Robust analysis Representation of uncertainties Let consider the example from (Sokestag & Postlewaite, 1996). G(s) = k 1 + τs e sh, 2 k, h, τ 3 Let us choose the nominal parameters as, k = h = τ = 2.5 and G the according nominal model. We can define the relative uncertainty, which is actually referred as a MULTIPLICATIVE UNCERTAINTY, as G(s) = G(s)(I + W m(s) (s)) with W m(s) = 3.5s+0.25 s+1 and 1

98 Magnitude (db) O. Sename [GIPSA-lab] 96/162 Example 2: unmodelled dynamcis Let us consider the system: 1 G(s) = G 0 (s) 1 + τs, τ τmax This can be modelled as: G(s) = G 0 (s)(i + W m(s) (s)) with W m(s) = τmaxjω and 1 1+τ maxjω 6 Robust analysis Representation of uncertainties W m Bode Diagram (Greal-Gnom)/Gnom This can be represented as Frequency (rad/s)

99 Example 3: parametric uncertainties 6 Robust analysis Representation of uncertainties Consider the first order system: Define now: Then it leads: G(s) = 1 s + a, a 0 b < a < a 0 + b a = a 0 + δ.b with δ < 1 1 s + a = 1 s + a 0 + δ.b = 1 (1 + δ.b ) 1 s + a 0 s + a 0 This can then be represented as a Multiplicative Inverse Uncertainty: z = y = 1 s+a 0 (w bu ) with O. Sename [GIPSA-lab] 97/162

100 O. Sename [GIPSA-lab] 98/162 6 Robust analysis Representation of uncertainties Example 3 (cont.) same example with state space formulation Let us first the transfer function G(s) = 1 s+a as { ẋ = ( a0 δ.b)x + w G : z = x (31) In order to use an LFT, let us define the uncertain input: u = δx, Then the previous system can be rewritten in the following LFR: where and y are given as: = [ δ ], y = (x) and N given by the state space representation: ẋ = a 0 x bu + w N : y = x (3 z = x

101 O. Sename [GIPSA-lab] 99/162 6 Robust analysis Representation of uncertainties Example 4: parametric uncertainties in state space equations Let us consider the following uncertain system: ẋ 1 = ( 2 + δ 1 )x 1 + ( 3 + δ 2 )x 2 G : ẋ 2 = ( 1 + δ 3 )x 2 + u (33) y = x 1 In order to use an LFT, let us define the uncertain inputs: u 1 = δ 1 x 1, u 2 = δ 2 x 2, u 3 = δ 3 x 2 Then the previous system can be rewritten in the following LFR: where and y are given as: = δ δ 2 0, y = x 1 x δ 3 x 2 and N given by the state space representation: ẋ 1 = 2x 1 3x 2 + u 1 + u 2 N : ẋ 2 = x 2 + u + u 3 y = x 1 (3

102 O. Sename [GIPSA-lab] 100/162 Towards LFR (LFT) 6 Robust analysis Representation of uncertainties The previous computations are in fact the first step towards an unified representation of the uncertainties: the Linear Fractional Representation (LFR). Indeed the previous schemes can be rewritten in the following general representation as: Figure: N structure This LFR gives then the transfer matrix from w to z, and is referred to as the upper Linear Fractional Transformation (LFT) : F u(n, ) = N 22 + N 21 (I N 11 ) 1 N 12 This LFT exists and is well-posed if (I N 11 ) 1 is invertible

103 O. Sename [GIPSA-lab] 101/162 6 Robust analysis Representation of uncertainties LFT definition In this representation N is known and (s) collects all the uncertainties taken into account for the stability analysis of the uncertain closed-loop system. (s) shall have the following structure: (s) = diag { 1 (s),, q(s), δ 1 I r1,, δ ri rr, ɛ 1 I c1,, ɛ ci cc } with i (s) RH k i k i, δ i R and ɛ i C. Remark: (s) includes q full block transfer matrices, r real diagonal blocks referred to as repeated scalars (indeed each block includes a real parameter δ i repeated r i times), c complex scalars ɛ i repeated c i times. Constraints: The uncertainties must be normalized, i.e such that: 1, δ i 1, ɛ i 1

104 O. Sename [GIPSA-lab] 102/162 6 Robust analysis Representation of uncertainties Uncertainty types We have seen in the previous examples the two important classes of uncertainties, namely: UNSTRUCTURED UNCERTAINTIES: we ignore the structure of, considered as a full complex perturbation matrix, such that 1. We then look at the maximal admissible norm for, to get Robust Stability and Performance. This will give a global sufficient condition on the robustness of the control scheme. This may lead to conservative results since all uncertainties are collected into a single matrix ignoring the specific role of each uncertain parameter/block. STRUCTURED UNCERTAINTIES: we take into account the structure of, (always such that 1). The robust analysis will then be carried out for each uncertain parameter/block. This needs to introduce a new tool: the Structured Singular Value. We then can obtain more fine results but using more complex tools. The analysis is provided in what follows for both cases. In Matlabthis analysis is provided in the tools robuststab and robustperf.

105 O. Sename [GIPSA-lab] 103/162 6 Robust analysis Definition of Robustness analysis 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

106 O. Sename [GIPSA-lab] 104/162 6 Robust analysis Definition of Robustness analysis Robustness analysis: problem formulation Since the analysis will be carried you for a closed-loop system, N should be defined as the connection of the plant and the controller. Therefore, in the framework of the H control, the following extended General Control Configuration is considered: Figure: P K structure and N is such that N = F l (P, K)

107 O. Sename [GIPSA-lab] 105/162 Robust analysis: problem definition 6 Robust analysis Definition of Robustness analysis In the global P K General Control Configuration, the transfer matrix from w to z (i.e the closed-loop uncertain system) is given by: z = F u(n, )w, with F u(n, ) = N 22 + N 21 (I N 11 ) 1 N 12. and the objectives are then formulated as follows: Nominal stability (NS): N is internally stable Nominal Performance (NP): N 22 < 1 and NS Robust stability (RS): F u(n, ) is stable, < 1 and NS Robust performance (RP): F u(n, ) < 1, < 1 and NS

108 O. Sename [GIPSA-lab] 106/162 Summary of the methodology 6 Robust analysis Definition of Robustness analysis Uncertainty definition Parameters, neglected dynamics... Unstructured case Model type (additive, multiplicative...) Small Gain theorem Sensitivity function template Structured case Structured Small Gain theorem RS (µ r (N 11 ) < 1?) RP (µ (N) < 1?) RS ( S < 1/ W?) RP ( NP + RS < 1?)

109 O. Sename [GIPSA-lab] 107/162 6 Robust analysis Robustness analysis: the unstructured case 1. Some definitions The H norm definition Stability issues 2. Introduction to Linear Matrix Inequalities Background in Optimisation LMI in control Some useful lemmas 3. What is the H performance? H norm as a measure of the system gain? How to compute the H norm? What is H control? 4. Why H control is adapted to control engineering? Performance analysis and specification using the sensitivity functions: the SISO case Performance analysis and specification using the sensitivity functions: the MIMO case The mixed sensitivity H control design Some performance limitations 5. How to solve an H control problem? The Static State feedback case The Dynamic Output feedback case The Riccati approach The LMI approach for H control design 6. Uncertainty modelling and robustness analysis Introduction Representation of uncertainties Definition of Robustness analysis Robustness analysis: the unstructured case analysis: the structured case Robust control design 7. What else in H approach? H 2 and multi-objective problems H observer design 8. Introduction to LPV systems and control What is a Linear Parameter Varying systems? Classes (models) of LPV systems How to approximate a nonlinear system by an LPV one? Stability of LPV systems Control & Observation The Static State feedback case The Dynamic Output feedback case LPV observer design Summary of LPV approach interests

110 O. Sename [GIPSA-lab] 108/162 Towards Robust stability analysis 6 Robust analysis Robustness analysis: the unstructured case Robust Stability= with a given controller K, we determine wether the system remains stable for all plants in the uncertainty set. According to the definition of the previous upper LFT, when N is stable, the instability may only come from (I N 11 ). Then it is equivalent to study the M structure, given as: Figure: M structure This leads to the definition of the Small Gain Theorem Theorem (Small Gain Theorem) Suppose M RH. Then the closed-loop system in Fig. 7 is well-posed and internally stable for all RH such that : δ(resp. < δ) if and only if M(s) < 1/δ(resp. M(s) 1)

111 O. Sename [GIPSA-lab] 109/162 Definition of the uncertainty types 6 Robust analysis Robustness analysis: the unstructured case Figure: 6 uncertainty representations

112 O. Sename [GIPSA-lab] 110/162 Robust stability analysis: additive case 6 Robust analysis Robustness analysis: the unstructured case Objective: applying the Small Gain Theorem to these unstructured uncertainty representations. Let us consider the following simple control scheme as: The objective is to obtain: Figure: Control scheme with the uncertain system Additive case: G(s) = G(s) + W A (s) A (s). Computing the N form gives ( ) WA KS N(s) = y W A KS y S y T y Output Multiplicative uncertainties: G(s) = (I + W O (s) O (s))g(s). Then it leads ( ) WO T N(s) = y W O T y S y T y