Robust and LPV control of MIMO systems Part 3: Robustness analysis

Robust and LPV control of MIMO systems Part 3: Robustness analysis Olivier Sename GIPSA-Lab Tecnologico de Monterrey, July 2016 Olivier Sename (GIPSA-Lab) Robust and LPV control - part 3 Tecnologico de Monterrey, July 2016 1 / 28

O. Sename [GIPSA-lab] 2/28 1. Introduction 2. Representation of uncertainties 3. Definition of Robustness analysis 4. Robustness analysis: the unstructured case 5. Robustness analysis: the structured case 6. Robust control design

O. Sename [GIPSA-lab] 3/28 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

O. Sename [GIPSA-lab] 4/28 Example 1: uncertainties 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

Magnitude (db) O. Sename [GIPSA-lab] 5/28 Representation of uncertainties Example 2: unmodelled dynamcis Let us consider the system: 1 G(s) = G(s) 1 + τs, τ τmax This can be modelled as: 0-50 -100 W m Bode Diagram G(s) = G 0 (s)(i + W m(s) (s)) with W m(s) = τmaxjω and 1 1+τ maxjω -150-200 (Greal-Gnom)/Gnom This can be represented as -250 10-8 10-6 10-4 10-2 10 0 10 2 10 4 Frequency (rad/s) with [ ] N11 (s) N N(s) = 12 (s) N 21 (s) N 22 (s) ( = 0 I G 0 W m(s) G 0 (s) )

Representation of uncertainties Example 3: parametric uncertainties Consider the first order system: G(s) = 1 s + a, a 0 b < a < a 0 + b Define now: Then it leads: 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: with z = y = 1 (w bu s+a 0 ) O. Sename [GIPSA-lab] 6/28

O. Sename [GIPSA-lab] 7/28 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 (1) 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 z = x

O. Sename [GIPSA-lab] 8/28 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 (3) 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 1 0 0 0 δ 2 0, y = x 1 x 2 0 0 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

O. Sename [GIPSA-lab] 9/28 Representation of uncertainties Towards LFR (LFT) 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

O. Sename [GIPSA-lab] 10/28 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

O. Sename [GIPSA-lab] 11/28 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.

O. Sename [GIPSA-lab] 12/28 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)

O. Sename [GIPSA-lab] 13/28 Definition of Robustness analysis Robust analysis: problem definition 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

O. Sename [GIPSA-lab] 14/28 Definition of Robustness analysis Towards Robust stability analysis 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. 3 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)

O. Sename [GIPSA-lab] 15/28 Robustness analysis: the unstructured case Definition of the uncertainty types Figure: 6 uncertainty representations

O. Sename [GIPSA-lab] 16/28 Robustness analysis: the unstructured case Robust stability analysis: additive 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 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

O. Sename [GIPSA-lab] 17/28 General results Robustness analysis: the unstructured case Theorem (Small Gain Theorem) Consider the different uncertainty types, and assume that NS is achieved, i.e M RH for each type. Then the closed-loop system is robustly stable, i.e. internally stable for all k RH (for k =A, 0, I, io, ii) such that : Additive : W A KS y 1 Additive Inverse: W ia S y 1 Output Multiplicative: W O T y 1 Input Multiplicative: W I T u 1 Output Inverse Multiplicative: W io S y 1 Input Inverse Multiplicative: W ii S u 1 This gives some robustness templates for the sensitivity functions. However this may be conservative.

O. Sename [GIPSA-lab] 18/28 Robustness analysis: the unstructured case Illustration on the SISO case Here Robust Stability is analyzed through the Nyquist plot. For illustration, let us consider the case of Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e Then the loop transfer function is given as: G = G(I + W m m) L = GK = GK(I + W m m) = L + W ml m; According to the Nyquist theorem, RS is achieved the the closed-loop system is stable for any L should not encircle, i.e L should not encircle -1 for all uncertainties. According to the figure, a sufficient condition is then: W ml < 1 + L, ω WmL 1+L < 1, ω W mt < 1 ω

O. Sename [GIPSA-lab] 19/28 Robustness analysis: the unstructured case A first insight in Robust Performance Objective: applying the Small Gain Theorem to these unstructured uncertainty representations. Let us consider the following simple control scheme as: Case of Output Multiplicative uncertainties: G(s) = (I + W O (s) O (s))g(s). Computing the N form gives N(s) = = y [ N11 (s) ] N 12 (s) N 21 (s) N 22 (s) W es y W es ( WO T y W O T y ) Figure: Control scheme We wish to get: The objectives are then formulated as follows: NS: N is internally stable NP: W es y < 1 and NS RS: W O T y < 1 and NS RP: F u(n, ) < 1, < 1, Sufficient condition: NS and σ(w O T y) + σ(w es y) < 1, ω

Robustness analysis: the unstructured case Illustration on the SISO case Here Robust Performance is analyzed through the Nyquist plot. For illustration, let us consider the case of Multiplicative uncertainties (Input and Output case are identical for SISO systems), i.e Then the loop transfer function is given as: G = G(I + W m m) L = GK = GK(I + W m m) = L + W ml m; First NP is achieved when: W es < 1 ω, W e < 1 + L, ω. Therefore RP is achieved if W e S < 1, S, ω W e < 1 + L, L, ω Since 1 + L 1 + L W ml m, a sufficient condition is actually: W e + W ml < 1 + L, ω W es + W mt < 1, ω O. Sename [GIPSA-lab] 20/28

O. Sename [GIPSA-lab] 21/28 The structured case Robustness analysis: the structured case = {diag{ 1,, q, δ 1 I r1,, δ ri rr, ɛ 1 I c1,, ɛ ci cc } C k k } (5) with i C k i k i, δ i R, ɛ i C, where i (s), i = 1,..., q, represent full block complex uncertainties, δ i (s), i = 1,..., r, real parametric uncertainties, and ɛ i (s), i = 1,..., c, complex parametric uncertainties. Taking into account the uncertainties leads to the following General Control Configuration, (s) ω v r P (s) z r e v K(s) y Figure: General control configuration with uncertainties where.

O. Sename [GIPSA-lab] 22/28 Robustness analysis: the structured case The structured singular value To handle parametric uncertainties, we need to introduce µ, the structured singular value, defined as: Definition (µ) For M C n n, the structure singular value is defined as: µ (M) := 1 min{σ( ) :, det(i M) 0} In other words, it allows to find the smallest structured which makes det(i M ) = 0. Theorem (The structured Small Gain Theorem) Let M(s) be a MIMO LTI stable system and (s) a LTI uncertain stable matrix, (i.e. RH ). The system in Fig. 3 is stable for all (s) in (5) if and only if: ω R µ (M(jω)) 1, with M(s) := N zv(s) More generally both following statements are equivalent For µ R, N(s) and (s) belong to RH, and ω R, µ (M(jω)) µ the system represented in figure 3 is stable for any uncertainty (s) of the form (5) such that :

O. Sename [GIPSA-lab] 23/28 Robustness analysis: the structured case Build the whole control scheme

O. Sename [GIPSA-lab] 24/28 Robustness analysis: the structured case Introduction of a fictive block Usually only real parametric uncertainties (given in r) are considered for RS analysis. RP analysis also needs a fictive full block complex uncertainty, as below, (s) f r ω v r N(s) z r e Figure: N [ N11 (s) N where N(s) = 12 (s) N 21 (s) N 22 (s) ], and the closed-loop transfer matrix is: T ew(s) = N 22 (s) + N 21 (s) (s)(i N 11 (s)) 1 N 12 (s) (6)

O. Sename [GIPSA-lab] 25/28 Robust analysis theorem Robustness analysis: the structured case For RS, we shall determine how large (in the sense of H ) can be without destabilizing the feedback system. From (6), the feedback system becomes unstable if det(i N 11 (s) = 0 for some s C, R(s) 0. The result is then the following. Theorem ([?]) Assume that the nominal system N ew and the perturbations are stable. Then the feedback system is stable for all allowed perturbations such that (s) < 1/β if and only if ω R, µ (N 11 (jω)) β. Assuming nominal stability, RS and RP analysis for structured uncertainties are therefore such that: NP σ(n 22 ) = µ f (N 22 ) 1, ω RS µ r (N 11 ) < 1, ω RP µ (N) < 1, ω, = [ ] f 0 0 r Finally, let us remark that the structured singular value cannot be explicitly determined, so that the method consists in calculating an upper bound and a lower bound, as closed as possible to µ.

O. Sename [GIPSA-lab] 26/28 Robustness analysis: the structured case Summary The steps to be followed in the RS/RP analysis for structured uncertainties are then: Definition of the real uncertainties r and of the weighting functions Evaluation of µ(n 22 ) f, µ(n 11 ) r and µ(n) Computation of the admissible intervals for each parameter Remark: The Robust Performance analysis is quite conservative and requires a tight definition of the weighting functions that do represent the performance objectives to be satisfied by the uncertain closed-loop system. Therefore it is necessary to distinguish the weighting functions used for the nominal design from the ones used for RP analysis.

O. Sename [GIPSA-lab] 27/28 Robust design Outline 1. Introduction 2. Representation of uncertainties 3. Definition of Robustness analysis 4. Robustness analysis: the unstructured case 5. Robustness analysis: the structured case 6. Robust control design

O. Sename [GIPSA-lab] 28/28 Robust design Brief overview In order to design a robust control, i.e. a controller for which the synthesis actually accounts for uncertainties, some of the methods are: Unstructured uncertainties: Consider an uncertainty weight (unstructured form), and include the Small Gain Condition through a new controlled output. For example, robustness face to Ouptut Multiplicative Uncertainties can be considered into the design procedure adding the controlled output e y = W O y, which, when tracking performance is expected, leads to the condition W O T y 1. Structured uncertainties: the design of a robust controller in the presence of such uncertainties is the µ synthesis. It is handled through an interactive procedure, referred to as the DK iteration. This procedure is much more involved than a "simple" H control design and often leads to an increase of the order of the controller (which is already the sum of the order of the plant and of the weighting functions). Use other mathematical representation of parametric uncertainties, [?], as for instance the polytopic model. In that case the set of uncertain parameters is assumed to be a polytope (i.e. a convex) set. The stability issue in that framework is referred to as the Quadratic stability i.e find a single Lyapunov function for the uncertainty set. While in the general case this is an unbounded problem, in the polytopic case (or in the affine case), the stability is to be analyzed only at the vertices of the polytope, which is a finite dimensional problem. This approach can then be applied to find a single controller, valid over the potyopic set. Note that this approach gives rise to the LPV design for polytopic systems, as described next.