Nonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México

Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October 14, 2014 Cancún, Quintana Roo, México Jaime A. Moreno Nonlinear Observers 1 / 363

Outline 1 Introduction Formulation of the Basic Observation Problem Motivation Jaime A. Moreno Nonlinear Observers 1 / 363

The objective of this course, divided in 5 lectures, is to give an overview of this vast topic, ranging from the basic linear theory, passing through the ubiquitous Extended Kalman Filter and culminating with a description of several methods of observer design for nonlinear systems. The mathematical developments are complemented by examples taken from different fields. Their behavior in closed loop, under sensor noise and their robustness under model uncertainties and perturbations is also discussed. Jaime A. Moreno Nonlinear Observers 2 / 363

Outline 1 Introduction Formulation of the Basic Observation Problem Motivation Jaime A. Moreno Nonlinear Observers 2 / 363

Basic Observation Problem I Consider a dynamical system ẋ = f (t, x, u), x (t 0 )=x 0 y = h (t, x) where x is the state, u is the known input, y is the measured output. For this deterministic and noise-free system the basic observation problem has two main questions: Observability: It asks for the theoretical (mathematical) possibility of reconstructing x(t) : Given a time interval t 2 [t 0, t 0 + T ], T > 0, is it possible to determine uniquely the actual value of x (t)using the available information (u, y) (also knowing f, h)? Jaime A. Moreno Nonlinear Observers 3 / 363

Basic Observation Problem II Observer: Does it exist a (finite, infinite dimensional) dynamical system = g (t,,u, y), (t 0 )= 0 x (t) =k (t,,u, y) that (asymptotically, in finite time) estimates (exactly, approximately) the value of x (t)? If yes, how to construct it? Jaime A. Moreno Nonlinear Observers 4 / 363

Basic Observation Problem III These two questions are related and they have both to be considered. There are many variations of the observation problem beyond this basic formulation: with unknown inputs, practical observers, robust observers, stochastic framework to deal with noises,... Jaime A. Moreno Nonlinear Observers 5 / 363

Outline 1 Introduction Formulation of the Basic Observation Problem Motivation Jaime A. Moreno Nonlinear Observers 5 / 363

Motivation 1 Observability Problem is basic in systems theory mainly because of two reasons: 1 Its relationship to the construction of observers, and 2 The realization Problem 2 Estimation of the Internal States of the system for (observer-based) Output Feedback Control 3 Parameter Identification 4 Simultaneous State and Parameter estimation, i.e. Adaptive Observation, for adaptive control. 5 Estimation of unmeasured perturbations 6 Fault Detection and Isolation Problems 7 Fault Tolerant Control 8 Reducing the cost of control by replacing expensive sensors with software sensors Jaime A. Moreno Nonlinear Observers 6 / 363

Lecture organization Part I: Observability and Observers for Linear Systems 2 Linear Time-Varying Systems Examples 3 Observability State Reconstruction Examples A Rank Condition for Observability Particular Types of Observability Uniform Complete Observability (UCO) Instantaneous Observability Differential Observability Linear Time-Invariant Systems Detectability 4 Observers Design for Linear Systems The Kalman Filter (Observer) The Luenberger Observer 5 Separation Principle Jaime A. Moreno Nonlinear Observers 7 / 363

Part II: Linearization Methods for Nonlinear Systems 6 Observability of Nonlinear Dynamical Systems 7 Taylor Linearization about an Operating Point 8 Taylor Linearization about a Trajectory Convergence Analysis 9 Exact Error Linearization Method Existence Conditions for the Transformation Some Particular Cases Output Feedback Control Linearization by Immersion Systems Affine in the Unmeasured States 10 Approximate Observer Error Linearization Method 11 Nonlinear Luenberger Observer Jaime A. Moreno Nonlinear Observers 8 / 363

Part III: High Gain Observers 12 The Linear Case 13 The Basic Nonlinear Case: system without input 14 Uniformly (in the input) Observable Nonlinear Systems 15 Output Feedback: A Case Example 16 Stabilization 17 Some Extensions of the Basic High-Gain Theory Perturbed State Affine Systems Design by Immersion Jaime A. Moreno Nonlinear Observers 9 / 363

Part IV: Dissipative Observer Design Method 18 Introduction 19 Dissipativity Theory A General Overview Important Special Cases 20 Dissipative Observer Design Motivation Observer Design: A simple case Observer Design: An Extended Case 21 Computational Issues 22 Relations With Other Design Methods 23 A Dissipative Approach to Unknown Input Observers Introduction and Problem Statement UIOs for Linear Time Invariant Systems System and Observers Considered Existence Conditions for an UIO Proof Example 24 Conclusions Jaime A. Moreno Nonlinear Observers 10 / 363

Part V: Sliding Mode Observers 25 Discontinuous Control A First Order Plant A Second Order Plant 26 Observation Problems Basic Observation Problem Nonlinear systems without inputs A Simple Observer and its Properties Sliding Mode Observer (SMO) 27 Observers a la Second Order Sliding Modes (SOSM) Generalized Super-Twisting Observers 28 Lyapunov Approach for Second-Order Sliding Modes GSTA without perturbations: ALE Jaime A. Moreno Nonlinear Observers 11 / 363

Part I Observability and Observers for Linear Systems Jaime A. Moreno Nonlinear Observers 12 / 363

Outline 2 Linear Time-Varying Systems Examples 3 Observability State Reconstruction Examples A Rank Condition for Observability Particular Types of Observability Uniform Complete Observability (UCO) Instantaneous Observability Differential Observability Linear Time-Invariant Systems Detectability 4 Observers Design for Linear Systems The Kalman Filter (Observer) The Luenberger Observer 5 Separation Principle Jaime A. Moreno Nonlinear Observers 13 / 363

LTI Systems I Consider a Linear Time Varying (LTV) System ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t), where x (t) 2 R n, u (t) 2 R p, y (t) 2 R q. (A, B, C) are known and u (t) and y (t), measured. It has a unique solution for every input u (t) and every initial condition x (t 0 )=x 0 where x (t) = (t, t 0 ) x 0 + t t 0 (t, ) B ( ) u ( ) d, y (t) =C (t) (t, t 0 ) x 0 + C (t) t t 0 (t, ) B ( ) u ( ) d, (t, t 0 ) is the State Transition Matrix of ẋ (t) =A (t) x (t), x (t 0 )=x 0, so that x (t) = (t, t 0 ) x 0. Jaime A. Moreno Nonlinear Observers 14 / 363

LTI Systems II It has some important properties: 1 (t, t) =I 2 (t, t 1 ) (t 1, t 0 )= (t, t 0 ) for all t 0, t 1, t. 3 1 (t, t 0 )= (t 0, t) 4 @ (t,t 0 ) @t = A (t) (t, t 0 ) 5 @ (t 1, t) @t = (t 1, t) A (t) 6 If A is constant where e At = 1X k=0 (t, t 0 )=e A(t t 0), 1 k! Ak t k = I + At + 1 2 A2 t 2 + Jaime A. Moreno Nonlinear Observers 15 / 363

LTI Systems III Proof. We only prove (5). Due to (3) (t 1, t) (t, t 1 )=I, and therefore @ @t { (t 1, t) (t, t 1 )} = @ (t 1, t) @t = @ (t 1, t) @t From this last equality (5) follows. (t, t 1 )+ (t 1, t) @ (t, t 1) @t (t, t 1 )+ (t 1, t) A (t) (t, t 1 )=0 Jaime A. Moreno Nonlinear Observers 16 / 363

Note that the total response (solution) of the system y (t) =C (t) (t, t 0 ) x 0 + C (t) t t 0 (t, ) B ( ) u ( ) d consists of two components: 1 The zero input response 2 and the zero state response ' (t, t 0, x 0, 0) =C (t) (t, t 0 ) x 0 ' t, t 0, 0, u [t0,t] = C (t) t t 0 (t, ) B ( ) u ( ) d. Jaime A. Moreno Nonlinear Observers 17 / 363

Example Consider the system apple ẋ (t) =A (t) x (t), A (t) = 1 e 2t 0 1. The transition matrix is (this can be obtained by direct integration) apple e (t ) 1 (t, )= 2 e t+ e t+3 0 e (t ). Note that @ @t apple (t, )= apple = e (t ) 1 2 e t+ + e t+3 0 e (t ), apple 1 e 2t e (t ) 1 2 e t+ e t+3 0 1 0 e (t ), = A (t) (t, ). Jaime A. Moreno Nonlinear Observers 18 / 363

Example Consider the system apple (t) (t) ẋ (t) =A (t) x (t), A (t) = 0 (t), or ẋ 1 (t) = (t) x 1 (t)+ (t) x 2 (t), ẋ 2 (t) = (t) x 2 (t). Jaime A. Moreno Nonlinear Observers 19 / 363

By direct integration x 1 (t) =exp = exp ˆ t t 0 ˆ t ˆ t + exp t 0 x 2 (t) =exp ( ) d t 0 ˆ t ˆ t t 0 ( ) d!! ( ) d ( ) d! ˆ t x 1 (t 0 )+ exp t 0 x 1 (t 0 ),! ˆ exp t 0 x 2 (t 0 ). ˆ t ( ) d ( ) d! ( ) d x 2 (t 0 ), ( ) x 2 ( ) d, Jaime A. Moreno Nonlinear Observers 20 / 363

The transition matrix is 2 (t, t 0 )= 4 exp t t t t 0 ( ) d, t 0 exp d + t 0 d t 0 exp ( ) d t 0 ( ) d 3 5. In the particular case when, and are constant, the transition matrix simplifies to (t, t 0 )=e A(t " # t0) e =, e t t 0 t t 0 e ( ) d. 0 e (t t 0) Jaime A. Moreno Nonlinear Observers 21 / 363

Or, if 6= apple e (t t 0 ), (t, t 0 )= e (t t 0) e (t t 0) 0 e (t t 0) and if = apple (t, t 0 )=e (t t 0) 1, (t t0 ) 0 1. Jaime A. Moreno Nonlinear Observers 22 / 363

Example Consider the system ẋ (t) = (t) Ax (t), where A is a constant matrix and : R! R is a scalar function. The transition matrix is! (t, t 0 )=exp ˆ t t 0 ( ) d A. Note that @ @t (t, t 0 )= (t) A exp ˆ t t 0 = (t) A (t, t 0 ). ( ) d A!, Jaime A. Moreno Nonlinear Observers 23 / 363

apple If for example A = 0, with,, constant and 6= (t, t 0 )= 2 4 e t t 0 ( )d, e 0 e t t 0 ( )d t t 0 ( )d e t t 0 ( )d 3 5. Jaime A. Moreno Nonlinear Observers 24 / 363

Observability When is it possible to uniquely determine x 0 = x (t 0 ) from y (t) and u (t) in the time interval t 2 [t 0, t 1 ], t 1 > t 0? Definition The dynamical system is called observable on [t 0, t 1 ] iff, for all inputs u [t0, t 1 ] and all corresponding outputs y [t0, t 1 ] the state x 0 at time t 0 is uniquely determined. Note that, if the initial state x 0 is known, then the actual state can be calculated as ˆ t x (t) = (t, t 0 ) x 0 + t 0 (t, ) B ( ) u ( ) d. Jaime A. Moreno Nonlinear Observers 25 / 363

Unobservable subspace From the general solution one obtains where C (t) (t, t 0 ) x 0 = ỹ (t), 8t 2 [t 0, t 1 ], ỹ (t) =y (t) C (t) t t 0 (t, ) B ( ) u ( ) d is known and can be calculated without the knowledge of x 0. Definition Indistinguishable states. Given two times t 1 > t 0 0, and an input u [t0, t 1 ], two initial states x 1 0 6= x 2 0 are indistinguishable by input u on [t 0, t 1 ] if y 1 (t) =y 2 (t), 8t 2 [t 0, t 1 ]. Jaime A. Moreno Nonlinear Observers 26 / 363

Since in the linear case y 1 (t) =C (t) y 2 (t) =C (t) (t, t 0 ) x0 1 + C (t) t t 0 (t, t 0 ) x0 2 + C (t) t t 0 (t, ) B ( ) u ( ) d, (t, ) B ( ) u ( ) d, subtracting the two equalities, it follows that the initial states x 1 0, x 2 0 are indistinguishable if and only if C (t) (t, t 0 ) x 1 0 x 2 0 = 0, 8t 2 [t 0, t 1 ]. Note that if x 1 0, x 2 0 are indistinguishable with an input u [t 0, t 1 ], then they are indistinguishable with any input (for example u [t0, t 1 ] = 0). Jaime A. Moreno Nonlinear Observers 27 / 363

For linear systems all indistinguishable states can be characterized by Definition Unobservable subspace. Given two times t 1 > t 0 0, the unobservable subspace on [t 0, t 1 ], UO[t 0, t 1 ], consists of all states x 2 R n for which C (t) (t, t 0 ) x = 0, 8t 2 [t 0, t 1 ]. Given an 0 6= x 2UO[t 0, t 1 ], the two states x 0 and x 0 + x are indistinguishable on [t 0, t 1 ] for every input u [t0, t 1 ]. Jaime A. Moreno Nonlinear Observers 28 / 363

Theorem Given (C ( ), A ( )), with A ( ) and C ( ) piece-wise continuous, we have the following equivalences: x 0 can be determined uniquely from y [t0,t 1 ] and u [t0,t 1 ], i.e. the system is observable in the time interval t 2 [t 0, t 1 ] () The linear operator F : R n!y q [t 0,t 1 ] defined by is injective. C (t) (t, t 0 ) x, 8t 2 [t 0, t 1 ] () The unobservable subspace contains only the zero vector, i.e. () det M (t 0, t 1 ) 6= 0, where M (t 0, t 1 ), ˆ t1 UO[t 0, t 1 ]={0} t 0 T (t, t 0 ) C T (t) C (t) (t, t 0 ) dt.

Example Consider the system apple ẋ (t) =A (t) x (t), A (t) = 1 e 2t 0 1 y (t) =C (t) x (t), C (t) = 0, e t., The transition matrix (given in a previous example) apple e (t ) 1 (t, )= 2 e t+ e t+3 0 e (t ), so that C (t) (t, t 0 )= 0, e (2t t 0), Jaime A. Moreno Nonlinear Observers 30 / 363

and the observability Gramian M (t 0, t 1 )= ˆ t1 t 0 = e 2t 0 apple ˆ t1 t 0 0 e 2t e t 0 0, e 2t e t 0 dt, apple 0 0 0 e 4t dt = 1 4 e2t 0 apple 0 0 0 e 4t 1 e 4t 0. Since rank (M (t 0, t 1 )) = 1 < 2 = n the system is not observable. apple Note that the null space of M (t 0, t 1 ) is given by the vectors of the form, for 2 R, i.e. 0 all unobservable states are of this form. Note that y (t) =C (t) (t, t 0 ) x 0 = 0, e apple (2t t 0) x 0 = 0 for all x 0 = 0. apple That is, none of the unobservable states can be distinguished from the 0 zero state. Jaime A. Moreno Nonlinear Observers 31 / 363

The observability of the linear system is independent of the input u (t), i.e. if it is observable for u (t) =0, then it is observable for every input. The Observability Gramian M (t 0, t 1 ) depends on the interval [t 0, t 1 ]. Jaime A. Moreno Nonlinear Observers 32 / 363

Theorem The matrix M (t 0, t 1 ) has the following properties: 1 M (t 0, t 1 ) is symmetric 2 M (t 0, t 1 ) is nonnegative definite for t 1 t 0, i.e. for every 2 R n T M (t 0, t 1 ) 0. 3 M (t 0, t 1 ) satisfies the linear matrix differential equation d dt M (t, t 1)= A T (t) M (t, t 1 ) M (t, t 1 ) A (t) C T (t) C (t), M (t 1, t 1 )=0. 4 M (t 0, t 1 ) satisfies the functional equation M (t 0, t 1 )=M (t 0, t)+ T (t, t 0 ) M (t, t 1 ) (t, t 0 ). Jaime A. Moreno Nonlinear Observers 33 / 363

Proof. (1) follows immediately from the definition. To establish (2) we observe that for any real constant vector 2 R n we have T M (t 0, t 1 ) = = ˆ t1 t 0 T T (t, t 0 ) C T (t) C (t) (t, t 0 ) dt, ˆ t1 t 0 kc (t) (t, t 0 ) k 2 dt 0. Jaime A. Moreno Nonlinear Observers 34 / 363

Proof. To obtain (3) we merely calculate the derivative of M (t 0, t 1 ) with respect to its first argument. Using Leibnitz s rule we have d dt M (t, t 1)= d ˆ t1 dt = + ˆ t1 t t T (, t) C T ( ) C ( ) (, t) d, T (t, t) C T (t) C (t) d dt (t, t)+ T (, t) C T ( ) C ( ) (, t) d, = C T (t) C (t) A T (t) ˆ t1 t ˆ t1 t T (, t) C T ( ) C ( ) T (, t) C T ( ) C ( ) (, t) d A (t). (, t) d Rearranging the terms gives the differential equation. The boundary condition is obvious. Jaime A. Moreno Nonlinear Observers 35 / 363

Proof. To obtain (4) we expand the integral defining M as follows M (t 0, t 1 )= = + ˆ t1 t 0 T (, t 0 ) C T ( ) C ( ) (, t 0 ) d, ˆ t t 0 T (, t 0 ) C T ( ) C ( ) (, t 0 ) d + ˆ t1 t T (, t 0 ) C T ( ) C ( ) (, t 0 ) d, = M (t 0, t)+ T (t, t 0 ) ˆ t1 = M (t 0, t)+ T (t, t 0 ) M (t, t 1 ) (t, t 0 ). t T (, t) C T ( ) C ( ) (, t) d (t, t 0 ), Jaime A. Moreno Nonlinear Observers 36 / 363

Remark It follows (from (4) in the last Theorem) that if (C ( ), A ( )) is observable in the interval [t 0, t 1 ] then it is also observable in the interval [t 0, t 2 ] for every t 2 t 1. Remark For the Linear Time Varying (LTV) System ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t). where C (t) (t, t 0 ) x 0 = ỹ (t), 8t 2 [t 0, t 1 ]. ỹ (t), y (t) The measurement energy t 1 t 0 C (t) t t 0 (t, ) B ( ) u ( ) d. ỹ T (t) ỹ (t) dt is given by ˆ t1 t 0 ỹ T (t) ỹ (t) dt = x T 0 M (t 0, t 1 ) x 0.

Gramian-Based Reconstruction Consider the Linear Time Varying (LTV) System ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t). where Premultipying by (C (t) C (t) (t, t 0 ) x 0 = ỹ (t), 8t 2 [t 0, t 1 ]. ỹ (t), y (t) C (t) t t 0 (t, ) B ( ) u ( ) d. (t, t 0 )) T and integrating ˆ t1 t 0 T (t, t 0 ) C T (t) C (t) (t, t 0 ) dt x 0 = t 1 t 0 T (t, t 0 ) C T (t) ỹ (t) dt. Jaime A. Moreno Nonlinear Observers 38 / 363

Fact If the system is observable on [t 0, t 1 ] then x 0 = M 1 (t 0, t 1 ) t 1 t 0 T (t, t 0 ) C T (t) ỹ (t) dt. Jaime A. Moreno Nonlinear Observers 39 / 363

Example Consider the system apple (t) ẋ (t) =A (t) x (t), A (t) = 0 y (t) =Cx (t), C =[1, 0],, where is constant and (t) = ( 0 if 0 apple t apple T 1 if T apple t for some T > 0. The transition matrix is (from a previous example) " e (t, t 0 )= (t t0), e # (t t 0) t t 0 ( ) d, 0 e (t t 0) Jaime A. Moreno Nonlinear Observers 40 / 363

so that and the observability Gramian M (t 0, t 1 )= C (t, t 0 )=e (t t 0) h 1, ˆ t1 t 0 e 2 (t t 0) = e 2 t 0 " ˆ t1 e 2 t t 0 t 2 4 t 0 1 ( ) d t t 0 # h 1, 1 t t 0 ( ) d, ( ) d t t t 0 t 0 t t 0 i, ( ) d ( ) d ( ) d i dt, 2 3 5 dt. Jaime A. Moreno Nonlinear Observers 41 / 363

Consider the interval [t 0, t 1 ]=[0, t 1 ] with t 1 < T. In this case (t) =0 and 8 " # >< 1 1 0 2 M (t 0, t 1 )= e2 t 1 1 if 6= 0 0 0, >: t 1 if = 0 that is not full rank, so that the system is unobservable during this interval. Jaime A. Moreno Nonlinear Observers 42 / 363

Now consider the interval [t 0, t 1 ]=[0, t 1 ] with t 1 > T. In this case " t1 # t1 M (t 0, t 1 )= 0 e2 t dt T e2 t (t T ) dt t1 t1 T e2 t (t T ) dt, T e2 t (t T ) 2, dt apple m11 m = 12. m 12, m 22 Consider (for simplicity) the case = 0, so that Since m 11 = t 1, m 12 = 1 2 (t 1 T ) 2, m 22 = 1 3 (t 1 T ) 3. m 11 > 0, m 11 m 22 m 2 12 = 1 4 1 3 t 1 + T (t 1 T ) 3 > 0, it follows that M (0, t 1 ) is regular and the system is observable. Jaime A. Moreno Nonlinear Observers 43 / 363

The same result can be obtained for 6= 0, where m 11 = 1 2 e2 t 1 1, 1 e 2 t 1 m 12 = t 1 T 2 2 m 22 = t 1 T 1 2 + e2 T 2 + 1 4 2 4 2,! e 2 t 1 2 e 2 T 4 3. In this system one has to wait at least T units of time to be able to reconstruct the initial (or the actual) state. Jaime A. Moreno Nonlinear Observers 44 / 363

Example Consider the system apple 0 (t) ẋ (t) =A (t) x (t), A (t) = 0 0 y (t) =Cx (t), C =[1, 0],, where (t) = ( 1 if 0 apple t apple T 0 if T apple t for some T > 0. The transition matrix is (from a previous example) " # t 1, (t, t 0 )= t 0 ( ) d, 0 1 Jaime A. Moreno Nonlinear Observers 45 / 363

so that h C (t, t 0 )= 1, and the observability Gramian " M (t 0, t 1 )= = ˆ t1 t 0 ˆ t1 t 0 2 4 t t 0 1 ( ) d t 1 t t 0 ( ) d, t 0 # h 1, ( ) d t t t 0 t 0 t t 0 i, ( ) d ( ) d ( ) d i dt, 2 3 5 dt. Jaime A. Moreno Nonlinear Observers 46 / 363

Consider the interval [t 0, t 1 ] with t 0 < t 1 < T. In this case ˆ t1 apple 1 (t t 0 ) M (t 0, t 1 )= t 0 (t t 0 ), (t t 0 ) 2 dt, apple 1 1 =(t 1 t 0 ) 2 (t 1 t 0 ) 1 2 (t 1 1 t 0 ), 3 (t 1 t 0 ) 2, which is positive definite, and therefore the system is observable. Note that the system is observable for every interval [t 0, t 0 + ] for arbitrary (small) >0, if t 0 < T. Jaime A. Moreno Nonlinear Observers 47 / 363

Now consider the interval [t 0, t 1 ] with T < t 0 < t 1. In this case (t) =0 and M (t 0, t 1 )= apple t1 t 0, 0 0, 0, that is singular, so that the system is unobservable during this interval. This system becomes unobservable if we start to observe it late, i.e. t 0 > T. We can estimate the initial state with the first measurements, but if there is a perturbation that changes the initial state after time T, it will not be possible to estimate again the state! Jaime A. Moreno Nonlinear Observers 48 / 363

Example Consider the system apple 0 (t) ẋ (t) =A (t) x (t), A (t) = 0 0 y (t) =Cx (t), C =[1, 0],, where (t) =e kt, k > 0. The transition matrix is (from a previous example) " # t apple 1, (t, t 0 )= t 0 e k d 1, 1 = k e kt e kt 0 0 1 0 1, so that C (t, t 0 )= 1, 1 k e kt e kt 0, Jaime A. Moreno Nonlinear Observers 49 / 363

and the observability Gramian " M (t 0, t 1 )= ˆ t1 ˆ = apple = t 0 0 " 1 1 k e kt e kt 0 1 k e kt e kt 0 1, e kt e kt k 2 0 2 # dt, 1 1 k e kt 0 e k 1 1 k e kt 0 e k 1 1, e 2kt k 2 0 e k 1 2 m 12 e kt 0 m 12 e kt 0, m22 e 2kt 0 = t t 0, =t 1 t 0, # d, where m 12 = 1 k 1 k 1 e k, m 22 = 1 k 2 1 2k e 2k + 2 1 k e k + 1 2k 2 1 k +. Jaime A. Moreno Nonlinear Observers 50 / 363

The eigenvalues of M (t 0, t 0 + whose roots are 1,2 = = apple p (s) =det )are the roots of the characteristic polynomial s m 12 e kt 0 m 12 e kt 0, s m22 e 2kt 0, = s 2 +m 22 e 2kt 0 s + m 22 m 2 12 e 2kt 0, +m 22 e 2kt 0 ± q +m 22 e 2kt 0 ± q ( + m 22 e 2kt 0 ) 2 2 4 m 22 m12 2 e 2kt 0, ( m 22 e 2kt 0 ) 2 + 4m12 2 e 2kt 0. 2 Jaime A. Moreno Nonlinear Observers 51 / 363

Fixing k > 0 and the time window length =t 1 t 0, the eigenvalues of M lim t 0!1 1 = 0, lim t 0!1 2 =. This means that M becomes very badly conditioned as time evolves, i.e. the system becomes less and less observable with time. Jaime A. Moreno Nonlinear Observers 52 / 363

A Rank Condition for Observability Checking the observability of a LTV system by means of the Observability Gramian is not a simple task, since it requires the knowledge of the transition matrix (t, t 0 ), that is seldom available. This is only possible (in most cases) numerically, or also solving the differential equations satisfied by the Gramians. However, if we assume smoothness properties stronger than continuity (or piecewise continuity) for the coefficient matrices, the Gramian condition leads to a sufficient condition that is easier to check. [72, Chapter 9], [69, 51], [70, Chapter 6], [20, Chapter 6] Jaime A. Moreno Nonlinear Observers 53 / 363

Consider the Linear Time Varying (LTV) System (without input) ẋ (t) =A (t) x (t), x (t0 )=x L : 0, y (t) =C (t) x (t). Assume that for some positive integer r matrix C (t) is r-times and matrix A (t) is (r 1)-times continuously differentiable for t 2 [t 0, t 1 ]. In this case y (t) =C (t) (t, t 0 ) x 0, 8t 2 [t 0, t 1 ] and y (t) is r-times continuously differentiable. For a 2 [t 0, t 1 ] the Taylor series of y (t) is y (t) =y ( )+ẏ ( )(t )+ +y (k) ( ) (t )k k! + +y (r) ( ) (t where y (k) (t) is the k-th time derivative and R r+1 is the rest. )r r! +R r+1, Jaime A. Moreno Nonlinear Observers 54 / 363

Definition Subject to the existence of the indicated derivatives, define a sequence of matrices L 0 (t), C (t), L 1 (t), L 0 (t) A (t)+ L 0 (t), L j (t), L j 1 (t) A (t)+ L j 1 (t), j = 1, 2, Jaime A. Moreno Nonlinear Observers 55 / 363

Since y (t) =C (t) (t, t 0 ) x 0 = L 0 (t) (t, t 0 ) x 0, ẏ (t) = Ċ (t)+c(t) A (t) (t, t 0 ) x 0 = L 1 (t) (t, t 0 ) x 0, y (2) (t) =L 2 (t) (t, t 0 ) x 0,. y (k) (t) =L k (t) (t, t 0 ) x 0. Jaime A. Moreno Nonlinear Observers 56 / 363

we have the mapping 2 6 4 y ( ) ẏ ( ). y (r) ( ) 3 2 7 5 = 6 4 L 0 ( ) L 1 ( ). L r ( ) 3 7 5 (,t 0) x 0. Jaime A. Moreno Nonlinear Observers 57 / 363

We obtain then the following result Theorem Under the stated assumptions, if for some positive integer r, and some 2 [t 0, t 1 ] the Observability matrix 2 O r ( ), 6 4 L 0 ( ) L 1 ( ). L r ( ) is injective (, rank (O r )=n), then the system is observable on [t 0, t 1 ]. 3 7 5 Jaime A. Moreno Nonlinear Observers 58 / 363

Proof. O r ( ) injective implies that the coefficients in the Taylor series of the output y (t) are different for different initial conditions, i.e. the map x 0! C (t) (t, t 0 ) x 0 is injective, or equivalently, the system is observable in the interval. When C (t) and A (t) are analytic, then the condition is also necessary for an arbitrary 2 [t 0, t 1 ]. Jaime A. Moreno Nonlinear Observers 59 / 363

Example Consider (again) the system apple 0 (t) ẋ (t) =A (t) x (t), A (t) = 0 0 y (t) =Cx (t), C =[1, 0],, where (t) =e kt, k > 0. The observability matrix is apple C (t) O 1 (t) = Ċ (t)+c(t) A (t) = apple 1 0 0 (t). This matrix has full rank, rank (O n 1 )=n for every t. This implies that the system is differentially observable for every t. Note, however, that the larger the time t the weaker becomes the observability (see example above)! Jaime A. Moreno Nonlinear Observers 60 / 363

Uniform Complete Observability In order to avoid the loss of the observability, the following observability concept was introduced by Kalman [41]: Definition The pair (C (t), A (t)) is Uniformly Completely Observable (UCO) if there exist constants 1, 2, >0 such that for all t 0 0 1I apple M (t 0, t 0 + ) = ˆ t0 + t 0 T (t, t 0 ) C T (t) C (t) (t, t 0 ) dt apple 2 I. Note: The inequality between symmetric matrices Q i = Q T i, i = 1, 2 Q 1 apple Q 2 () Q 2 Q 1 0 () Q 2 Q 1 is a positive semidefinite matrix. Jaime A. Moreno Nonlinear Observers 61 / 363

Instantaneous Observability Systems for which the state can be estimated in an arbitrary small time window are particularly well suited to design observers that converge arbitrarily fast. Definition The pair (C (t), A (t)) is (Uniformly) Instantaneously Observable if for every positive constant >0 there exist constants 1, 2 > 0 such that for all t 0 0 1I apple M (t 0, t 0 + ) = ˆ t0 + t 0 T (t, t 0 ) C T (t) C (t) (t, t 0 ) dt apple 2 I. As we will see below, an example of Instantaneously observable systems are Linear Time Invariant ones, i.e. with constant matrices (C, A). Jaime A. Moreno Nonlinear Observers 62 / 363

Differential Observability The previous discussion motivates the following definition Definition Assume that matrix C (t) is n-times and matrix A (t) is (n 1)-times continuously differentiable for t 2 [t 0, t 1 ]. System (or the pair) (C (t), A (t)) is differentially observable (totally observable) on [t 0, t 1 ] if O n 1 ( ) is injective (, rank (O n 1 )=n) for every (almost all) 2 [t 0, t 1 ]. Differential observability is a strong property. It implies instantaneous observability Theorem If System (or the pair) (C (t), A (t)) is differentially observable on [t 0, t 1 ] then it is instantaneously observable on [t 0, t 1 ]. Jaime A. Moreno Nonlinear Observers 63 / 363

Proof. From differential observability follows that for every 2 [t 0, t 1 ] 2 3 y ( ) ẏ ( ) Y n 1 ( ), 6 4 7. 5 = O n 1 ( ) (,t 0 ) x 0. y (n 1) ( ) Premultiplying by O T n 1 ( ) and solving for x 0 one obtains O T n 1 ( ) O n 1 ( ) (,t 0 ) 1 O T n 1 ( ) Y n 1 ( ) =x 0. Jaime A. Moreno Nonlinear Observers 64 / 363

Linear Time-Invariant Systems If the system is Linear and Time Invariant (LTI) (consider u (t) =0 without loss of generality) ẋ (t) =Ax (t), x (0) =x 0, y (t) =Cx (t) with A, C constant. By differentiating the output w.r.t. time and evaluating at t = 0 2 6 4 y (0) ẏ (0) ÿ (0). y (k) (0) 3 2 C CA CA 2 3 = x 7 6 7 0. 5 4. 5 CA k {z } O k Jaime A. Moreno Nonlinear Observers 65 / 363

From the injectivity of a linear map x 0 can be uniquely determined if and only if for some value of k Matrix O k is injective () rank (O k )=n From the Cayley-Hamilton Theorem it suffices to check k = n 1. Jaime A. Moreno Nonlinear Observers 66 / 363

Theorem Observability of the LTI system is equivalent to each one of the following: 1 2 rank (O n 2 1 )=rank 6 4 C CA () The system is differentially observable. M (t 1, t 0 )= ˆ t1 t 0. CA n 1 3 7 5 = n e A T (t t 0 ) C T Ce A(t t0) dt is positive definite for every t 1 > t 0. () System is infinitesimally observable. 3 Hautus-Belevitch-Popov Criterion apple si rank C A = n, 8s 2 C.

Detectability Detectability is a weaker condition than observability, that is fullfilled in a continuous-time LTI if and only if all the unobservable modes are stable [35]. Definition The LTI system is detectable if y(t) =0 for t > 0 =) x(t)! 0 as t!1. Theorem (Conditions for Detectability) A continuous-time LTI system, where A 2 R n, is detectable if si A rank = n, 8 s 2 C with Re(s) 0. C Theorem (Implications on Observers Design) A LTI system has an asymptotic observer, i.e. the estimation error is assymptotically stable, if and only if it is detectable. Jaime A. Moreno Nonlinear Observers 68 / 363

Observers Design for Linear Systems I In the Linear case ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t). there are two main classes of observers: Kalman and Luenberger. Jaime A. Moreno Nonlinear Observers 69 / 363

A deterministic look at the KF I We follow a deterministic approach to derive a state estimator (observer), that brings us quite close to the true stochastic Kalman Filter. We consider for simplicity that u = 0. An alternative approach is the Minimum Energy Estimation (see [37, Lecture 23.2]). For the LTV system L one has that C ( ) (, t) x (t) =y ( ) 8 2 [t 0, t]. Premultipying by (C ( ) (, t)) T and integrating we obtain where N (t 0, t) ˆx (t) = t t 0 T (, t) C T ( ) y ( ) d, N (t 0, t) = t t 0 T (, t) C T ( ) C ( ) (, t) d. When t > t 0 is such that the Constructibility Gramian is invertible, then ˆx (t) =N 1 (t 0, t) t t 0 T (, t) C T ( ) y ( ) d. Jaime A. Moreno Nonlinear Observers 70 / 363

A deterministic look at the KF II We construct a recursive version of this algorithm as follows: 1 Find a recursive expression (Differential Equation (DE)) for N (t 0, t): d dt N (t 0, t) = A T (t) N (t 0, t) N (t 0, t) A (t)+c T (t) C (t), N (t 0, t 0 )=0. 2 Find a recursive expression (DE) for the estimate z (t), N (t 0, t) ˆx (t) : ˆ t T (t, t 0 ) z (t) = T (t, t 0 ) T (, t) C T ( ) y ( ) d, = = t 0 ˆ t t 0 ( (, t) (t, t 0 )) T C T ( ) y ( ) d, ˆ t t 0 T (, t 0 ) C T ( ) y ( ) d. Taking the derivative with respect to t we obtain (A (t) (t, t 0 )) T z (t)+ T (t, t 0 ) ż (t) = T (t, t 0 ) C T (t) y (t) Jaime A. Moreno Nonlinear Observers 71 / 363

A deterministic look at the KF III and therefore T (t, t 0 ) ż (t) = T (t, t 0 ) A T (t) z (t)+ T (t, t 0 ) C T (t) y (t) or, since T (t, t 0 ) is invertible ż (t) = A T (t) z (t)+c T (t) y (t), z (t 0 )=0. 3 Find a recursive expression for N 1 (t 0, t): Taking the time derivative of the identity (valid for a t such that N (t 0, t) is invertible!) we obtain N 1 (t 0, t) N (t 0, t) =I, dn 1 (t 0, t) N (t 0, t)+n 1 (t 0, t) dn (t 0, t) = 0. dt dt Jaime A. Moreno Nonlinear Observers 72 / 363

A deterministic look at the KF IV Using the recursive expression for N it follows that dn 1 (t 0, t) = N 1 (t 0, t) dn (t 0, t) N 1 (t 0, t), dt dt = N 1 ( ) A T (t) N ( ) N ( ) A (t)+c T (t) C (t) N 1 ( ), = N 1 ( ) A T (t)+a (t) N 1 ( ) N 1 ( ) C T (t) C (t) N 1 ( ). 4 Find a recursive expression for ˆx (t): with z (t), N (t 0, t) ˆx (t) and therefore N (t 0, t) d ˆx (t) dt d ˆx (t) ż (t) =N (t 0, t) + dn (t 0, t) ˆx (t), dt dt = A T (t) z (t)+c T (t) y (t), + A T (t) N (t 0, t) N (t 0, t) A (t)+c T (t) C (t) ˆx (t), = A T (t) N (t 0, t) ˆx (t)+c T (t) y (t). Jaime A. Moreno Nonlinear Observers 73 / 363

A deterministic look at the KF V This simplifies to d ˆx (t) dt = A (t) ˆx (t) N 1 (t 0, t) C T (t)(c (t) ˆx (t) y (t)). Jaime A. Moreno Nonlinear Observers 74 / 363

In synthesis, we get the recursive estimator: (Defining (t), N 1 (t 0, t)) d ˆx (t) = A (t) ˆx (t) (t) C T (t) {C (t) ˆx (t) y (t)}, dt d (t) = (t) A T (t)+a(t) (t) (t) C T (t) C (t) (t). dt or, alternatively, (Defining (t), N (t 0, t)) d ˆx (t) = A (t) ˆx (t) 1 (t) C T (t) {C (t) ˆx (t) y (t)}, dt d (t) = A T (t) (t) (t) A (t)+c T (t) C (t). dt (t) satisfies a nonlinear Riccatti Differential Equation, (t) satisfies a linear Lyapunov Differential Equation. Jaime A. Moreno Nonlinear Observers 75 / 363

The Kalman Filter I For the system (with noise) ẋ (t) =A (t) x (t)+b(t) u (t)+v (t), E {x (t0 )} = 0, Ln : y (t) =C (t) x (t)+w (t), where E {x (t 0 )} = 0, n o E [x (t 0 ) E {x (t 0 )}][x (t 0 ) E {x (t 0 )}] T = P 0 > 0, Jaime A. Moreno Nonlinear Observers 76 / 363

The Kalman Filter II and v (t), w (t) are white Gaussian noises with E {v (t)} = 0, E v (t) v T ( ) = Q (t) (t ) Q (t) =Q T (t) 0, E {w (t)} = 0, E w (t) w T ( ) = R (t) (t ) R (t) =R T (t) > 0, E x (t 0 ) v T (t) = 0, E x (t 0 ) w T (t) = 0, E v (t) w T ( ) = 0, the Kalman (Filter) Observer is d dt ˆx (t) =A (t) ˆx (t)+b(t) u (t)+h (t)(y (t) C (t) ˆx (t)), ˆx (t 0)=0, d dt P (t) =P (t) AT (t)+a(t) P (t) P (t) C T (t) R 1 (t) C (t) P (t)+q(t),, H (t) =P (t) C T (t) R 1 (t), P (t 0 )=P 0 > 0, Jaime A. Moreno Nonlinear Observers 77 / 363

The Kalman Filter III where P (t) =E n[x o (t) ˆx (t)] [x (t) ˆx (t)] T is the estimation error covariance matrix, that satisfies a Differential Riccati Equation. Note the observer s structure: Plant s model and Output Injection Jaime A. Moreno Nonlinear Observers 78 / 363

Riccati Differential Equation Let Q (t) and R (t) be symmetric, positive definite and bounded matrices, i.e. 0 < q 1 I apple Q (t) < q 2 I, 0 < r 1 I apple R (t) < r 2 I, and let P (t) the (symmetric) solution of the RDE, P (t 0 )=P 0 > 0 d dt P (t) =P (t) AT (t)+a (t) P (t) P (t) C T (t) R 1 (t) C (t) P (t)+q (t). If (A (t), C (t)) es UCO, then P (t) exists for all t t 0 and it satisfies 0 < p 1 I apple P (t) < p 2 I =) 0 < p 3 I apple P 1 (t) < p 4 I. Jaime A. Moreno Nonlinear Observers 79 / 363

Convergence of the Kalman Filter I The (filtered) estimation error satisfies the differential equation e = ˆx d dt e (t) =[A (t) H (t) C (t)] e (t)+ (t), e (t 0)=e 0, d dt P (t) =P (t) AT (t)+a(t) P (t) P (t) C T (t) R 1 (t) C (t) P (t)+q(t), H (t) =P (t) C T (t) R 1 (t), P (t 0 )=P 0 > 0, (t), H (t) w (t) v (t). Consider the Lyapunov Function Candidate V (t, e (t)) = e T (t) P 1 (t) e (t), that is positive definite and decrescent if the pair (A (t), C (t)) is UCO, i.e. p 3 ke (t)k 2 apple V (t, e (t)) = e T (t) P 1 (t) e (t) apple p 4 ke (t)k 2. x Jaime A. Moreno Nonlinear Observers 80 / 363

Convergence of the Kalman Filter II Its time derivative along the trajectories is V = ė T P 1 e + e T P 1 ė + e T d dt P 1 e. Recalling that it follows that d dt P 1 (t) = P 1 (t) dp (t) P 1 (t). dt V = e T (A HC) T P 1 e + e T P 1 (A HC) e + e T P 1 + T P 1 e e T P 1 dp dt P 1 e, V = e T (A HC) T P 1 + P 1 (A HC) P 1 dp dt P 1 e+ + 2e T P 1, Jaime A. Moreno Nonlinear Observers 81 / 363

Convergence of the Kalman Filter III V = e T P 1 P (A HC) T +(A HC) P + 2e T P 1, dp dt P 1 e+ V = e T P 1 P A PC T R 1 C T + A PC T R 1 C P + 2e T P 1, dp dt P 1 e+ V = e T P 1 PA T + AP PC T R 1 CP PC T R 1 CP + 2e T P 1, dp dt P 1 e+ V = e T P 1 dp dt + 2e T P 1, PA T AP + PC T R 1 CP P 1 e e T C T R 1 Ce+ Jaime A. Moreno Nonlinear Observers 82 / 363

Convergence of the Kalman Filter IV = e T P 1 QP 1 e e T C T R 1 Ce + 2e T P 1, apple c 1 kek 2 2 + 2c 2 kek 2 k k 2. It follows that the observation error is Input-to-State Stable (ISS) with respect to : if = 0 then e = 0 is a globally, uniformly, asymptotically stable (GUAS) equilibrium point. When k k!0 then kek!0 and if is bounded then e will be bounded. Jaime A. Moreno Nonlinear Observers 83 / 363

The Original Idea Of Luenberger (See [53, 54]). For simplicity we consider a Linear Time-Invariant system without input, with x 2 R n ẋ (t) =Ax (t), x (0) =x0, L : y (t) =Cx (t). Consider a different linear system with state z 2 R m driven by the output y of the plant L O : ż (t) =Fz (t)+hy (t), z (0) =z 0, where F is a Hurwitz matrix, i.e. all its eigenvalues have negative real parts. We look for a matrix T 2 R m n such that lim (z (t) Tx (t)) = 0. t!1 Jaime A. Moreno Nonlinear Observers 84 / 363

To achieve this, set e = z Tx, with dynamics If the matrix equation ė = Fz + HCx TAx = Fe +(FT TA + HC) x. TA = FT + HC (1) then ė = Fe and lim t!1 e (t) =0, and O estimates asymptotically Tx (t), a linear function of x (t). Note that (1) is linear in T. Given A, C, F, H for arbitrary m, n (not necesarily equal), there exists a unique solution T to the equation (1) if A and F have different eigenvalues. Thus any system O, having different eigenvalues from A, is an observer for L, i.e. it estimates certain linear function Tx of the plant s state x. This result is useful to construct observers that estimate linear functions of the state, or to construct reduced order observers (m < n). If we are interested in estimating the full state x, then it is required that the matrix T that solves (1) be injective (left invertible), so that from z it is possible to recover x. Jaime A. Moreno Nonlinear Observers 85 / 363

When m = n this is equivalent to having T invertible (regular). In this case there is an invertible solution to (1) if (A, C) is observable and (F, H) is controllable. An interesting case is to ask that T = I (the identity matrix). This is called the identity observer. Jaime A. Moreno Nonlinear Observers 86 / 363

Plant: Proposed Observer: Condition: ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t). L : d dt ˆx (t) =F (t) ˆx (t)+g (t) u (t)+h (t) y (t), ˆx (t 0)=ˆx 0. for all x (t 0 ) and ˆx (t 0 ). For the estimation error e (t) =ˆx (t) lim (ˆx (t) x (t)) = 0 t!1 x (t) we have ė (t) =F (t) e (t)+[g (t) B (t)] u (t)+[h (t) C (t) x (t) A (t)+f (t)] x (t), Setting F (t) =A (t) H (t) C (t), G (t) =B (t), Jaime A. Moreno Nonlinear Observers 87 / 363

the estimation error converges to zero if F (t) is asymptotically stable. So the observer is d L : ˆx dt (t) =A (t) ˆx (t)+b(t) u (t)+h (t)[y (t) ŷ (t)], ˆx (t 0)=ˆx 0, ŷ (t) =C (t) ˆx (t), i.e. a copy of the plant and an output injection. Design of the output injection gain H (t): convergence of the estimation error ė (t) =[A (t) H (t) C (t)] e (t), e (t 0 )=ˆx (t 0 ) x (t 0 ). (2) Design of H (t) is the dual of the control problem. If H (t) adequately designed, then the algorithm is also robust, that is one advantage of using feedback in the observer. For Linear Time Invariant systems design H such that (A Pole assignement (e.g. Ackermann s formula) Solving an LQR Problem (an stationary Kalman Filter) AP + PA T PC T R 1 CP + Q = 0, an Algebraic Riccati Equation, with P > 0. H = PC T R 1, HC) Hurwitz. Jaime A. Moreno Nonlinear Observers 88 / 363

Consider the LTI system with scalar output (for simplicity without input) ẋ (t) =Ax (t), x (0) =x0, L : y (t) =Cx (t). and assume that it is observable, i.e. the (square n n) Kalman observability matrix 2 3 C CA O = 6 7 4. 5 CA n 1 has full rank. To design an observer by pole assignement we first bring the system to the observer form, what is possible if and only if the system is observable. Jaime A. Moreno Nonlinear Observers 89 / 363

Observer Form Consider a state transformation z = Tx, with T an invertible square matrix. In these coordinates the system dynamics becomes ż (t) =TAT L : 1 z (t), z (0) =Tx 0, y (t) =CT 1 z (t). Jaime A. Moreno Nonlinear Observers 90 / 363

Theorem The (single output) pair (A, C) is transformable by a state transformation to the Observer Form (A o, C o )= TAT 1, CT 1, with 2 3 0 0 0 a 0 1 0 0 a 1 A o = 6 7 4..... 5, C o = 0 0 0 1, 0 0 1 a n 1 where a i,i= 0, 1,, n polynomial of A, i.e. 1, are the coefficients of the characteristic det (si A) =s n + a n 1 s n 1 + + a 1 s + a 0, if and only if the Kalman observability matrix is regular. Jaime A. Moreno Nonlinear Observers 91 / 363

Proof I Sufficiency. Let the vector r be the unique solution to the linear equation 2 3 2 3 2 3 C Cr 0 CA 7 6 CAr 7 6 0 7 that is Or = 6 4. CA n 1 By the Cayley-Hamilton Theorem 7 5 r = 6 4 2 r = O 1 6 4. CA n 0 0. 1 3 7 5 1 r 7 5 = 6 4 A n = a n 1 A n 1 a 1 A a 0 I.. 1 7 5 Jaime A. Moreno Nonlinear Observers 92 / 363

Proof II The definition of r implies that CA i r = 0, 0 apple i apple n 2, CA n 1 r = 1, which in turn implies that the matrix 2 3 C CA N = 6 7 r Ar A n 1 r, 4. 5 CA n 1 2 Cr CAr CA n 1 3 2 3 r 0 0 1 = 6.... 7 4 CA n 2 r CA n 1 r CA 2n 3 r 5 = 6.... 7 4 0 1? 5. CA n 1 r CA n r CA 2n 2 r 1?? Jaime A. Moreno Nonlinear Observers 93 / 363

Proof III is regular. Since O is nonsingular it follows that the n n matrix T 1 = r Ar A n 1 r is also nonsingular. In the new coordinates z = Tx the system is in the observer form. For example and CT 1 = C r Ar A n 1 r = Cr CAr CA n 1 r = C o TAT 1 = A O, Ar A 2 r A n r = r Ar A n 1 r A o what is true. Necessity: Since observability is invariant under a state transformation, and the pair (A O, C O ) is observable, then the pair (A, C) has to be observable. Jaime A. Moreno Nonlinear Observers 94 / 363

Pole Assignement For the system in Observer Form ż (t) =AO z (t), z (0) =z L : 0, y (t) =C O z (t), consider the observer ẑ (t) =A O : O ẑ (t)+h (y ŷ), ẑ (0) =ẑ 0, ŷ (t) =C O ẑ (t), where H = T h 0 h 1 h n 1 2 R n is a column vector. The dynamics of the estimation error e = ẑ z is given by Jaime A. Moreno Nonlinear Observers 95 / 363

2 3 0 0 0 (a 0 + h 0 ) 1 0 0 (a 1 + h 1 ) ė =(A O HC O ) e = 6 7 4..... 5 e. 0 0 1 (a n 1 + h n 1 ) Since the characteristic polynomial of the matrix (A O HC O ) is given by det (si A O + HC O )=s n +(a n 1 + h n 1 ) s n 1 + +(a 1 + h 1 ) s +(a 0 + h 0 ), these coefficients can be assigned arbitrarily by selection of the vector H, and therefore the eigenvalues (poles) of the convergence dynamics of the observer can be freely assigned. Jaime A. Moreno Nonlinear Observers 96 / 363

Separation Principle I Plant: ẋ (t) =A (t) x (t)+b(t) u (t), x (t0 )=x L : 0, y (t) =C (t) x (t). State Feedback u = K (t) x (t)+v (t) such that the origin of the closed loop system is GUAS (Globally Uniformly Asymptotically Stable) when v (t) =0 ẋ (t) =[A (t) B (t) K (t)] x (t)+b(t) v (t), x (t0 )=x CLS : 0, y (t) =C (t) x (t). State Observer: d L : ˆx dt (t) =A (t) ˆx (t)+b(t) u (t)+h (t)[y (t) ŷ (t)], ˆx (t 0)=ˆx 0, ŷ (t) =C (t) ˆx (t), such that e = 0 is GUAS for ė (t) =[A (t) H (t) C (t)] e (t), e (t 0 )=ˆx (t 0 ) x (t 0 ). Jaime A. Moreno Nonlinear Observers 97 / 363

Separation Principle II Closed Loop with Observer 8 ẋ (t) =A (t) x (t) B (t) K (t) ˆx (t)+b(t) v (t), x (t 0 )=x 0, >< d CLO : ˆx dt (t) =A (t) ˆx (t)+b(t) u (t)+h (t)[y (t) ŷ (t)], ˆx (t 0)=ˆx 0, y (t) =C (t) x (t), >: ŷ (t) =C (t) ˆx (t), or 8 >< CLO : >: d dt " x (t) e (t) # " A (t) B (t) K (t) B (t) K (t) = 0 A (t) H (t) C (t) " # B (t) + v (t). 0 #" x (t) e (t) #, If B (t) K (t) bounded then e (t) and x (t) converge exponentially. If matrices constant then apple A BK BK = {A BK }[ {A HC}. 0 A HC Jaime A. Moreno Nonlinear Observers 98 / 363

Separation Principle III Note that the state e is uncontrollable from v. Warning: The separation principle does not imply that the closed loop with observer is robust. In fact it can be very sensitive to uncertainties in the plant! [22, 23] A possible solution in the literature is the Loop Transfer Recovery (LTR) (see e.g. [56, Chapter 5][22, 23, 24, 71][37, Lecture 23.5]). Jaime A. Moreno Nonlinear Observers 99 / 363

Part II Linearization Methods for Nonlinear Systems Jaime A. Moreno Nonlinear Observers 100 / 363

Outline 6 Observability of Nonlinear Dynamical Systems 7 Taylor Linearization about an Operating Point 8 Taylor Linearization about a Trajectory Convergence Analysis 9 Exact Error Linearization Method Existence Conditions for the Transformation Some Particular Cases Output Feedback Control Linearization by Immersion Systems Affine in the Unmeasured States 10 Approximate Observer Error Linearization Method 11 Nonlinear Luenberger Observer Jaime A. Moreno Nonlinear Observers 101 / 363

Observability of Nonlinear Systems We consider a Nonlinear System ẋ (t) =f (x (t), u (t)), x (t 0 )=x 0, y (t) =h (x (t)). x (t) 2 R n, u (t) 2 R p, y (t) 2 R q. The functions f, h are known, and u (t) and y (t) are measurable. Its solution is represented as x (t) =' t, t 0, x 0, u [t0,t] y (t) =h ' t, t 0, x 0, u [t0,t] where ' t, t 0, x 0, u [t0,t] is the State Transition Function. Some important properties are: 1 ' t 0, t 0, x 0, u [t0,t 0 ] = x 0 2 ' t, t 1,' t 1, t 0, x 0, u [t0,t 1 ], u [t1,t] = ' t, t 0, x 0, u [t0,t] for all t 0 apple t 1 apple t. Jaime A. Moreno Nonlinear Observers 102 / 363

Is it possible to determine uniquely x 0 (or equivalently x(t)) from the knowledge of y (t) and u (t) in the time interval t 2 [t 0, t 1 ]? Note that y (t) =h ' t, t 0, x 0, u [t0,t] so that there exists a function H that, given an input signal u [t0,t], assigns to each initial state x 0 an output signal y [t0,t] in the time interval [t 0, t] It can be shown that Theorem y [t0,t] = H u[t0,t] (x 0) During the time interval [t 0, t], given an input u [t0,t], it is possible to determine uniquely x 0 from y [t0,t 1 ], i.e. the system is observable for the input u [t0,t], if and only if the map H u[t0,t] (x 0) is injective. Note that Observability of non linear systems depends in general on the input! Jaime A. Moreno Nonlinear Observers 103 / 363

Example apple apple ẋ1 1 u (t) = ẋ2 0 1 apple x1 x 2 y = 1 0 apple x 1 x 2 If u (t) =1 then the resulting system is LTI and it is observable since rank (O 1 )=rank apple 1 0 1 1 = 2 If u (t) =0 then the resulting system is LTI and not observable since rank (O 1 )=rank apple 1 0 1 0 = 1 < 2 Jaime A. Moreno Nonlinear Observers 104 / 363

To determine the observability of a NL system is in general a difficult task. For smooth systems the idea of differentiating the output leads to sufficiency conditions. For simplicity consider a nonlinear system without inputs Differentiating the output y (t) =h (x (t)), ẏ (t) = d dt ÿ (t) = @L f h (x) @x. ẋ (t) =f (x (t)), x (t 0 )=x 0, y (t) =h (x (t)). @h (x) @h (x) h (x (t)) = ẋ (t) = @x @x f (x) :=L f h (x), ẋ (t) = @L f h (x) f (x) :=L 2 f @x h (x), y (k) (t) = @Lk 1 f h (x) ẋ (t) = @Lk 1 f h (x) f (x) :=L k f @x @x h (x). where L k f h (x) are Lie s derivatives of h along f. Jaime A. Moreno Nonlinear Observers 105 / 363

Evaluating at t = 0 2 6 4 y (0) ẏ (0) ÿ (0). y (k) (0) 3 2 = 7 6 5 4 h (x 0 ) L f h (x 0 ) L 2 f h (x 0). L k f h (x 0) This is a finite dimensional map and therefore Theorem 3 := o 7 k (x 0 ) 5 The NL system is observable if for some k the observability map o k (x 0 ) is injective. In general there is no value of k to stop. Jaime A. Moreno Nonlinear Observers 106 / 363

Example I Consider the system apple apple ẋ1 x2 = ẋ2 0 y = x 3 1., Observability maps are o 1 (x) = apple x 3 1 3x 2 1 x 2 2, o 2 (x) = 4 x 3 1 3x 2 1 x 2 6x 1 x 2 2 3 5, Jaime A. Moreno Nonlinear Observers 107 / 363

Example II o 3 (x) = 2 6 4 x 3 1 3x 2 1 x 2 6x 1 x 2 2 6x 3 2 3 7 5, o k (x) = 2 6 4 x 3 1 3x 2 1 x 2 6x 1 x 2 2 6x 3 2 0. 3, 7 5 Note that o 1 (x) is not injective, since o 1 0 1 = o1 0 2. The same happens to o 2 (x). But o 3 (x) is injective, so the system is globally observable. Jaime A. Moreno Nonlinear Observers 108 / 363

A sufficient criterion for local observability á la Kalman can be derived from the observability map: Theorem The NL system is locally observable at x 0 if for some value of k the Observability Matrix O k (x 0 )= @o k (x 0 ) @x is injective, i.e. rank (O k (x 0 )) = n. 2 6 4 dh (x 0 ) dl f h (x 0 ) dl 2 f h (x 0). dl k f h (x 0) 3 7 5 Jaime A. Moreno Nonlinear Observers 109 / 363

Detectability and the existence of Observers I Consider the Differential-Algebraic System where ė = F (e; x, u), e (t 0 )=e 0, 0 = H (e; x, u), F (e; x, u) =f (x + e, u) f (x, u), H (e; x, u) =h (x + e, u) h (x, u). Observability is equivalent to the fact that its only solution is e (t) =0, 8t 0. A weaker condition, called Detectability, is that all solutions converge to zero, i.e. lim e (t) =0. t!1 It is possible to show that Detectability is necessary for the existence of asymptotic convergent observers. Jaime A. Moreno Nonlinear Observers 110 / 363

Operating Point (OP): a constant triplet (ū, x, ȳ) such that Deviation from OP: 0 = f ( x, ū), ȳ = h ( x). x (t) =x (t) x, y (t) =y (t) ȳ, u (t) =u (t) ū, If the LTI Linearization ẋ (t) =Ax (t)+bu (t), x (t 0 )=x 0, y (t) =Cx (t), @f ( x,ū) @f ( x,ū) A = @x, B = @u, C = @h( x) @x. is observable (detectable) then the NL system will be locally observable (detectable) at OP, and a local observer can be designed. There are two possibilities: Jaime A. Moreno Nonlinear Observers 111 / 363

1) A Linear Observer for the linearized system d dt ˆx (t) =Aˆx (t)+bu (t)+h [y (t) ŷ (t)], ˆx (0) =x 0, ŷ (t) =Cˆx (t), (A HC) is Hurwitz. It operates correctly for the Nonlinear Plant as far as its trajectories stay in a neighborhood of the OP, i.e. for small values of kx (0)k, kˆx (0)k, ku (t)k. Jaime A. Moreno Nonlinear Observers 112 / 363

d dt ˆx (t) =Aˆx (t)+bu (t)+h [y (t) Cˆx (t)], ˆx (0) =x 0, u (t) = K ˆx (t), u (t) =ū K ˆx (t), (A HC) is Hurwitz. The Closed Loop system has an OP at and its linearization at this OP is x = x, x = ˆx x = 0 d dt apple x (t) x (t) = apple A BK BK 0 A HC apple x (t) x (t). Since it is (by design) exponentially stable, the Linearization Method of Lyapunov assures that the OP for the Nonlinear Closed Loop is also (locally) exponentially stable. However, the size of the attraction region can be very small. Jaime A. Moreno Nonlinear Observers 113 / 363

2) A Nonlinear Observer d NL : ˆx dt (t) =f (ˆx (t), u (t)) + H [y (t) ŷ (t)], ˆx (t 0)=ˆx 0, ŷ (t) =h (ˆx (t)), such that A HC Hurwitz. The interconnected system Plant-Observer has an operating point at x = x, u = ū, ˆx = x. Introducing the observation error x = ˆx x, the Plant-Observer system can be rewritten in the variables (x, x) Jaime A. Moreno Nonlinear Observers 114 / 363

where and it satisfies ẋ (t) =f (x (t), u (t)), x (t0 )=x : 0, y (t) =h (x (t)). d : dt x (t) =g ( x (t), x (t), u (t)), x (t 0)=ˆx 0 x 0. g ( x, x, u), f (x + x, u) f (x, u)+h [h (x) h (x + x)] g (0, x, u) =0, 8x, u. Jaime A. Moreno Nonlinear Observers 115 / 363

The linearization of the observation error system at the equilibrium point is x = x, u = ū, x = 0 d x =(A dt HC) x. that is (by design of H) exponentially stable. Lyapunov s Linearization Theorem implies that the observer will perform appropriately if the trajectories stay in a neighborhood of the OP, i.e. for small values of k x (0)k, kx (t) xk, ku (t) ūk. Exercise What happens to the Output Feedback Controller implemented with this observer? Jaime A. Moreno Nonlinear Observers 116 / 363

Taylor Linearization about a Trajectory I Fix ū (t), x (t) and ȳ (t) such that it is a trajectory of the system, i.e. d dt x (t) =f ( x (t), ū (t)), ȳ (t) =h ( x (t)), represent the deviations from this nominal trajectory as x (t) =x (t) x (t), y (t) =y (t) ȳ (t), u (t) =u (t) ū (t). If the LTV linearization along (ū (t), x (t), ȳ (t)) ẋ (t) =A (t) x (t)+b(t) u (t), x (t 0 )=x 0, y (t) =C (t) x (t), A (t) =, B (t) = @f ( x(t),ū(t)) @x @f ( x(t),ū(t)) @u, C (t) = @h( x(t)) @x. is Observable (UCO) then the NL system will be Locally Observable around (ū (t), x (t), ȳ (t)) and a local observer can be constructed based on the linearization. If a Kalman Filter is designed the Extended Kalman Filter is obtained Jaime A. Moreno Nonlinear Observers 117 / 363

Taylor Linearization about a Trajectory II ẋ (t) =f (x (t), u (t)) + w (t), x (t0 )=x : 0, y (t) =h (x (t)) + v (t). 8 d ˆx dt (t) =f (ˆx (t), u (t)) + H (t)(y (t) h (ˆx (t))), ˆx (t 0)=ˆx 0, >< H (t) =P (t) C T (t) R 1 (t), d EKF : dt P (t) =P (t) AT (t)+a(t) P (t)+ P (t) C >: T (t) R 1 (t) C (t) P (t)+q(t), A (t) = @ @x f (ˆx (t), u (t)), C (t) = @ @x h (ˆx (t)). For implementation: Calculate matrices A (t) and C (t) depending on (ˆx (t), u (t)). Solve the Differential Riccati Equation (RDE) Calculate H (t) Note that the RDE and the Observer DE build a DE of order n n + 2 n 2 + n = 1 2 n2 + 3 2n, that grows very strongly with the order of the system n. These DEs have to be solved simultaneously, since A (t) y C (t) depend on (ˆx (t), u (t)). Jaime A. Moreno Nonlinear Observers 118 / 363

Convergence Analysis Convergence can be assured (in general) only locally. See [49] and the references therein. The dynamics of the observation error x = ˆx x is given by d dt = f (ˆx, u) f (x, u)+h (h (x) h (ˆx)) +, x (t 0)= x 0, H = PC T R 1, d dt P = PAT + AP PC T R 1 CP + Q, = Hv w, A (t) = @ @x f (ˆx (t), u (t)), C (t) = @ @x h (ˆx (t)). Assuming the x (t), u (t), H (t), v (t), w (t) are bounded, and that f and h are twice continuously differentiable, then, by a Taylor series development around the equilibrium point x = 0, one can write Jaime A. Moreno Nonlinear Observers 119 / 363

where d dt x =[A (t) H (t) C (t)] x + ( x, t)+ (t), x (t 0)= x 0, H = PC T R 1, d dt P = PAT + AP PC T R 1 CP + Q. ( x, t) =f (ˆx, u) f (x, u) H (h (ˆx) h (x)) [A (t) H (t) C (t)] x, (0, t) =0. and k ( x, t)k applek 1 k xk 2, k (t)k applek 2. Jaime A. Moreno Nonlinear Observers 120 / 363

Proof By the mean value theorem f (ˆx, u) f (x, u) A (t) x = f (ˆx, u) f (ˆx x, u) A (t) x, = = = ˆ1 0 ˆ1 0 ˆ1 0 @ @x f (ˆx x, u) d x @ f (ˆx (t), u (t)) x, @x apple @ @x f (ˆx x, u) @ f (ˆx (t), u (t)) d @x x, apple @ @x f (ˆx x, u) @ f (ˆx (t), u (t)) d @x x. Jaime A. Moreno Nonlinear Observers 121 / 363

For each component of f, i = 1, 2,, n, and since f is twice continuously differentiable, one can apply again the mean value theorem @ @x f i (ˆx x, u) ˆ1 @ @x f @ 2 i (ˆx (t), u (t)) = @x 2 f i (ˆx 0 x, u) d x so that kf (ˆx, u) f (x, u) A (t) xk applek 11 k xk 2. By the same arguments kh (t) C (t) x H (h (ˆx) h (x))k applek 12 k xk 2, and therefore k ( x, t)k applek 1 k xk 2. Jaime A. Moreno Nonlinear Observers 122 / 363

Lemma System d dt x =[A (t) H (t) C (t)] x + ( x, t)+ (t), x (t 0)= x 0, H = PC T R 1, d dt P = PAT + AP PC T R 1 CP + Q, is Input-to-State-Stable (ISS) of small gain [44, Chapter 4] with respect to (t), i.e., for small initial condition and small perturbations (t), if = 0 then x = 0 is a locally exponentially stable equilibrium point. Moreover, if! 0 then x! 0 and if is bounded then x remains bounded. Jaime A. Moreno Nonlinear Observers 123 / 363

Proof As in the linear case we use as a Lyapunov function candidate V (t, x) = x T P 1 x, that is decrescent and positive definite if the pair (A (t), C (t)) is UCO. Then V = x T P 1 PA T + AP PC T R 1 CP PC T R 1 CP + 2 x T P 1 ( ( x, t)+ ), dp dt P 1 x+ = x T P 1 QP 1 x x T C T R 1 C x + 2 x T P 1 ( x, t)+2 x T P 1, apple c 1 k xk 2 2 + c 2 k xk 3 2 + c 3 k xk 2 k k 2. Exercise What can be said about the output feedback control with this observer? Jaime A. Moreno Nonlinear Observers 124 / 363

Exact Error Linearization Method I For a NL system [15, 47, 48, 86, 57] ẋ (t) =f (x (t), u (t)), x (t 0 )=x 0, y (t) =h (x (t)), x (t) 2 R n, u (t) 2 R p, y (t) 2 R q, look for transformations (diffeomorphisms = invertible and differentiable) = T (x), = (y), such that transformed system is linear up to an input-output injection (t) =A (t)+' ( (t), u (t)), (t 0 )= 0, (t) =C (t). (3) If (C, A) observable, this is an Observer Form, and an observer d ˆ dt (t) =Aˆ (t)+'( (t), u (t)) + H (t) ˆ (t), ˆ (t 0 )=ˆ 0, ˆ (t) =C ˆ (t), Jaime A. Moreno Nonlinear Observers 125 / 363

Exact Error Linearization Method II leads to a linear observation error dynamics ė (t) =(A HC) e (t) that converges exponentially if (A ˆx (t) =T HC) Hurwitz, and 1 ˆ (t). The difficult part: to find the transformations. Moreover, they exist only for a very restrictive class of systems. Jaime A. Moreno Nonlinear Observers 126 / 363

Existence Conditions for the Transformation For simplicity let us consider the case of single output systems and where only the state diffeomorphism is used. :ẋ = f (x)+g (x, u), x (0) =x 0, y = h (x), (4) where x 2 R n, u 2 R p, y 2 R, f, g are smooth vector fields with f (0) =0 and g (x, 0) =0, 8x, and h is a smooth function with h (0) =0. Jaime A. Moreno Nonlinear Observers 127 / 363

Theorem I There exists a local diffeomorphism T ( ) in a neighborhood U 0 of the origin that transforms the system (4) to the form (3) if and only if in U 0 : n o h i 1 (i) rank dh (x), dl f h (x),, dl n 1 f h (x) = rank @ (x) @x = n, i 2 (ii) had i f r, ad jf r = 0,0apple i, j apple n 1, i 3 (iii) hg, ad jf r = 0,0apple j apple n 2,8u 2 R m where (x) = h (x), L f h (x), L n 1 f h (x) T and r is the unique smooth vector field that is solution of Lr h (x), L r L f h (x), L r L n 1 f h (x) T @ (x) = @x r (x) =[0 1]T. (5) The diffeomorphism is global, i.e. U 0 = R n, if and only if the conditions (i)-(iii) are satisfied in R n and, additionally, 4 (iv) ad i f r,0apple i apple n 1, are complete vector fields. Jaime A. Moreno Nonlinear Observers 128 / 363

Theorem II The coordinate change = T (x) is given by @T (x) @x [r 1, r 2,, r n ]=I, where r i =( 1) i 1 ad i 1 f r, 1 apple i apple n. Jaime A. Moreno Nonlinear Observers 129 / 363

Remark L f h (x) = @h(x) @x vector field f, and L i f h (x) =L f f (x) denotes the Lie derivative of the function h along the h (x). L i 1 f The differential dh of a smooth function h : U h R n! R can be written in local coordinates as a row vector of the gradient @h(x) @h(x) @x 1,, @x n i. For two vector fields f and g a new vector field is defined by the Lie bracket [f, g] = @g g, in local coordinates. @x f @f @x For iterative Lie brackets the operator ad is defined as h i ad 0 f g = g, ad i f g = f, ad i 1 f g. A vector field f is said to be complete if the solutions of the differential equation ẋ = f (x) are defined for all times t 2 R. Jaime A. Moreno Nonlinear Observers 130 / 363

Remark The conditions of the Theorem are very strong and are generically not satisfied. Condition (i) corresponds to the local observability for every x for the system without inputs, and it is generically satisfied. The commutativity condition (ii) is very restrictive. Remark This is the simplest form of the observer error linearization problem. This can be extended in different forms. For example, allowing the transformation T to be a semi-diffeomorphism (the inverse is not required to be differentiable) it is possible to design continuous observers when condition (i) in the Theorem is not satisfied [87, 73]. Extensions include to use an output transformation [48], a time scale transformation [32, 67], and immersion, where the transformed system is allowed to have a larger dimension than the plant [52, 40]. Additionally, it is possible to consider forms with derivatives of the input and the output [89, 33]. Jaime A. Moreno Nonlinear Observers 131 / 363

Example I Consider the system apple ẋ = y = x 1. 1 (x 1 )+x 2 f 2 (x) apple 0 + 1 u, Then apple apple h (x) x (x) = = 1, L f h (x) 1 (x 1 )+x 2 " # @ (x) 1 0 = d @x 1 (x 1 ), dx 1 1 apple @ (x) rank = 2, 8x, @x apple " # apple @ (x) 0 @x r (x) = 1 0 0 =) d 1 1 (x 1 ) r (x) = dx 1 1 1, Jaime A. Moreno Nonlinear Observers 132 / 363

Example II r (x) = " ad f r =[f, r] = 1 0 d 1 (x 1 ) dx 1 1 # 1 apple 0 1 @f (x) @x r = [r, ad f r]= @ (ad f r) r = @x " = " 1 0 d 1 (x 1 ) " d 1 (x 1 ) dx 1 1 @f 2 (x) @f 2 (x) @x 1 @x 2 0 0 @ 2 f 2 (x) @x 1 @x 2 @ 2 f 2 (x) @x 2 2 dx 1 1 # apple 0 1 # apple 0 1 # apple 0 1 = = apple 0 = 1 " " 1 @f 2 (x) @x 2 0 @ 2 f 2 (x) @x 2 2, # #,, [r, ad f r]=0 () @2 f 2 (x) @x 2 2 = 0 () f 2 (x) = 2 (x 1 )+x 2 3 (x 1 ), [g, r] =0(g and r are constant vector fields). Therefore, all conditions are satisfied. r 1 =( 1) 0 ad 0 f r = r = apple 0 1, Jaime A. Moreno Nonlinear Observers 133 / 363

Example III apple r 2 =( 1) 1 ad 1 f r = ad f r = " @T1 (x) @x 1 @T 1 (x) @x 2 @T 2 (x) @T 2 (x) @x 1 @x 2 " @T1 (x) @T 1 (x) @x 1 @x 2 @T 2 (x) @T 2 (x) @x 1 @x 2 " @T1 (x) @T 1 (x) @x 1 @x 2 @T 2 (x) @T 2 (x) @x 1 @x 2 @T (x) [r 1, r 2 ]=I, @x # apple 0 1 1 3 (x 1 ) # = # apple = 1 3 (x 1 ) = apple 0 1 1 3 (x 1 ) 3 (x 1 ) 1 1 0, apple 1 0 0 1 1,,, T 1 (x) =x 2 T 2 (x) =x 1, ˆ x1 0 3 ( ) d, Jaime A. Moreno Nonlinear Observers 134 / 363

Example IV = apple apple 1 x2 = 2 x1 0 3 ( ) d x 1, apple = y = 2, 2 (x 1 )+x 2 3 (x 1 ) 3 (x 1 )( 1 (x 1 )+x 2 )+u 1 (x 1 )+x 2, = apple 0 0 1 0 y = 0 1. apple + 2 (y) 1 (y) 3 (y)+u 1 (y)+ y 0 3 ( ) d, Jaime A. Moreno Nonlinear Observers 135 / 363

Some Particuar Cases I [48, 62] For systems in the Observability Normal Form, i.e. ẋ = x 2, x n n (x) T, x (0) =x0, y = x 1, there exist an state transformation z = T (x) and an output transformation ȳ = (y) that brings the system to the Observer Form (3) if and only if: If n = 2: 2 (x) has the polynomial form 2 (x) =k 0 (y)+k 1 (y) x 2 + k 2 (y) x 2 2. In this case the output transformation can be calculated from 00 (y) = k 2 (y) 0 (y), (0) =0, where 0 (y), d (y) dy. The output injection terms in (3) are defined by ' 0 1 (y) =k 1 (y) 0 (y),' 1 (0) =0, ' 2 (y) =k 0 (y) 0 (y),' 2 (0) =0. Jaime A. Moreno Nonlinear Observers 136 / 363

Some Particuar Cases II If n = 3: 3 (x) has the polynomial form 3 (x) =k 0 (y)+k 1 (y) x 2 + k 2 (y) x 3 + k 3 (y) x 2 2, + k 4 (y) x 2 x 3 + k 5 (y) x 3 2, where the coefficients satisfy the relations: 3k 5 (y) =k 0 4 (y) k 3 (y) =k 0 2 (y) 1 3 k 4 2 (y), 1 3 k 2 (y) k 4 (y). In this case the output transformation can be calculated from 00 (y) = 1 3 k 4 (y) 0 (y), (0) =0. Jaime A. Moreno Nonlinear Observers 137 / 363

Some Particuar Cases III The output injection terms are determined from the equations ' 0 1 (y) =k 2 (y) 0 (y),' 1 (0) =0, ' 0 2 (y) =k 1 (y) 0 (y),' 2 (0) =0, ' 3 (y) =k 0 (y) 0 (y),' 3 (0) =0. If an output transformation is not used, then additional restrictions are required for the coefficients, that can be derived imposing 0 (y) =1 in the previous relations. Jaime A. Moreno Nonlinear Observers 138 / 363

Example I ẋ = y = x 1, apple x2 x 1 x 2 x1 2 x2 2 apple + x 1 x 1 x 2 u, can be transformed to apple apple 0 1 ż = z + 0 0 ȳ = z 1. by the transformations ȳ u (ȳ + 1) ln ȳ + 1 ln ȳ + 1 (1 + ȳ)(1 + ln ȳ + 1 + u), ȳ = (y) =exp (y) 1, y = 1 (ȳ) =ln ȳ + 1, : R! ( 1, 1), apple exp (x1 ) 1 z = T (x) =, T : R 2! ( 1, 1) R exp (x 1 )(x 2 + 1) 1 apple ln x = T 1 z1 + 1 (z) =. z 2 +1 z 1 +1 1 Jaime A. Moreno Nonlinear Observers 139 / 363

Output Feedback Control I Suppose that u = (x) is a state feedback that globally stabilizes the equilibrium point at x = 0, i.e. ẋ (t) =Ax (t)+' (y (t), u (t)), x (0) =x 0, y (t) =Cx (t), u = (x). is GUAS. The dynamic output control law, using the observer is given by d dt ˆx (t) =Aˆx (t)+' (y (t), u (t)) + H (y (t) Cˆx (t)), ˆx (0) =ˆx 0, u = (ˆx). The Plant-Observer system, in the coordinates (x, e), where e = ˆx observation error, is given by x is the ẋ = Ax + ' (y, (x + e)), x (0) =x 0, ė =(A HC) e, e (0) =e 0, Jaime A. Moreno Nonlinear Observers 140 / 363

Output Feedback Control II or ẋ = Ax + ' (y, (x)) + [' (y, (x + e)) ' (y, (x))], x (0) =x 0, ė =(A HC) e, e (0) =e 0. This is a cascade (series) system, where The origin of the master system (Observation error) ė =(A HC) e, e (0) =e 0 is GES (globally and exponentially stable), and The origin of the slave system, without the input of the master system (e = 0), that is, the controlled plant ẋ = Ax + ' (y, (x)), x (0) =x 0 is GAS (globally, asymptotically stable). Jaime A. Moreno Nonlinear Observers 141 / 363

Output Feedback Control III Then the origin of the cascade (x, e) =(0, 0) is GAS if (sufficient condition) the slave system ẋ = Ax + ' (y, (x)+e), x (0) =x 0 is ISS from e! x. A particular case is, for example, when ' (y, u) is globally Lipschitz in (y, u), and the slave system without input ẋ = Ax + ' (y, (x)) has a GES (globally, exponentially stable) equilibrium point [44, Lemma 4.6]. Jaime A. Moreno Nonlinear Observers 142 / 363

Drawbacks: Transformations difficult to find: system of PDEs Conditions are very restrictive Extensions: Add Output [48] and time-scale transformations [32, 67]. Use immersions [52, 40]. Consider input and output derivatives [89, 33]. Use Semi-diffeomorphisms (continuous observers) [87, 73]. Approximate Observer Error Linearization Methods. Jaime A. Moreno Nonlinear Observers 143 / 363

Linearization by Immersion I For the system ẋ (t) =f (x (t), u (t)), x (t 0 )=x 0, y (t) =h (x (t)). x (t) 2 R n, u (t) 2 R p, y (t) 2 R q, look for an immersion such that = T (x), 2 R m, m > n (t) =A (t)+' ( (t), u (t)), (t 0 )= 0, (t) =C (t). If (C, A) observable, then the observer d ˆ dt (t) =Aˆ (t)+'( (t), u (t)) + H ˆ (t) =C ˆ (t), (t) leads to a linear observation error dynamics (e = ˆ ) ˆ (t), ˆ (t 0 )=ˆ 0, ė (t) =(A HC) e (t) Jaime A. Moreno Nonlinear Observers 144 / 363

Linearization by Immersion II that is convergent if (A HC) Hurwitz. The estimate of the original state is ˆ ˆx (t) =T I (t), where T I is a pseudo-inverse of T. Jaime A. Moreno Nonlinear Observers 145 / 363

Example I ẋ 1 = x 1 + g (x 2 ), ẋ 2 = ax 2, y = x 1, where g (x 2 )=x 2 x 2 2 1. Observability maps apple x1 o 1 (x) = : not injective. x1 + g (x2) 2 3 x 1 o 2 (x) = 4 y + x2 3 x 2 5 : injective. 3ay (3a + 1) ẏ 2ax 2 If a 6= 0 global observability. If a = 0 no observability because there are indistinguishable states. Jaime A. Moreno Nonlinear Observers 146 / 363

Example II Differentiating once more y (3) (t) = 3aẏ (t) (3a + 1) ÿ (t)+2a 2 x 2, = (4a + 1) ÿ (t) a (4 + 3a) ẏ (t) 3a 2 y, the system has an observable linear representation 2 3 2 d 1 0 1 0 4 2 5 = 4 0 0 1 dt 3 3a 2 a (4 + 3a) (4a + 1) 2 y = 1 0 0 4 1 2 3 3 5. 3 2 5 4 1 2 3 3 5, Jaime A. Moreno Nonlinear Observers 147 / 363

Example III The state can be estimated by a linear (higher order) observer and the projection map ˆx 1 = ˆ 1, ˆx 2 = 1 2a 3aˆ 1 (3a + 1) ˆ 2 ˆ 3. Jaime A. Moreno Nonlinear Observers 148 / 363

Systems Affine in the Unmeasured States I For ẋ (t) =f (x (t), u (t)), x (t 0 )=x 0, y (t) =h (x (t)), x (t) 2 R n, u (t) 2 R p, y (t) 2 R q, similarly to the Exact Error Linearization Method, look for transformations = T (x), = (y) so that (t) =A (y (t), u (t)) (t)+' ( (t), u (t)), (t 0 )= 0, (t) =C (t). (6) Jaime A. Moreno Nonlinear Observers 149 / 363

If A (y (t), u (t)) is uniformly bounded and (C, A (y (t), u (t))) is uniformly observable, then the Kalman-like observer d ˆ dt (t) =A (y, u) ˆ (t)+'(,u)+h (t) (t) C ˆ (t), ˆ (t 0 )=ˆ 0, d dt P (t) =P (t) AT (y, u)+a (y, u) P (t) P (t) C T R 1 CP (t)+q, H (t) =P (t) C T R 1, P (t 0 )=P 0, converges and ˆx (t) =T ˆ 1 (t). Jaime A. Moreno Nonlinear Observers 150 / 363

Approximate Obs. Error Lin. Methods I See [55], [11, 59]. ẋ = f (x, u)+g (x, u), x (0) =x0, : y = h (x). By State Transformation z = T (x) System (f, h) =) ONF Approximate Nonlinear Observer Form (ANOF) 8 < ż = Az + G (, y, u)+ (y, u), z (0) =z 0, aonf : y = Cz, : = Hz. N.B.: Transformation and perturbation term are not uniquely determined. Jaime A. Moreno Nonlinear Observers 151 / 363

Full order observer 8 >< ẑ = Aẑ + L (ŷ y)+g (ˆ + N (ŷ y), y, u)+ (y, u), : ŷ = Cẑ, >: ˆ = Hẑ. Matrices L, and N to be designed. Defining the error variables by z, ẑ z, ỹ, ŷ y,, ˆ,, (H + NC) z = + Nỹ, and a new nonlinearity (,,y, u), (, y, u) ( +,y, u), Note that (0, ; y, u) =0 for all, y, u. Jaime A. Moreno Nonlinear Observers 152 / 363

Error dynamics z = A L z G (H N z,,y, u), z (0) = z 0, where A L, A + LC, and H N, H + NC. L is selected so that the matrix (A The error equation, z = ẑ LC) is Hurwitz. z, is then approximately linear. Select T, and so that the "size" of the error perturbation term in a compact region of the state space is minimized. No certainty of convergence. Jaime A. Moreno Nonlinear Observers 153 / 363

Nonlinear Luenberber Observer (See [68, 42, 46, 45, 1, 3, 2]). For simplicity we consider a Time Invariant system without input, with x 2 R n ẋ (t) =f (x (t)), x (0) =x0, NL : y (t) =h (x (t)). Consider a different system with state z 2 R m driven by the output y of the plant NL O : ż (t) =Fz (t)+h (y (t)), z (0) =z 0, where F is a Hurwitz matrix, i.e. all its eigenvalues have negative real parts, and H ( ) is a nonlinear function. We look for a function T : R n! R m such that if z (0) =T (x (0)) then the solution of the cascade NL O satisfies z (t) =T (x (t)) for all t 0. Jaime A. Moreno Nonlinear Observers 154 / 363

This will be the case if the condition @T (x) f (x) =FT (x)+h (h (x)) (7) @x is satisfied. For an arbitrary initial condition of z we have therefore that e = z T (x) @T (x) ė = ż ẋ = Fz (t)+h (y (t)) FT (x (t)) H (h (x (t))), @x = F [z (t) T (x (t))] = Fe, and, since F is Hurwitz lim (z (t) Tx (t)) = 0. t!1 O estimates asymptotically T (x (t)), a function of x (t). Note that (7) is a linear Partial Differential Equation for T. It reduces to the matrix equation TA = FT + HC in the linear case. Generically there exists always a solution T ( ) of (7) for arbitrary f, h, F, H and m, n (not necessarily equal). Jaime A. Moreno Nonlinear Observers 155 / 363

If we are interested in estimating the full state x, then it is required that the function T ( ) is injective, i.e. it posses a (continuous) left-inverse T I ( ), i.e. T I (T (x)) = x. In this case ˆx (t) =T I (z (t)) is an estimate of the state. This is possible under some observability conditions on the system (see [1, 3, 2]). An interesting case is to ask that T = I (the identity map). This is called the identity observer. Jaime A. Moreno Nonlinear Observers 156 / 363

A full state observer Suppose that there exists a mapping T (x), solution of (7), and invertible, i.e. m = n and there exists the inverse function T 1 (z) for every z 2 R n. This inverse satisfies T 1 (T (x)) = T T 1 (x) = x, 8x 2 R n, @T 1 1 (z) @T (x) =, z = T (x). @z @x In this case the observer ż (t) =Fz (t)+h (y (t)), z (0) =z 0, ˆx (t) =T 1 (z (t)), Jaime A. Moreno Nonlinear Observers 157 / 363

can be written also as 1 @T (ˆx (t)) ˆx (t) =f (ˆx (t)) + [H (y (t)) H (h (ˆx (t)))], ˆx (0) =ˆx 0. @ˆx Note that this latter form of the observer has the usual structure of being a copy of the plant and a nonlinear output injection term. Jaime A. Moreno Nonlinear Observers 158 / 363

Proof. Deriving ˆx (t) from the first version of the observer we obtain ˆx (t) = d dt T 1 (z (t)) = @T 1 (z (t)) ż (t), @z = @T 1 (z (t)) Fz (t)+ @T 1 (z (t)) H (y (t)), @z @z 1 1 @T (ˆx (t)) @T (ˆx (t)) = FT (ˆx (t)) + H (y (t)), @ˆx @ˆx 1 1 @T (ˆx (t)) @T (ˆx (t)) = f (ˆx (t)) H (h (ˆx (t))) + H (y (t)), @ˆx @ˆx 1 @T (ˆx (t)) = f (ˆx (t)) + [H (y (t)) H (h (ˆx (t)))]. @ˆx Jaime A. Moreno Nonlinear Observers 159 / 363

Part III High-Gain Observers Jaime A. Moreno Nonlinear Observers 160 / 363

Outline 12 The Linear Case 13 The Basic Nonlinear Case: system without input 14 Uniformly (in the input) Observable Nonlinear Systems 15 Output Feedback: A Case Example 16 Stabilization 17 Some Extensions of the Basic High-Gain Theory Perturbed State Affine Systems Design by Immersion Jaime A. Moreno Nonlinear Observers 161 / 363

The basic idea of the High-Gain Observer was presented around the same time by different groups. 1 In France by the group around J.P. Gauthier and H. Hammouri [30, 31, 40]. Many developments have been obtained starting from this group (see the monographs [31, 13]). In particular, the nice contributions of G. Besançon [13, 14, 12]. 2 In the USA the group around H. Khalil [25, 9, 10], see also [81, 28]. 3 The group around L. Praly has also made fundamental contributions to the Observer Theory, in general, and to the High-Gain Observers in particular [76, 77]. Jaime A. Moreno Nonlinear Observers 162 / 363

Consider the LTI system with scalar output (for simplicity) ż (t) =Az (t)+bu (t), z (0) =z0, L : y (t) =Cz (t). and assume that it is observable, i.e. the (square n n) Kalman observability matrix 2 3 C CA O = 6 7 4. 5 CA n 1 has full rank. To design a High-Gain observer we first bring the system to the observability form, what is possible if and only if the system is observable. Jaime A. Moreno Nonlinear Observers 163 / 363

Observability Form Theorem The (single output) pair (A, C) is transformable by x = Oz to the Observability Form (A ob, C ob )= OAO 1, CO 1, with 2 A ob = 6 4 3 0 1 0 0 0 0 0 0....., C 7 ob = 1 0 0, 0 0 0 1 5 a 0 a 1 a n 2 a n 1 where a i are the coefficients of the characteristic polynomial of A, i.e. det (si A) =s n + a n 1 s n 1 + + a 1 s + a 0, if and only if the Kalman observability matrix is regular. Jaime A. Moreno Nonlinear Observers 164 / 363

Consider the observable system in the observability form 8 < ẋ (t) =A ob x (t)+bu (t), x (0) =x 0, L : = A 0 x + b 0 g (x)+bu, : y (t) =C 0 x (t). where 2 A 0 = 6. 4 0 1 0 0 0 0 0 0.... 0 0 0 1 0 0 0 0 3 2, b 7 0 = 6 5 4 0 0. 0 1 3, 7 5 C 0 = C ob, g (x) = a 0 x 1 a n 2 x n 1 a n 1 x n. Jaime A. Moreno Nonlinear Observers 165 / 363

We can write this in a more explicit way as ẋ 1 = x 2 + B 1 u, ẋ 2 = x 3 + B 2 u,. ẋ n 1 = x n + B n 1 u, ẋ n = g (x)+b n u, y = x 1, that is (basically) a chain of integrators with a perturbation term g (x) in the last equation. We will design an observer for the system in three steps: 1 Design an observer for the nominal system. 2 Analyze the effect of the perturbation and measurement noise, and 3 Strength the robustness by increasing the gain. Jaime A. Moreno Nonlinear Observers 166 / 363

1. Observer Design for the nominal plant I Consider the nominal system ẋ (t) =A0 x (t), x (0) =x NL : 0, y (t) =C 0 x (t), and the proposed observer d ˆx dt (t) 2 =A 0ˆx (t)+ L H 0 (y 3 (t) C 2 0ˆx (t)), 3 ˆx (0) =ˆx 0, L 0 0 h 0 0 L 2 0 L = 6.. 4..... 7 0 5, H h 1 0 = 6 7 4. 5, L > 0. 0 0 0 L n h n 1 The observation error, ˆx x satisfies (t) =(A 0 L H 2 0 C 0 ) (t), (0) =ˆx 0 x 0, 3 Lh 0 1 0 0 L 2 h 1 0 0 0 A 0 L H 0 C 0 = 6...... 7 4 L n 1 h n 2 0 0 1 5 L n h n 1 0 0 0 Jaime A. Moreno Nonlinear Observers 167 / 363

1. Observer Design for the nominal plant II Note that the characteristic polynomial of (A 0 L H 0 C 0 ) is det (si (A 0 L H 0 C 0 )) = s n + Lh 0 s n 1 + + L n 1 h n 2 s + L n h n 1. If the eigenvalues of (A 0 L H 0 C 0 ) when L = 1, i.e.(a 0 H 0 C 0 ), are the complex numbers (A 0 H 0 C 0 )={ 1, 2,, n }, then the eigenvalues of (A 0 L H 0 C 0 ) are (A 0 L H 0 C 0 )={L 1, L 2,, L n }. If H 0 is selected such that (A 0 H 0 C 0 ) is Hurwitz, then (A 0 L H 0 C 0 ) will be Hurwitz for every L > 0. By increasing the value of the gain L the eigenvalues of a Hurwitz matrix (A 0 H 0 C 0 ) move deeper into the left half plane. Jaime A. Moreno Nonlinear Observers 168 / 363

1. Observer Design for the nominal plant III Performing the variable change (with >0) 2 2 3 L 1 0 0, 1 L = 0 L 2 0 6 4..... 7 0 5 = 6 4 0 0 0 L n one obtains 1 L 2 L 2 3 L 3. n L n 3 7 5 = 1 L = 1 L (A 0 L H 0 C 0 ) L (t), (t 0 )= 0. Note that 1 L A 0 L = LA 0, C 0 L = LC 0, Jaime A. Moreno Nonlinear Observers 169 / 363

1. Observer Design for the nominal plant IV and therefore = L (A 0 H 0 C 0 ) (t), (t 0 )= 0. Since (A o H o C o ) is Hurwitz, then for any symmetric, positive definite matrix Q = Q T > 0 there exists exactly one symmetric and positive definite matrix P = P T > 0 that is solution to the Algebraic Lyapunov Equation For the Lyapunov function candidate (A o H o C o ) T P + P (A o H o C o )= Q. V ( ) = T P the time derivative along the solutions of the estimation error DE is V h i = L T (A o H o C o ) T P + P (A o H o C o ) = L T Q. Jaime A. Moreno Nonlinear Observers 170 / 363

1. Observer Design for the nominal plant V Since for any symmetric, positive definite matrix M = M T > 0 min (M) k k 2 2 apple T M apple max (M) k k 2 2, where min (M) and max (M) are the smallest (largest) eigenvalue of M, respectively. Therefore where V = L T Q apple L min (Q) k k 2 2 apple L V, Using the comparison principle [44] = min (Q) max (P). V ( (t)) = T (t) P (t) apple e L (t t 0) V ( (t 0 )) = e L (t t 0) T (t 0 ) P (t 0 ). Therefore k (t)k 2 2 apple max (P) min (P) e L (t t 0) k (t 0 )k 2 2 Jaime A. Moreno Nonlinear Observers 171 / 363

1. Observer Design for the nominal plant VI converges exponentially to zero for every L > 0. For the original error variables, since = 1 L 1 L 2 1 = 2 T 2 L = 2 2 L 2 2 1 + 1 L 4 2 2 + + and therefore, for every L 1 1 L 2(n 1) 2 n 1 + 1 L 2n 2 n We obtain then 1 L 2n 2 k k 2 1 2 apple L 2 apple 2 2 L 2 k k2 2. 1 L 2n 2 k (t)k 2 2 apple max (P) min (P) e L (t t 0) 2 L 2 k (t 0)k 2 2 and finally k (t)k 2 2 apple max (P) min (P) L2(n 1) k (t 0 )k 2 2 e L (t t 0). Jaime A. Moreno Nonlinear Observers 172 / 363

1. Observer Design for the nominal plant VII Note: The convergence velocity of the observation error increases with increasing L. The bound on the initial error (when t! t 0 ) also grows with L: This is the Peaking Phenomenon! Jaime A. Moreno Nonlinear Observers 173 / 363

2. Effect of Perturbation and Noise I Consider the perturbed nominal system ẋ (t) =A0 x (t)+b NL : 0 ṽ (t)+bu (t), x (0) =x 0, y (t) =C 0 x (t)+w (t), where ṽ (t) and w (t) are additive system and measurement perturbations, respectively. We propose the observer d dt ˆx (t) =A 0ˆx (t)+bu + L H 0 (y (t) C 0ˆx (t)), ˆx (0) =ˆx 0. The estimation error for the observer is (t) =(A 0 L H 0 C 0 ) (t)+ L H 0 w (t) b 0 v (t), (0) =ˆx 0 x 0 where v (t) =ṽ (t). For the transformed error variable, since = 1 L = 1 L (A 0 L H 0 C 0 ) 1 L + 1 L ( L H 0 w b 0 v), (0) = 1 L 0, = L (A 0 H 0 C 0 ) + H 0 w 1 L b 0v. Jaime A. Moreno Nonlinear Observers 174 / 363

2. Effect of Perturbation and Noise II The derivative of the Lyapunov-like function V ( ) = T P is V = L T Q + 2 T P H 0 w 1 L b 0v. We will assume that the perturbations are bounded, i.e. kw (t)k applew 1, 8t t 0, kv (t)k applev 1, 8t t 0, and that L 1. We have therefore that V = L T Q + 2 T P H 0 w L n b 0 v, apple L min (Q) k k 2 + 2 max (P) k k kh 0 k w 1 + L n v 1, apple max (P) L V + 2 p min (P) kh p 0k w 1 + L n v 1 V. Jaime A. Moreno Nonlinear Observers 175 / 363

2. Effect of Perturbation and Noise III Introducing the function W = p V it follows that Then Ẇ = V 2 p V apple 1 2 L W + max (P) p min (P) kh 0k w 1 + L n v 1. W (t) apple e 1 2 L (t t0) W 0 + p apple e 1 2 L (t t 0) W 0 + 2 L It follows that 8t t 0 max (P) min (P) 2 (kh 0 k w 1 + L n v 1 ) L 1 e 1 2 L (t t 0), max (P) p min (P) kh 0k w 1 + L n v 1 8t t 0. k (t)k applec P e 1 2 L (t t 0) k (t 0 )k + 2c2 P L kh 0 k w 1 + L n v 1. Jaime A. Moreno Nonlinear Observers 176 / 363

2. Effect of Perturbation and Noise IV q where c P = max(p) 1. For the original error variables ( = min(p) L ) 1 1 k (t)k apple Ln L (t), apple c P e 1 2 L (t t 0) 1 L k (t 0)k + 2c2 P L kh 0 k w 1 + L n v 1, that is Note: k (t)k applec P e 1 2 L (t t 0) L n 1 k (t 0 )k + 2c2 P Ln 1 kh 0 k w 1 + 2c2 P L v 1. The convergence velocity of the observation error increases with increasing L. The effect of the system s perturbation v (t) is attenuated for increasing gain L. Jaime A. Moreno Nonlinear Observers 177 / 363

2. Effect of Perturbation and Noise V The effect of the measurement noise is amplified for a large gain. The bound on the initial error (when t! t 0 ) also grows with L: This is the Peaking Phenomenon! High-Gain Idea: Make the gain L large to dominate the perturbation term! Jaime A. Moreno Nonlinear Observers 178 / 363

3. Strength the robustness by increasing the gain I Consider the perturbed nominal system ẋ (t) =A0 x (t)+b NL : 0 ṽ (t)+bu (t), x (0) =x 0, y (t) =C 0 x (t)+w (t), where ṽ (t) =g (x (t)) and w (t) are additive system and measurement perturbations, respectively. We propose the observer d dt ˆx (t) =A 0ˆx (t)+b 0 g (ˆx (t)) + Bu + L H 0 (y (t) C 0ˆx (t)), ˆx (0) =ˆx 0. The estimation error for the observer is (t) =(A 0 L H 0 C 0 ) (t)+ L H 0 w (t) b 0 v (t), (0) =ˆx 0 x 0, where v (t) =g ( (t)). For the transformed error variable, since = 1 L, = 1 L (A 0 L H 0 C 0 ) 1 L + 1 L ( L H 0 w b 0 v), (0) = 1 L 0, = L (A 0 H 0 C 0 ) + H 0 w 1 L b 0v. Jaime A. Moreno Nonlinear Observers 179 / 363

3. Strength the robustness by increasing the gain II The derivative of the Lyapunov-like function V ( ) = T P is V = L T Q + 2 T P H 0 w 1 L b 0v. We will assume that the measurement perturbation is bounded, i.e. kw (t)k applew 1, 8t t 0 and that L 1. Note also that 1 L b 0g 1 L = 1 L b 0 a0 a n 2 a n 1 1 L, = b 0 1 1 a L n 1 0 L a n 2 a n 1, 1 = b 0 L n 1 a 0 1 + + 1 L a n 2 n 1 + a n 1 n = b 0 g L n., Jaime A. Moreno Nonlinear Observers 180 / 363

3. Strength the robustness by increasing the gain III For L 1 the absolute value of the scalar function g (L n L ) is bounded v u! 2 g L n apple t 1 L n 1 (a 0 1 ) 2 + + 1 L 2 (a n 2 n 1 ) 2 +(a n 1 n ) 2, apple r (a 0 1 ) 2 + +(a n 2 n 1 ) 2 +(a n 1 n ) 2, apple g ( ) applek k k, for some k = max 1appleiapplen { a i } > 0, depending on the values of a i. We have therefore that V = L T Q + 2 T P H 0 w b 0 g L n, apple [L min (Q) 2k max (P)] k k 2 + 2 max (P) kh 0 kk k w 1, max (P) apple (L 2k) V + 2 p min (P) kh 0k p Vw 1. Jaime A. Moreno Nonlinear Observers 181 / 363

3. Strength the robustness by increasing the gain IV By considering W = p V it follows that Ẇ = V 2 p V apple 1 L 2 2k max (P) W + p min (P) kh 0k w 1. Selecting L > L 0 = 2k, what is always possible since P, Q, k are independent of L, then W (t) apple e 1 2 (L L 0) (t t 0 ) W 0 + p max (P) min (P) apple e 1 2 (L L 0) (t t 0 )(t t 0 ) 2 W 0 + (L L 0 ) 2 kh 0 k w 1 (L L 0 ) 1 e 1 2 (L L 0) (t t 0 ), max (P) p min (P) kh 0k w 1 8t t 0. Jaime A. Moreno Nonlinear Observers 182 / 363

3. Strength the robustness by increasing the gain V It follows that 2c 2 P k (t)k applec P e (L L 0) 2 (t t 0 ) k (t 0 )k + (L L 0 ) kh 0 k w 1. q max(p) 1 where c P =. For the original error variables ( = min(p) L ) 1 1 k (t)k apple Ln L (t), 2c 2 P apple c P e (L L 0) 2 (t t 0 ) 1 L k (t 0)k + (L L 0 ) kh 0 k w 1, that is k (t)k applec P e (L L 0) 2 (t t 0 ) L n 1 k (t 0 )k + 2c2 P Ln (L L 0 ) kh 0 k w 1. Note: Jaime A. Moreno Nonlinear Observers 183 / 363

3. Strength the robustness by increasing the gain VI The convergence velocity of the observation error increases with increasing L. The effect of the perturbation g (x) is dominated for a gain L sufficiently large (L > L 0 ). The effect of the measurement noise is amplified for a large gain. The bound on the initial error (when t! t 0 ) also grows with L: This is the Peaking Phenomenon! High-Gain Idea: Make the gain L large to dominate the perturbation term! Jaime A. Moreno Nonlinear Observers 184 / 363

The Basic Nonlinear Case I Consider a nonlinear single output nonlinear system ẋ (t) =f (x (t)), x (t 0 )=x 0, y (t) =h (x (t)), where x (t) 2 R n, y (t) 2 R. Suppose that the observability map of order n 1 1 is a global diffeomorphism, 2 is globally Lipschitz. 2 O n 1 (x) = 6 4. h (x) L f h (x) L 2 f h (x) L n 1 f h (x) 3 7 5 Jaime A. Moreno Nonlinear Observers 185 / 363

The Basic Nonlinear Case II Then with = T (x) =O n 1 (x), we observe that 1 = d dt h (x) =L f h (x) = 2, 2 = d dt L f h (x) =L 2 f h (x) = 3,. n 1 = d dt Ln 2 f h (x) =L n 1 f h (x) = n, n = d dt Ln 1 f h (x) =L n f h (x), y = 1. Jaime A. Moreno Nonlinear Observers 186 / 363

The Basic Nonlinear Case III Since O n 1 (x) is a diffeomorphism, there exists a smooth inverse and so there is a smooth function x = O 1 n 1 ( ), ' ( (t)) = L n f h O 1 n 1 ( ). Therefore, the system is represented in the new coordinates as (t) =A 0 (t)+b 0 ' ( (t)), (t 0 )= 0, y (t) =C 0 (t). It will be assumed that ' ( ) is globally Lipschitz in, i.e. there exists a constant k (independent of ) such that ' ( ) ' apple k, 8, 2 R n. Under these conditions the observer [30, 31] Jaime A. Moreno Nonlinear Observers 187 / 363

The Basic Nonlinear Case IV d dt ˆ (t) =A o ˆ (t)+b 0 ' ˆ (t) + L H o y (t) ˆx (t) =O 1 n 1 ˆ (t), C o ˆ (t), ˆ (t 0 )=ˆ 0, converges exponentially when (A o H o C o ) is Hurwitz, and L is sufficiently large. Moreover, the convergence velocity is proportional to L. The estimation error for the observer is where (t) =(A 0 L H 0 C 0 ) (t)+ L H 0 w (t) b 0 v (t), (0) =ˆx 0 x 0 v (t) =' ( (t)+ (t)) ' ( (t)). For the transformed error variable = L (A 0 H 0 C 0 ) + H 0 w 1 L b 0v, (0) = 1 L 0. Jaime A. Moreno Nonlinear Observers 188 / 363

The Basic Nonlinear Case V The derivative of the Lyapunov-like function V ( ) = T P is V = L T Q + 2 T P H 0 w 1 L b 0v. Recall that it is assumed that ' ( ) is globally Lipschitz in, i.e.there exists a k such that for all, z ' ( + z) ' ( ) applek kzk. This implies that 1 b 0 ' + 1 L ' ( ) = b 0 L n ' + 1 L ' ( ) Jaime A. Moreno Nonlinear Observers 189 / 363

The Basic Nonlinear Case VI satisfies ' + 1 L n L ' ( )] apple k L n 1 L, s L apple k 2 L 1 L n 2 2 L + n 2 2 + + n, s 2 2 1 1 apple k L n 1 1 + L n 2 2 + +( n ) 2, apple k k k. The rest of the proof is identical to the linear case, since the perturbation has a linear growth in k k. Jaime A. Moreno Nonlinear Observers 190 / 363

Another realization of the observer is given by d ˆx (t) =f (ˆx (t)) + dt 1 @On 1 + (ˆx (t)) L H o (y (t) h (ˆx (t))), ˆx (t 0 )=ˆx 0. @ˆx The advantage is that it does not require the inversion of the mapping O n 1 (ˆx (t)), but only to take the matrix inverse of the Jacobian Matrix of the mapping, a much easier task. Jaime A. Moreno Nonlinear Observers 191 / 363

Consider now a nonlinear single output system affine in the known inputs ẋ (t) =f (x (t)) + g (x (t)) u (t), x (t 0 )=x 0, y (t) =h (x (t)), where x (t) 2 R n, u (t) 2 R p, y (t) 2 R. Suppose that 1 The observability map of order n 1, when u = 0 2 3 h (x) L f h (x) O n 1 (x) = L 2 f h (x) 6 7 4. 5 h (x) 1 is a global diffeomorphism, 2 is globally Lipschitz. L n 1 f 2 System is globally observable for every input (uniformly observable). This means that the system is observable for every input, what is a strong property for nonlinear systems. Jaime A. Moreno Nonlinear Observers 192 / 363

In this case it has been shown by [29] (see also [30]) that the transformation brings the system to the form = T (x) =O n 1 (x), (t) =A 0 (t)+' ( (t), u (t)), (t 0 )= 0, y (t) =C 0 (t), where 2 3 2 0 1 0 0 0 0 1 0 A 0 =. 6......,'(, u) = 7 6 4 0 0 0 1 5 4 0 0 0 0 C 0 = 1 0 0. ' 1 ( 1, u) ' 2 ( 1, 2, u) ' 3 ( 1, 2, 3, u). ' n (,u) If ' (,u) is globally Lipschitz in, uniformly in u, then the observer [30, 31] 3, 7 5 d ˆ dt (t) =A o ˆ (t)+' ˆ (t), u (t) + H o y (t) ˆx (t) =O ˆ 1 (t). C o ˆ (t), ˆ (t 0 )=ˆ 0, Jaime A. Moreno Nonlinear Observers 193 / 363

converges exponentially when (A o H o C o ) is Hurwitz, and L is sufficiently large. The convergence velocity is proportional to L. Again, the proof of this result is similar to the previous one. The estimation error for the observer is where (t) =(A 0 L H 0 C 0 ) (t)+ L H 0 w (t) v (t), (0) =ˆx 0 x 0, v (t) =' ( (t)+ (t), u (t)) ' ( (t), u (t)). For the transformed error variable = L (A 0 H 0 C 0 ) + H 0 w 1 L v, (0) = 1 L 0. The derivative of the Lyapunov-like function V ( ) = T P is V = L T Q + 2 T P H 0 w 1 L v. Jaime A. Moreno Nonlinear Observers 194 / 363

Recall that it is assumed that each component of the vector ' (, u) is globally Lipschitz in, uniformly in u, i.e. for each i = 1, 2,, n there exists a positive constant k i, independent of (, u), such that for all, z ' i ( 1 + z 1,, i + z i, u) ' i ( 1,, i, u) applek i k[z 1,, z i, 0,, 0]k. And therefore the components of ' (t)+ 1 L (t), u (t) 1 L L i ' i 1 + L 1,, i + Li i, u ' ( (t), u (t)) satisfy ' i ( 1,, i, u) apple k i L i 1 L [ 1,, i, 0,, 0] T. Jaime A. Moreno Nonlinear Observers 195 / 363

For L 1, and with the Euclidean norm one has L i 1 L [ 1,, i, 0,, 0] T = L i This means that = L i s L 1 = apple s 1 apple L 1, L2 2, Li i, 0,, 0, 2 2 Li i + +, 2 2 2 + + + i 2, L i 1 q 2 1 + + 2 i, L i 2 = k[ 1,, i, 0,, 0]k. 1 L v apple k k k, k = max {k i}. 0appleiapplen With this inequality, the proof remains the same as in the previous cases. Jaime A. Moreno Nonlinear Observers 196 / 363

Finally, another realization of the observer is given by d ˆx (t) =f (ˆx (t)) + g (ˆx (t)) u (t)+ dt 1 @On 1 + (ˆx (t)) L H o (y (t) h (ˆx (t))), ˆx (t 0 )=ˆx 0. @ˆx The advantage is that it does not require the inversion of the mapping O n 1 (ˆx (t)), but only to take the matrix inverse of the Jacobian Matrix of the mapping, a much easier task. Jaime A. Moreno Nonlinear Observers 197 / 363

Output Feedback I For system [44, Chapter 14] ẋ 1 = x 2, ẋ 2 = (x, u), y = x 1, suppose the controller u = (x) stabilizes the origin of A HG observer ẋ 1 = x 2, ẋ 2 = (x, (x)). ˆx 1 = ˆx 2 + l 1 (y ˆx 1 ), ˆx 2 = 0 (ˆx, u)+l 2 (y ˆx 1 ), where 0 (x, u) is a nominal model of (x, u). The observation error x 1 = ˆx 1 x 1, x 2 = ˆx 2 x 2, Jaime A. Moreno Nonlinear Observers 198 / 363

Output Feedback II x 1 = l 1 x 1 + x 2, ˆx 2 = l 2 x 1 + (x, x), (x, x) = 0 (ˆx, (ˆx)) (x, (ˆx)). apple apple l1 l Design L = so that A l 0 = 1 1 is Hurwitz. 2 l 2 0 The transfer matrix from! x is apple 1 1 G o (s) = s 2. + l 1 s + l 2 s + l 1 Let us design L to make sup!2r kg o (j!)k as small as possible. Select l 1 = 1 ", l 2 = 2 " 2,">0, apple " G o (s) = ("s) 2 + 1 "s + 2 " "s + 1. Jaime A. Moreno Nonlinear Observers 199 / 363

Output Feedback III The eigenvalues of the observer are ( 1 /") and ( 2 /"), where 1, 2 are the roots of 2 + 1 + 2 = 0, Note that sup kg o (j!)k = O (").!2R lim G o (s) =0, "!0 the attenuation effect is higher for small values of ". Define 1 = x 1 ", 2 = x 2 and so " 1 = 1 1 + 2, " 2 = 2 1 + " (x, x). Then: 1 The final bound of is of the order of O (") 2 decays faster than an exponential function e at/", a > 0. Jaime A. Moreno Nonlinear Observers 200 / 363

The Peeking Phenomenon I x 1 (0) 6= ˆx 1 (0) =) 1 (0) =O (1/"). The solution contains a term of the form 1 " e at/" that approximates an impulse function when "! 0. Jaime A. Moreno Nonlinear Observers 201 / 363

Example I State Feedback Control Output Feedback Control ẋ 1 = x 2, ẋ 2 = x 3 2 + u, y = x 1. u = x 3 2 x 1 x 2. ˆx 1 = ˆx 2 + 2 " (y ˆx 1), ˆx 2 = 1 " 2 (y ˆx 1 ), u = ˆx 3 2 ˆx 1 ˆx 2. Jaime A. Moreno Nonlinear Observers 202 / 363

Example II u 0.2 1 0 0.5 0.2 0 0.4 x 1 x 2 0.5 0.6 0.8 1 SF OF: ε=0.1 OF: ε = 0.01 OF: ε = 0.005 1 1.5 SF OF: ε = 0.1 OF: ε = 0.01 OF: ε = 0.005 1.2 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 2 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 0 50 100 8 7 6 5 HGO: ε=0.1 HGO: ε =0.01 HGO: ε =0.005 150 x 2e x 2 4 3 200 250 SF OF: ε = 0.1 OF: ε = 0.01 OF: ε = 0.005 2 1 0 300 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) Jaime A. Moreno Nonlinear Observers 203 / 363

Example III u 0.2 0 OF: ε=0.004 0 50 0.2 100 x 1 0.4 x 2 150 0.6 200 0.8 250 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time (sec) 300 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time (sec) 2000 10 9 1500 8 7 1000 6 500 x 2e x 2 5 4 3 0 2 1 0 500 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time (sec) 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) Figure : Instability caused by the peeking phenomenon when " = 0.004. Jaime A. Moreno Nonlinear Observers 204 / 363

u 0 1 2 3 4 5 6 7 8 9 10 0.1 0.08 0.06 0.04 SF OF: ε=0.1 OF: ε = 0.01 OF: ε = 0.001 0.04 0.02 0 0.02 x 1 x 2 0.04 0.02 0 0.02 0.06 0.08 0.1 SF OF: ε = 0.1 OF: ε = 0.01 OF: ε = 0.001 0.04 Time (sec) 0.12 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 30 0.2 0 0.2 0.4 x 2e x 2 25 20 15 HGO: ε=0.1 HGO: ε =0.01 HGO: ε =0.001 0.6 SF OF: ε = 0.1 0.8 OF: ε = 0.01 1 OF: ε = 0.001 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) 10 5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) Figure : Closed Loop with saturated control u = sat ˆx 3 2 ˆx 1 ˆx 2 Jaime A. Moreno Nonlinear Observers 205 / 363

Figure : Attraction region with state feedback Jaime A. Moreno Nonlinear Observers 206 / 363

Figure : Attraction region with Output Feedback. " = 0.1 (dashed), " = 0.05 (dashed-dot), Jaime A. Moreno Nonlinear Observers 207 / 363

Analysis of the closed loop system: ẋ 1 = x 2 ẋ 2 = (x, (x x)) " 1 = 1 1 + 2 " 2 = 2 1 + " (x, x) Figure : Closed loop system s Analysis: ẋ 1 = x 2, ẋ 2 = (x, (x x)), " 1 = 1 1 + 2, " 2 = 2 1 + " (x, x) Jaime A. Moreno Nonlinear Observers 208 / 363

Measurement noise effect I The High-Gain Observer is an approximate differentiator. The transfer function from y! ˆx (with 0 = 0) is 2 apple 1 +(" 1 / 2 ) s ("s) 2 + 1 "s + 2 s! apple 1 s when "! 0. The differentiation amplifies the measurement noise. Jaime A. Moreno Nonlinear Observers 209 / 363

Measurement noise effect II y = x 1 + v, k n = sup " opt = O s k n k d!. t 0 v (t) < 1, k d = sup ẍ 1 (t), t 0 Jaime A. Moreno Nonlinear Observers 210 / 363

Consider the system ẋ = Ax + B (x, z, u), ż = (x, z, u), y = Cx, = q (x, z), u 2 R p, y 2 R m, 2 R s, x 2 R, z 2 R`. A, B, C are Block-diagonal matrices where 2 3 0 1 0 0 0 0 1 0 A i =. 6...... 7 4 0 0 0 1 5 0 0 0 0 i i C i = 1 0 0 0 1 i, = 2, B i = 6 4 mx i. i=1 0 0. 0 1 3 7 5 i 1, Jaime A. Moreno Nonlinear Observers 211 / 363

The measured variables are y,. This happens, e.g. for systems in the normal form of the systems input/output linearizable mechanical and electromechanical systems Example Magnetic suspension system ( ẋ1 = x 2, x : k ẋ 2 = g ( " z : ẋ 3 = 1 L (x 1 ) y = x 1, = x 3. m x 2 L 0 ax 2 3 2m(a+x 1 ) 2, Rx 3 + L 0ax 2 x 3 (a + x 1 ) 2 + u #, Jaime A. Moreno Nonlinear Observers 212 / 363

We will proceed in two steps to design an output feedback control: 1 Design of a partial state feedback controller, using the measurements of x and to stabilize the origin. 2 Design of a High-Gain observer to estimate x from y. The state feedback controller can be a dynamic system of the form # = (#, x, ), u = (#, x, ), where and are locally Lipschitz functions in their arguments, in the domain of interest, and they are functions globally bounded in x. Moreover, (0, 0, 0) =0, (0, 0, 0) =0. Or, as a particular case of the previous one, a static controller of the form u = (x, ). The closed loop system with state controller is written as X = f (X ), X =(x, z,#). The origin of X = f (X ) is asymptotically stable. Jaime A. Moreno Nonlinear Observers 213 / 363

The output controller is taken as # = (#, ˆx, ), u = (#, ˆx, ), where ˆx is generated by the High-Gain observer The observer gain L is selected as ˆx = Aˆx + B 0 (ˆx,,u)+L (y Cˆx), L = blockdiag [L 1,, L m ], L i = 6 4 2 i 1 /" i 2 /"2. i i 1 /" i 1 i i /" i where " is a positive constant to be specified. The positive constants i j are selected such that 3 7 5 i 1, s i + i 1s i 1 + + i i 1s + i i Jaime A. Moreno Nonlinear Observers 214 / 363

are Hurwitz. 0 (x,,u) is a nominal model of (x, z, u), and it is globally bounded in x, and 0 (0, 0, 0) =0. Jaime A. Moreno Nonlinear Observers 215 / 363

Nonlinear Separation Principle I Suppose that the origin of X = f (X ) is asymptotically stable and R is its attraction region. Let S be any compact set in the interior of R and Q any compact subset of R. Then 9 " 1 > 0 such that, for all 0 <"apple " 1, the solutions (X (t), ˆx (t)) of the closed loop system, starting in S Q, are bounded for all t 0. For any given µ>0, 9 " 2 > 0 and T 2 > 0, both depending on µ, such that, for all 0 <"apple " 2, the solutions (X (t), ˆx (t)) of the closed loop system, starting in S Q, satisfy kx (t)k appleµ, kˆx (t)k appleµ, 8 t T 2. For any given µ>0, 9 " 3 > 0, depending on µ, such that, for all 0 <"apple " 3, the solutions (X (t), ˆx (t)) of the closed loop, starting in S Q, satisfy kx (t) X r (t)k appleµ, 8 t 0, where X r (t) is the solution of X = f (X ), starting in X (0). Jaime A. Moreno Nonlinear Observers 216 / 363

Nonlinear Separation Principle II If the origin of X = f (X ) is exponentially stable, then 9 " 4 > 0 such that, for all 0 <"apple " 4, the origin of the closed loop system is exponentially stable, and S Qis a subset of its attraction region. Jaime A. Moreno Nonlinear Observers 217 / 363

Fundamental ideas of the proof: The closed loop is represented as a system with singular perturbations, where X is the slow state and (the scaled estimation error) is the fast state. The converse Lyapunov theorem is used to construct positive invariant sets. The global boundedness of ˆx is used to show that reaches O (") while X is inside a positive invariant set. Local Analysis versus non local analysis. A novel aspect is the performance recovery, that has three aspects: 1 Recovery of the exponential stability. 2 Recovery of the attraction region in the sense that any compact set in its interior can be recovered. 3 The solution X (t) under output control approaches the state feedback solution when "! 0. Jaime A. Moreno Nonlinear Observers 218 / 363

Example I m`2 + mg 0` sin + k 0`2 = u. Stabilization at =, = 0. A sliding mode control is u = k sat Suppose that only is measured: a 1 ( ˆ =ˆ! + 2 " ˆ! = 0 ˆ, u ˆ, u u = 0 0 k sat @ a 1 ˆ )+!. µ ˆ, + 1 ˆ, " 2 = â sin ˆ + ĉu, 1 +ˆ! A, µ Jaime A. Moreno Nonlinear Observers 219 / 363

Example II or also u = a1 ( k sat )+ˆ!. µ Jaime A. Moreno Nonlinear Observers 220 / 363

Some Extensions of the Basic HG Theory There are many possible extension of the presented theory of HG Observers. Here we give but some examples. Jaime A. Moreno Nonlinear Observers 221 / 363

Perturbed State Affine Systems I [12] Consider systems in the form (t) =A (y (t), u (t)) (t)+' ( (t), u (t)), (t 0 )= 0, (t) =C (t). (8) where 2 0 a 1 (y, u) 0 0 0 0 a 2 (y, u) 0 A (y, u) =. 6...... 4 0 0 0 a n 1 (y, u) 0 0 0 0 3 7 5, C 0 = 1 0 0. Jaime A. Moreno Nonlinear Observers 222 / 363

Perturbed State Affine Systems II and 2 ' (, u) = 6 4 ' 1 ( 1, u) ' 2 ( 1, 2, u) ' 3 ( 1, 2, 3, u). ' n (,u) 3. 7 5 Jaime A. Moreno Nonlinear Observers 223 / 363

If A (y (t), u (t)) is uniformly bounded and (C, A (y (t), u (t))) is uniformly observable, then the Kalman-like observer d ˆ dt (t) =A (y, u) ˆ (t)+' ˆ,u L S 1 C C T ˆ (t) y, ˆ (t 0 )=ˆ 0, d dt S (t) = L S AT (y, u) S SA (y, u)+c T C, S (0) > 0. converges for L and large enough. In the case that 0 < a im apple a i (y, u) apple a im, then it is possible to construct an observer with fixed gain. The Differential Riccati Equation is therefore not necessary in this case [31]. Jaime A. Moreno Nonlinear Observers 224 / 363

Design by Immersion I Consider a nonlinear single output nonlinear system ẋ (t) =f (x (t)), x (t 0 )=x 0, y (t) =h (x (t)), x (t) 2 R n, y (t) 2 R. Suppose that the observability map of order m 1 > n 1 2 O m 1 (x) = 6 4. h (x) L f h (x) L 2 f h (x) L m 1 f h (x) is injective from an open set X R n!o m 1 (X) R m. Then with = T (x) =O m 1 (x), 3 7 5 Jaime A. Moreno Nonlinear Observers 225 / 363

Design by Immersion II we observe that 1 = d dt h (x) =L f h (x) = 2, 2 = d dt L f h (x) =L 2 f h (x) = 3,. m 1 = d dt Ln 2 f h (x) =L n 1 f h (x) = m, m = d dt Lm 1 f h (x) =L m f h (x), y = 1. Since O m 1 (x) is injective, there exists a left inverse O m 1 : Rm! X R n such that x = O m 1 (O m 1 (x)), Jaime A. Moreno Nonlinear Observers 226 / 363

Design by Immersion III and so there is a function ' m : R m! R such that ' m (O m 1 (x)) = L m f h (x), 8x 2 X. Note that the functions O m 1 and ' m are uniquely defined on the set O m 1 (X) R m, but there are many possibilities for their definition outside from this set. Now assume that O m 1 and ' m can be defined continuous on R m. Therefore, the system is immersed in the new coordinates as Note that the set = O m (t) =A 0 (t)+b 0 ' m ( (t)), (t 0 )= 0, y (t) =C 0 (t). 1 (x) is an invariant manifold of the system ẋ = f (x), (t) =A 0 + b 0 ' m ( ). It will be assumed that ' m ( ) is globally Lipschitz in, i.e. there exists a constant k (independent of ) such that ' ( ) ' apple k, 8, 2 R m. Jaime A. Moreno Nonlinear Observers 227 / 363

Design by Immersion IV Under these conditions the observer [31] d dt ˆ (t) =A o ˆ (t)+b 0 ' ˆ (t) + L H o y (t) ˆx (t) =O m 1 ˆ (t), C o ˆ (t), ˆ (t 0 )=ˆ 0, converges asymptotically when (A o H o C o ) is Hurwitz, and L is sufficiently large. Similar results apply to systems with inputs, when they are uniformly observable (in the input). Jaime A. Moreno Nonlinear Observers 228 / 363

Part IV Dissipative Observer Design Method Jaime A. Moreno Nonlinear Observers 229 / 363

Outline 18 Introduction 19 Dissipativity Theory A General Overview Important Special Cases 20 Dissipative Observer Design Motivation Observer Design: A simple case Observer Design: An Extended Case 21 Computational Issues 22 Relations With Other Design Methods 23 A Dissipative Approach to Unknown Input Observers Introduction and Problem Statement UIOs for Linear Time Invariant Systems System and Observers Considered Existence Conditions for an UIO Proof Example 24 Conclusions Jaime A. Moreno Nonlinear Observers 230 / 363

Introduction I Observer design is usually a two step procedure: System Transformation to a "simpler" system Observer design for the transformed system If transformed system is Linear + Nonlinearity, the nonlinearity is compensated by High Gain, for example. Our proposal: generalized and unified view of dealing with the nonlinearity using dissipativity theory. This eliminates restrictions and opens new design possibilities. Since Dissipative Theory acts like a unifying theory in control, one expects that the same can be true for observer design. The Dissipative Method can provide a unified framework to understand and compare different design methods, to design robust observers and to integrate them in control and other interconnected loops. Jaime A. Moreno Nonlinear Observers 231 / 363

Dissipativity Theory I It was developed in the 60 s and 70 s especially by J.C. Willems as a unified framework to understand the stability properties of interconnected systems. It has been motivated by the works on the absolute stability problem, Popov s criterion, Yakubovich s work, Kalman-Yakubovich Lemma, Passivity,... It has become a very general framework for studying stability-like properties, in particular through the work of Sontag. It has been also physically motivated by the energy considerations in physics, thermodynamics and electrical circuits. It can be seen as a generalization of Lyapunov theory for systems with interconnection ports. It is useful for Robustness Analysis and Design Jaime A. Moreno Nonlinear Observers 232 / 363

Energy Balance: dv (x) dt apple s(y, u) d(x, u), V (x(t)) V (x(0)) apple t s(y( ), u( ))d. 0 Jaime A. Moreno Nonlinear Observers 233 / 363

Famous Dissipative actors I Passivity: V (x) apple y T u (kxk), ( ) 2K 1 ISS: V (x) apple 1 (kxk)+ 2 (kuk), i ( ) 2K 1 iiss: V (x) apple 1 (kxk)+ 2 (kuk), 1 ( ) 2K, 2 ( ) 2K 1 IOSS: V (x) apple 1 (kxk)+ 2 (kuk)+ 3 (kyk), i ( ) 2K 1 Input Output Stability: V (x) apple 1 (kuk) 2 (kyk), i ( ) 2K 1 Jaime A. Moreno Nonlinear Observers 234 / 363

Properties I Generally speaking (be careful) the interconnection of dissipative systems is dissipative! Important Examples: The cascade interconnection of two ISS systems is ISS. Passivity Theorem: The negative feedback interconnection of two Passive systems is Passive. (modern version of) The Small Gain Theorem: The negative feedback interconnection of two ISS systems is ISS, if the gains are small enough. Jaime A. Moreno Nonlinear Observers 235 / 363

Relationship of (open) dissipative systems with the Lyapunov stability of closed systems dv Dissipativity: i (x i ) dt apple s i (y i, u i ) d i (x i, u i ) i = 1, 2. Interconnection: u 2 = y 1, u 1 = y 2 Negative Feedback Interconnection. Total Energy: V (x) =V 1 (x 1 )+ V 2 (x 2 ). Energy Balance: dv (x) dt apple s 1 (y 1, y 2 )+ s 2 (y 2, y 1 ) (d 1 (x 1, y 2 )+ d 2 (x 2, y 1 )). Internal stability condition: s 1 (y 1, y 2 )+ s 2 (y 2, y 1 ) apple d 1 (x 1, y 2 )+ d 2 (x 2, y 1 ). Jaime A. Moreno Nonlinear Observers 236 / 363

Useful for Analysis I Stability analysis of interconnected systems Robustness Analysis Small Gain Theorem, Passivity Theorem, Absolute Stability, IQC, etcetera. Jaime A. Moreno Nonlinear Observers 237 / 363

Useful for Design: Stabilization by Interconnection I Jaime A. Moreno Nonlinear Observers 238 / 363

Useful for Design: Stabilization by Interconnection I More sophisticated: Decomposition and aggregation! Jaime A. Moreno Nonlinear Observers 239 / 363

Special Case: LTI Systems I System L : ẋ = Ax + Bu, x (0) =x0, x 2 R n, u 2 R q, y = Cx, y 2 R m, is state strictly dissipative (SSD) w.r.t. the supply rate apple y! (y, u) = u T apple Q S T apple S y R u, if there exists a storage function V (x) =x T Px such that dv (x (t)) dt apple! (y (t), u (t)) x T Px, for some >0. Jaime A. Moreno Nonlinear Observers 240 / 363

Dissipativity: Dynamic System L is state strictly dissipative (SSD) w.r.t. the supply rate! (y, u), or for short (Q, S, R)-SSD, if there exist a matrix P = P T > 0, and >0 such that m apple PA + A T P + P, PB B T P 0 apple C T QC S T C C T S R apple 0. Replacing P with I gives an equivalent problem. The inequality is an LMI in P and. Jaime A. Moreno Nonlinear Observers 241 / 363

If m = q, passivity corresponds to the supply rate! (y, u) =y T u. Kalman-Yakubovich-Popov Lemma: If (A, B) controllable, and (A, C) observable, then the previous MI (for passivity) is equivalent to G (s) =C (si A) 1 B being strictly positive real (SPR). Jaime A. Moreno Nonlinear Observers 242 / 363

Dissipativity: Static Nonlinearity I A time-varying memoryless nonlinearity :[0, 1) R q! R m, y = (t, u), piecewise continuous in t and locally Lipschitz in u, such that (t, 0) =0, is said to be dissipative w.r.t. the supply rate! (y, u) or for short (Q, S, R)-D, if for every t 0, and u 2 R q! (y, u) =! ( (t, u), u) = T Q + 2 T Su + u T Ru 0. Generalizes the classical (m = q) concept of a Sector nonlinearity [44]. If is in the sector [K 1, K 2 ], i.e. (y K 1 u) T (K 2 u y) 0, then it is (Q, S, R)-D, with (Q, S, R) = I, 1 (K 1 2 1 + K 2 ), K T 2 1 K 2 + K2 T K 1. If is in the sector [K 1, 1], i.e. (y K 1 u) T u 0, then it is 0, 1 I, 1 K 2 2 1 + K1 T -D. Jaime A. Moreno Nonlinear Observers 243 / 363

Dissipativity: Static Nonlinearity II Figure : Nonlinearity in the sector [0, k] or ( 1, k/2, 0)-D. A local (or regional) version is also of interest. Jaime A. Moreno Nonlinear Observers 244 / 363

Feedback Interconnection Lemma Consider the feedback interconnection ẋ = Ax + Bu, x (0) =x 0, y = Cx, u = (t, y). If the linear system (C, A, B) is (Q, S, R)-SSD, then the equilibrium point x = 0 is globally exponentially stable for every R, S T, Q -D nonlinearity. Proof: dv dt apple! (y, u) V =! (y, ) V apple V. Jaime A. Moreno Nonlinear Observers 245 / 363

Jaime A. Moreno Nonlinear Observers 246 / 363

The main motivation in [58] was: Observer design is usually a two step procedure: 1 System Transformation (immersion) to a "simpler" system, 2 Observer design for the transformed system. The useful "simpler" systems (forms) are usually hard to meet. Simple forms consist usually of a Linear System + Nonlinearity. The design will be done for the Linear System. If the Nonlinearity is of special forms (triangular for HG observers, for example), it will be compensated. The methods are not able to deal with cases where the transformation does not lead to the required form. The aim of the Dissipative Approach is to provide a systematic means to deal with cases not necessarily considered in the previous approaches. Furthermore, and far beyond this initial motivation, it provides a unified framework to design observers; to compare their performance; to analyze and improve their robustness; to include different objectives in the observer design. Jaime A. Moreno Nonlinear Observers 247 / 363

Consider the Nonlinear system ẋ (t) =f (x (t), u (t)), x (t 0 )=x 0, y (t) =h (x (t)), where x (t) 2 R n, u (t) 2 R p, y (t) 2 R q. Suppose that some transformation (immersion) has been applied to the system so that it becomes (t) =A (t)+g (,, u)+' (,u), (t 0 )= 0, (t) =C (t), (t) =H (t), where 2 R r is a linear function of the state. Here the dissipativity theory will be used to propose a Nonlinear Observer for the system in the last form. Jaime A. Moreno Nonlinear Observers 248 / 363

Observer Design: Simple Case I 8 < ẋ = Ax + G ( )+'(t, y, u), x 2 R n, u 2 R q, : y = Cx, y 2 R m, : = Hx, 2 R r. Proposed observer for 8 >< ˆx = Aˆx + L (ŷ y) + G (ˆ + N (ŷ y))+' (t, y, u), : ŷ = Cˆx, >: ˆ = H ˆx. Matrices L 2 R n p, and N 2 R r p are to be designed. Jaime A. Moreno Nonlinear Observers 249 / 363

Jaime A. Moreno Nonlinear Observers 250 / 363

Estimation Error Dynamics I Define the errors e, ˆx x, ỹ, ŷ y, and, ˆ, and the new nonlinearity (z, ), ( ) ( + z). Note that (0, )=0for all. The error dynamic is given by 8 < ė = A L e + G, e (0) =e 0, : z = H N e, : = (z, ), where A L, A + LC, and H N, H + NC. In turn, plant s state. is the influence of the Jaime A. Moreno Nonlinear Observers 251 / 363

Jaime A. Moreno Nonlinear Observers 252 / 363

Observer Design I Assumption (z, ) is (Q, S, R)-D for some non positive semidefinite quadratic form! (, z) = T Q + 2 T Sz + z T Rz 0 8. Theorem Suppose that the Assumption is satisfied. If there are matrices L and N such that the linear subsystem of is R, S T, Q -SSD, then is a global exponential observer for, i.e. there exist constants apple, > 0 such that for all e (0) ke (t)k appleapple ke (0)k exp ( t). Jaime A. Moreno Nonlinear Observers 253 / 363

Observer Design: Extended Case I Assumption (z, ) is (Q i, S i, R i )-D for i = 1, 2,, M. =) is P M i=1 i (Q i, S i, R i )-D for every i 0, i.e. is dissipative w.r.t. the supply rate! (, z) = P M i=1 i! i (, z). It is clear that it is necessary that the quadratic forms are independent. Jaime A. Moreno Nonlinear Observers 254 / 363

Example of a Multivariable Nonlinearity I Consider a lower triangular nonlinearity : R n R m! R n (x, u) = 1 (x 1, u), n 1 (x 1,, x n 1, u), n (x, u) T, (9) with (0, u) =0 for all u. Assume that each component is (globally) Lipschitz, uniformly in u (or for u in a compact set). i.e. i x i, u i y i, u apple k i x i y i, i = 1,, n, where k i > 0 is the Lipschitz constant of i, and x i = x 1 x i T. By defining (z, x, u) = (x, u) (x + z, u), the Lipschitz condition on implies for each component of that i z i, x i, u apple k i z i, i = 1,, n. Jaime A. Moreno Nonlinear Observers 255 / 363

Example of a Multivariable Nonlinearity II Considering the Euclidean norm this implies 2 i (z 1,, z i, x 1,, x i, u) apple ki 2 z1 2 + + zi 2, i = 1,, n. These inequalities show that is (Q i, S i, R i )-dissipative for every i = 1, 2,, n, with (Q i, S i, R i )= b i bi T, 0, k i I i, where b i are the basis vectors of R n, I i = diag (I i, 0 n i ), and I p is the identity matrix of dimension p. Jaime A. Moreno Nonlinear Observers 256 / 363

Theorem Suppose that the assumption is satisfied. If there are matrices L and N, and a vector =( 1,, M ), i 0, such that the linear subsystem of is R, S T, Q -SSD, (Q, S, R )= MX i (Q i, S i, R i ), i=1 then is a global exponential observer for. Jaime A. Moreno Nonlinear Observers 257 / 363

Computational Issues I Observer design: find matrices L and N, a vector =( 1,, M ), i matrix P = P T > 0, and >0such that the inequality apple PAL + A T L P + I + HT N R H N, PG HN T ST G T apple 0 P S H N Q 0, a is satisfied. In general it is a nonlinear matrix inequality feasibility problem. If N = 0 it is an LMI in, P, PL,. If M = 1 and R = 0, then it is an LMI in, P, PL, N. LMIs can be effectively solved by several algorithms in the literature [17]. There is place for optimization, since a feasible LMI has usually a set of solutions. Jaime A. Moreno Nonlinear Observers 258 / 363

Relations with other design methods I Circle criterion design (Arcak and Kokotovic 1999)[6, 27]: Corresponds to the simple case with square and monotonic nonlinearities. Lipschitz observer design (Thau 1973; Rajamani 1998)[65, 79]: Corresponds to the simple case with nonsquare general nonlinearities, and N = 0. ẋ = Ax + (x, u), x (0) =x 0, y = Cx. (x, u) is Lipschitz in x, unif. in u, T + z T z 0, where (z, x, u) = (x, u) (x + z, u),, is ( I, 0, I)-dissipative. Jaime A. Moreno Nonlinear Observers 259 / 363

High-Gain observer design (Gauthier et al. 1992) [30, 31]: It corresponds to the extended case with nonsquare triangular nonlinearities, and N = 0. 2 3 2 3 x 2 1 (x 1, u) ẋ = 6. 7 4 x n 5 + 6. 7 4 n 1 (x 1,, x n 1, u) 5, 0 n (x, u) y = x 1. (x, u) Lip. ) 2 i (z 1,, z i, x 1,, x i ) apple k 2 i z 2 1 + + z2 i, is (Q i, S i, R i )-dissipative, i = 1, 2,, n. Jaime A. Moreno Nonlinear Observers 260 / 363

The proposed method is always applicable when the high-gain methodology is. Moreover, it represents a generalization of the HG method, and can deliver a solution when the HG does not. Since the HG represents only one possible solution, the proposed method can find better solutions, and is not constrained to high-gain ones. Moreover, other optimization and design criteria can be included to find a better solution. A simple and immediate generalization of HG is to use the additional injection depending on matrix N. The system class for which this method is applicable is much bigger, the conservativeness of the results is reduced, and the quality of the results increased. This follows from the fact that, in the proposed method, the particular characteristics of the nonlinearities can be taken into account using different supply rates and not only the triangular and Lipschitzness characteristics used by the HG method. Jaime A. Moreno Nonlinear Observers 261 / 363

Perspective: A More General Version of the Dissipative Observer Basic Idea Decompose the observation error dynamics in two (or more) interconnected subsystems. The Output injection will be used to provide (if possible) complementary dissipativeness. Some references for the Dissipative Observers are: [58, 94, 92, 95, 97, 100, 102, 93, 96, 91, 90, 99, 101, 98, 105, 104, 103] Jaime A. Moreno Nonlinear Observers 262 / 363

Jaime A. Moreno Nonlinear Observers 263 / 363

Unknown Input Observers (UIO) I ẋ = f (x, u)+g (x) w, x (0) =x0 Plant: : y = h (x) (10) x 2 R n : State Vector u 2 R p : Known Input Vector w 2 R q : Arbitrary Unknown Input Vector y 2 R m : Output Vector Jaime A. Moreno Nonlinear Observers 264 / 363

Figure : lim t!1 (x (t) ˆx (t)) = 0 Jaime A. Moreno Nonlinear Observers 265 / 363

Possible Applications of UIOs I State Estimation and Control Robust State Estimation and Control Jaime A. Moreno Nonlinear Observers 266 / 363

Possible Applications of UIOs II Fault Detection and Isolation (FDI) Nonlinear Robust Observer Design Decentralized Control Jaime A. Moreno Nonlinear Observers 267 / 363

UIOs for Linear Time Invariant Systems I ẋ = Ax + Bu + Dw, Plant: lin : y = Cx. dim (x) =n, dim (u) =p, dim (w) =q := rank(d), dim (y) =m := rank(c), Jaime A. Moreno Nonlinear Observers 268 / 363

There exists an UIO for lin [Hautus, 83] m ẋ = Ax + Bu + Dw, y = Cx. apple si A D rank C 0 = n + q, 8 s 2 C + 0 and rank (CD) =rank (D) =q = dim (w). Minimum Phaseness w! y Relative Degree rd = 1 C + 0 denotes the closed right half complex plane. Jaime A. Moreno Nonlinear Observers 269 / 363

The Relative Degree condition rankcd = rankd = q implies dim(y) dim(w). the existence of regular output and state transformations ȳ = apple ȳ1 ȳ 2 = apple H1 H 2 y, apple ȳ1 = apple H1 C M x = Tx, such that in new coordinates lin ȳ 1 = Ā11ȳ 1 + Ā12 + B 1 u + w, = Ā21ȳ 1 + Ā22 + B 2 u, ȳ 2 = C 22. The Minimum Phaseness condition is equivalent to: apple si A D rank C 0 8 < 8 s 2 C + 0, Ā22, C 22 Detectable, = n + q, : 8 s 2 C, Ā22, C 22 Observable. Jaime A. Moreno Nonlinear Observers 270 / 363

A Reduced Order observer is ˆ = Ā21ȳ 1 + Ā22 ˆ + B 2 u + R C22 ˆ ȳ2, ˆx = T 1 apple ȳ1ˆ. The estimation error = ˆ satisfies = Ā22 + R C 22 that converges asymptotically to zero if Ā 22 + R C 22 is Hurwitz. This convergence is assignable when Ā22, C 22 is Observable. Jaime A. Moreno Nonlinear Observers 271 / 363

The class of plants 8 < ẋ = Ax + G ( )+'(t, y, u) Bw, x (0) =x 0, : y = Cx, : = Hx, x 2 R n : states u 2 R p : known inputs w 2 R q : unknown inputs y 2 R m : outputs 2 R r : linear function of the state (not measured) Jaime A. Moreno Nonlinear Observers 272 / 363

Proposed UIO (0) = 0. 8 = A + G (ˆ + N (ŷ y)) + L (ŷ y)+' (t, y, u), >< ŷ = C, : ˆ = H, >: ˆx = D + Ey, Jaime A. Moreno Nonlinear Observers 273 / 363

Estimation error dynamics obs e : 8 ẋ = Ax + G ( )+'(t, y, u) ė =(A + LC) e + G + Bw, >< z =(H + NC) e, ỹ = Ce, = Hx, >: = (z, ), Bw, x (0) =x 0, e (0) =e 0, e = x, ỹ = ŷ y, (z, ), ( ) ( + z). Jaime A. Moreno Nonlinear Observers 274 / 363

Theorem (Rocha & Moreno, 2004, 2010) If 2 4 (a) is dissipative for some! (, z) = T Q + 2 T Sz + z T Rz, i.e.! (, z) 0 8, with Q apple 0. (b) 9 P = P T > 0, L, N,S and >0, such that PA L + A T L P + P PG PB G T P 0 0 B T P 0 0 then there exists an UIO for. 3 5 2 4 H T N RH N H T N ST C T S T SH N Q 0 SC 0 0 3 5 apple 0, Jaime A. Moreno Nonlinear Observers 275 / 363

Dissipativity Property I The time derivative of the (candidate) Storage Function V (e, x) =e T Pe along trajectories of obs e is 2 V = 4 where A L =(A + LC). e w 3 5 T 2 4 PA L + A T L P PG PB G T P 0 0 B T P 0 0 3 2 5 4 e w 3 5, Jaime A. Moreno Nonlinear Observers 276 / 363

Dissipativity Property II From condition (b) 2 V apple 4 = e w " z 3 5 T 2 4 HT N RH N P HN T S T C T S T SH N Q 0 SC 0 0 # T " #" # Q S S T R z 3 2 5 4 V (e)+2ỹ T S T w. e w 3 5, with H N =(H + NC). Because of (a) V apple V (e)+2w T Sỹ, {z} passive output Jaime A. Moreno Nonlinear Observers 277 / 363

Transformation I Fromapple Condition (b) it follows the existence of a transformation SC = e = Te that takes the observation error obs M e to the form 1 = Ā11 + L 1 C1 1 + Ā12 + L 1 C2 2 + Ḡ1 + B T PBw, 2 = Ā21 + L 2 C1 1 + Ā22 + L 2 C2 2 + Ḡ2, z = H + N C = H1 + N C 1, H2 + N C 2, ỹ = C, Sỹ = 1, = (z, ). Jaime A. Moreno Nonlinear Observers 278 / 363

x 1, x 2 : position and velocity of m 1 x 3, x 4 : position and velocity of m 2 A nl : non linear friction, F Anl = c nl sign (x 2 ) ln (1 + x 2 ) u : known force applied to m 1 w : unknown force applied to m 2 The measured states are (x 1, x 4 ) and (x 2, x 3 ) are to be estimated. Jaime A. Moreno Nonlinear Observers 279 / 363

2 ẋ = 6 4 2 + 6 4 0 1 0 0 k 1 +k 2 m 1 b 1 +b 2 m 1 k 2 m 1 b 2 m 1 0 0 0 1 k 2 b 2 k 2 b 2 m 2 m 2 m 2 m 2 0 1 m 1 0 0 3 7 5 u + y 1 =x 1, y 2 = x 4. 2 6 4 0 1 m 1 0 0 3 7 5 ( )+ 3 7 5 x+ 2 6 4 ( )=c nl sign ( ) ln (1 + ), = x 2. 0 0 0 1 m 2 3 7 5 w, Jaime A. Moreno Nonlinear Observers 280 / 363

(z, )=c nl sign ( ) ln (1 + ) c nl sign ( + z) ln (1 + + z ) is in sector [ c nl, 0], i.e., Q = 1, S = 1 2 c nl, R = 0. Parameter values: m 1 = 5 (kg), m 2 = 1 (kg), k 1 = 30 (N/m), k 2 = 10 (N/m), b 1 = 4 (Ns/m), b 2 = 2 (Ns/m), c nl = 5 (N). Jaime A. Moreno Nonlinear Observers 281 / 363

After the transformation b2 h i ȳ 2 = ȳ 2 + k2 b 2 k 2 m m 2 m 2 m 2 + 1 w, 2 m 2 2 3 2 3 2 0 0 1 0 = 4 b 2 5 m 1 ȳ 2 + 4 k 1 +k 2 b 1 +b 2 k 2 5 m 1 m 1 m 1 + 4 1 0 0 0 ȳ 1 = 1 0 0. The observer has the form 0 1 m 1 0 3 5, = A + G (ˆ + N T (ŷ y)) + L T (ŷ y)+' (t, y, u), ŷ = C, ˆ = H, ˆx = D + Ey, and the error (e = x) dynamics ẋ = Ax + G ( )+'(t, y, u) ė =(A + L T C) e + G + Bw, z =(H + N T C) e, = (z, ), Bw, Jaime A. Moreno Nonlinear Observers 282 / 363

is dissipative with storage function Moreover 2 V (") =e T 4 12 0.6 8.1 0 0.6 3.7 1.8 0 8.1 1.8 13.1 0 0 0 0 1 3 5e. V (e) apple 0.85 kek 2 + w T 0 1 e. Jaime A. Moreno Nonlinear Observers 283 / 363

Entradas 10 5 0 conocida u desconocida w no linealidad Estados Error= x xestim 5 0 2 4 6 8 10 12 14 16 18 20 Tiempo (seg) 2 1 0 1 2 0 2 4 6 8 10 12 14 16 18 20 Tiempo (seg) 4 x1 x1estim 2 x2 x2estim x3 x3estim 0 x4 x4estim 2 4 0 2 4 6 8 10 12 14 16 18 20 Tiempo (seg) x1 x2 x3 x4 Simulation of the UIO with u = 5 (N), w = 4 sin (t)+2 (N). The states x 1, x 3 are in (m), x 2, x 4 in (m/s) Some references for the Dissipative design of Unknown Input Observers are: [93, 96, 91]. Jaime A. Moreno Nonlinear Observers 284 / 363

Conclusions The proposed observer design methodology generalizes and unifies several standard design methods. It is based on the dissipativity theory. It eliminates several restrictions: Lipschitz, monotonic and square nonlinearities; observability of the system; High Gain... It opens new design possibilities. For example, using the design degrees of freedom for optimization. It is computable. It can be generalized to more general class of nonlinear systems (at cost of the computability). Jaime A. Moreno Nonlinear Observers 285 / 363

Part V Sliding Mode Observers Jaime A. Moreno Nonlinear Observers 286 / 363

Outline 25 Discontinuous Control A First Order Plant A Second Order Plant 26 Observation Problems Basic Observation Problem Nonlinear systems without inputs A Simple Observer and its Properties Sliding Mode Observer (SMO) 27 Observers a la Second Order Sliding Modes (SOSM) Generalized Super-Twisting Observers 28 Lyapunov Approach for Second-Order Sliding Modes GSTA without perturbations: ALE Jaime A. Moreno Nonlinear Observers 287 / 363

A First Order Plant Consider the plant = + u, (0) =1, (11) where 2 ( 1, 1) is a perturbation. Continuous (linear) Control = k, k > 0, (0) =1. (12) Comments: 0 = sign(0)? RHS of DE continuous (linear). If = 0 exponential (asymptotic) convergence to = 0. If 6= 0 practical convergence. Jaime A. Moreno Nonlinear Observers 288 / 363

Example: Discontinuous control Discontinuous Control = sign( ), (0) =1, with 2 ( 1, 1). >0 ) =< 0 <0 ) => 0 and (t) 0, 8t T. Comments: 0 = sign(0)? RHS of DE is discontinuous. After arriving at = 0, sliding on 0. 1 0.5 0 1 0.5 0 0.5 1 1.2 1 0.8 0.6 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Finite-Time convergence. Differential Inclusion. 2 [, ] sign( ) ūu Jaime A. Moreno Nonlinear Observers 289 / 363

Example: Discontinuous control 2 1.5 Lineal Discontínuo Lineal Discontínuo 1.5 1 0.5 1 0 σ 0.5 u 0.5 0 1 1.5 0.5 2 1 0 5 10 15 20 t 2.5 0 5 10 15 20 t Notice the chattering = infinite switching of the control variable! Jaime A. Moreno Nonlinear Observers 290 / 363

A Second Order Plant Consider the system ẋ 1 = x 2 ż 2 = z 3.. ẋ 2 = (x 1, x 2 )+u y = x 1 with a perturbation/uncertainty. If = 0 two alternative output controllers: Continuous (linear) output controller Discontinuous output controller ẋ 1 = x 2 ẋ 2 = k 1 y y = x 1 ẋ 1 = x 2 ẋ 2 = k 1 sign(y) y = x 1 Jaime A. Moreno Nonlinear Observers 291 / 363

3 Linear controller 5 Discontinuous controller x 1 x 2 4 x 1 x 2 2 3 1 2 1 x 0 x 0 1 1 2 2 3 4 3 0 5 10 15 20 25 30 35 40 t 5 0 5 10 15 20 25 30 35 40 t Both Oscillate! It is impossible to stabilize a double (or triple etc) integrator by static output feedback! Jaime A. Moreno Nonlinear Observers 292 / 363

A State Feedback If = 0 two alternative state feedback controllers: Continuous (linear) state feedback controller ẋ 1 = x 2 ẋ 2 = k 1 x 1 k 2 x 2 y = x 1 Linear+Discontinuous state feedback controller ẋ 1 = x 2 ẋ 2 = k 1 sign(x 1 ) k 2 x 2 y = x 1 Jaime A. Moreno Nonlinear Observers 293 / 363

3 Linear controller Discontinuous controller x 1 x 1 2.5 x 2 0.25 x 2 0.2 2 0.15 1.5 0.1 1 0.05 x x 0 0.5 0.05 0 0.1 0.5 0.15 0.2 1 0.25 1.5 0 5 10 15 20 25 30 35 40 t 0 5 10 15 20 25 30 35 40 t Linear controller stabilize exponentially Linear+Discontinuous also stabilizes exponentially! Jaime A. Moreno Nonlinear Observers 294 / 363

Perturbed case 3 Linear controller Discontinuous controller 2.5 2 x 1 x 2 0.25 0.2 0.15 x 1 x 2 1.5 0.1 1 0.05 x x 0 0.5 0.05 0 0.1 0.5 0.15 0.2 1 0.25 1.5 0 10 20 30 40 t 0 10 20 30 40 t Linear controller is not insensitive to perturbation Linear+Discontinuous is insensitive to perturbation! Jaime A. Moreno Nonlinear Observers 295 / 363

The classical First Order Sliding Mode Controller ẋ 1 = x 2 ẋ 2 = k 2 sign(x 2 + k 1 x 1 ) y = x 1 can be rewritten as a first order system with a stable (first order) zero dynamics: with = x 2 + k 1 x 1 sliding variable then ẋ 1 = k 1 x 1 + = k 2 sign( ) y = x 1 Jaime A. Moreno Nonlinear Observers 296 / 363

Phase plane Jaime A. Moreno Nonlinear Observers 297 / 363

Behavior with perturbation 3 Linear controller Discontinuous controller 2.5 2 x 1 x 2 0.25 0.2 0.15 x 1 x 2 1.5 0.1 1 0.05 x x 0 0.5 0.05 0 0.1 0.5 0.15 0.2 1 0.25 1.5 0 10 20 30 40 t 0 10 20 30 40 t Linear controller stabilize exponentially and is not insensitive to perturbation. SM control also stabilizes exponentially but is insensitive to perturbation! Jaime A. Moreno Nonlinear Observers 298 / 363

The Twisting Controller A discontinuous controller able to obtain finite-time convergence and insensitivity to perturbations: ẋ 1 = x 2 ẋ 2 = k 1 sign(x 1 ) k 2 sign(x 2 ) y = x 1 3 Linear controller 3 Discontinuous controller 2.5 x 1 x 2 x 1 x 2 2 2 1.5 1 1 x x 0 0.5 0 1 0.5 2 1 1.5 0 10 20 30 40 t 3 0 10 20 30 40 t Jaime A. Moreno Nonlinear Observers 299 / 363

Recapitulation What is good in sliding-mode (discontinuous) control? Simple Precise. More than robust (insensitive) against a class of perturbations. What is bad in sliding-mode (discontinuous) control? Infinite switching = Chattering (, whic can be reduced via Higher Order Sliding Modes). Jaime A. Moreno Nonlinear Observers 300 / 363

Basic Observation Problem Jaime A. Moreno Nonlinear Observers 302 / 363

An important Property: Observability Consider a nonlinear system without inputs, x 2 R n y 2 R Differentiating the output y (t) =h (x (t)). ẏ (t) = d dt ÿ (t) = @L f h (x) @x. y (n 1) (t) = @Ln 2 f @x ẋ (t) =f (x (t)), x (t 0 )=x 0, y (t) =h (x (t)). @h (x) @h (x) h (x (t)) = ẋ (t) = @x @x f (x) :=L f h (x). h (x) ẋ (t) = @L f h (x) f (x) :=L 2 f @x h (x). ẋ (t) = @Ln 2 f @x h (x) f (x) :=L n 1 f h (x). Jaime A. Moreno Nonlinear Observers 303 / 363

Evaluating at t = 0 2 6 4 y (0) ẏ (0) ÿ (0). y (k) (0) O n (x): Observability map Theorem 3 2 = 7 6 5 4 h (x 0 ) L f h (x 0 ) L 2 f h (x 0). L k f h (x 0) 3 := O 7 n (x 0 ) 5 If O n (x) is injective (invertible)! The NL system is observable. Jaime A. Moreno Nonlinear Observers 304 / 363

Observability Form In the coordinates of the output and its derivatives z = O n (x), x = On 1 (z) the system takes the (observability) form ż 1 = z 2 ż 2 = z 3. ż n = (z 1, z 2,...,z n ) y = z 1 So we can consider a system in this form as a basic structure. Jaime A. Moreno Nonlinear Observers 305 / 363

A Simple Observer and its Properties Linear Plant with unknown input and a Linear Observer Plant: ẋ 1 = x 2, ẋ 2 = w(t) Observer: ˆx1 = l 1 (ˆx 1 x 1 )+ˆx 2, ˆx 2 = l 2 (ˆx 1 x 1 ) Estimation Error: e 1 = ˆx 1 x 1, e 2 = ˆx 2 x 2 ė 1 = l 1 e 1 + e 2, ė 2 = l 2 e 1 w (t) Figure : Linear Plant with an unknown input and a Linear Observer. Jaime A. Moreno Nonlinear Observers 306 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) 4 2 Estimation error e 1 3 2 1 0 Linear Observer Estimation error e 2 1.5 1 0.5 0 0.5 1 Linear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Linear Observer without unknown input. Jaime A. Moreno Nonlinear Observers 307 / 363

State x 1 140 120 100 80 60 40 20 0 0 20 40 60 Time (sec) Statet x 2 4 3.5 3 2.5 2 1.5 1 0.5 0 20 40 60 Time (sec) 4 2 Estimation error e 1 3 2 1 0 1 Linear Observer 2 0 20 40 60 Time (sec) Estimation error e 2 1.5 1 0.5 0 0.5 1 Linear Observer 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Linear Observer with unknown input. Jaime A. Moreno Nonlinear Observers 308 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Linear Observer without UI with large initial conditions. Jaime A. Moreno Nonlinear Observers 309 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Linear Observer without UI with very large initial conditions. Jaime A. Moreno Nonlinear Observers 310 / 363

Recapitulation. Linear Observer for Linear Plant If no unknown inputs/uncertainties: it converges exponentially fast. If there are unknown inputs/uncertainties: no convergence. At best bounded error. Convergence time depends on the initial conditions of the observer Is it possible to alleviate these drawbacks? Jaime A. Moreno Nonlinear Observers 311 / 363

Sliding Mode Observer (SMO) Figure : Linear Plant with an unknown input and a SM Observer. Jaime A. Moreno Nonlinear Observers 312 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the SM Observer without unknown input. Jaime A. Moreno Nonlinear Observers 313 / 363

State x 1 140 120 100 80 60 40 20 0 0 20 40 60 Time (sec) Statet x 2 4 3.5 3 2.5 2 1.5 1 0.5 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the SM Observer with unknown input. Jaime A. Moreno Nonlinear Observers 314 / 363

Recapitulation Sliding Mode Observer for Linear Plant If no unknown inputs/uncertainties: e 1 converges in finite time, and e 2 converges exponentially fast. If there are unknown inputs/uncertainties: no convergence. At best bounded error. Only e 1 converges in finite time! Convergence time depends on the initial conditions of the observer It is not the solution we expected! None of the objectives has been achieved! Jaime A. Moreno Nonlinear Observers 316 / 363

Super-Twisting Algorithm (STA) Plant: Observer: ẋ 1 = x 2, ẋ 2 = w(t) ˆx 1 = l 1 e 1 1 2 sign (e 1 )+ˆx 2, ˆx 2 = l 2 sign (e 1 ) Estimation Error: e 1 = ˆx 1 x 1, e 2 = ˆx 2 x 2 Solutions in the sense of Filippov. ė 1 = l 1 e 1 1 2 sign (e 1 )+e 2 ė 2 = l 2 sign (e 1 ) w (t), Jaime A. Moreno Nonlinear Observers 318 / 363

Figure : Linear Plant with an unknown input and a SOSM Observer. Jaime A. Moreno Nonlinear Observers 319 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer without unknown input. Jaime A. Moreno Nonlinear Observers 320 / 363

State x 1 140 120 100 80 60 40 20 0 0 20 40 60 Time (sec) Statet x 2 4 3.5 3 2.5 2 1.5 1 0.5 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer with unknown input. Jaime A. Moreno Nonlinear Observers 321 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer without UI with large initial conditions. Jaime A. Moreno Nonlinear Observers 322 / 363

Recapitulation. Super-Twisting Observer for Linear Plant If no unknown inputs/uncertainties: e 1 and e 2 converge in finite-time! If there are unknown inputs/uncertainties: e 1 and e 2 converge in finite-time! Observer is insensitive to perturbation/uncertainty! Convergence time depends on the initial conditions of the observer. This objective is not achieved! Jaime A. Moreno Nonlinear Observers 323 / 363

Generalized Super-Twisting Algorithm (GSTA) Plant: Observer: ẋ 1 = x 2, ẋ 2 = w(t) ˆx 1 = l 1 1 (e 1 )+ˆx 2, ˆx 2 = l 2 2 (e 1 ) Estimation Error: e 1 = ˆx 1 x 1, e 2 = ˆx 2 x 2 Solutions in the sense of Filippov. ė 1 = l 1 1 (e 1 )+e 2 ė 2 = l 2 2 (e 1 ) w (t), 1 (e 1 )=µ 1 e 1 1 2 sign (e 1 )+µ 2 e 1 3 2 sign (e 1 ),µ 1,µ 2 0, 2 (e 1 )= µ2 1 2 sign (e 1)+2µ 1 µ 2 e 1 + 3 2 µ2 2 e 1 2 sign (e 1 ), Jaime A. Moreno Nonlinear Observers 324 / 363

Figure : Linear Plant with an unknown input and a Non Linear Observer. Jaime A. Moreno Nonlinear Observers 325 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer without unknown input and large initial conditions. Jaime A. Moreno Nonlinear Observers 326 / 363

60 2 State x 1 50 40 30 20 10 Statet x 2 1.5 1 0.5 0 0 20 40 60 Time (sec) 0 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer without unknown input and very large initial conditions. Jaime A. Moreno Nonlinear Observers 327 / 363

State x 1 140 120 100 80 60 40 20 0 0 20 40 60 Time (sec) Statet x 2 4 3.5 3 2.5 2 1.5 1 0.5 0 20 40 60 Time (sec) Estimation error e 1 4 3 2 1 0 Linear Observer Nonlinear Observer Estimation error e 2 2 1.5 1 0.5 0 0.5 1 Linear Observer Nonlinear Observer 1 0 20 40 60 Time (sec) 1.5 0 20 40 60 Time (sec) Figure : Behavior of Plant and the Non Linear Observer with UI with large initial conditions. Jaime A. Moreno Nonlinear Observers 328 / 363

Effect: Convergence time independent of I.C. 16 NSOSMO GSTA with linear term 14 STO 12 Convergence Time T 10 8 6 4 2 0 10 1 10 2 10 3 10 4 norm of the initial condition x(0) (logaritmic scale) Figure : Convergence time when the initial condition grows. Jaime A. Moreno Nonlinear Observers 329 / 363

Recapitulation Generalized Super-Twisting Observer for Linear Plant If no unknown inputs/uncertainties: e 1 and e 2 converge in finite-time! If there are unknown inputs/uncertainties: e 1 and e 2 converge in finite-time! Observer is insensitive to perturbation/uncertainty! Convergence time is independent of the initial conditions of the observer!. All objectives were achieved! How to prove these properties? Jaime A. Moreno Nonlinear Observers 330 / 363

Lyapunov functions: 1 We propose a Family of strong Lyapunov functions, that are Quadratic-like 2 This family allows the estimation of convergence time, 3 It allows to study the robustness of the algorithm to different kinds of perturbations, 4 All results are obtained in a Linear-Like framework, known from classical control, 5 The analysis can be obtained in the same manner for a linear algorithm, the classical ST algorithm and a combination of both algorithms (GSTA), that is non homogeneous. Jaime A. Moreno Nonlinear Observers 332 / 363

Generalized STA Solutions in the sense of Filippov. ẋ 1 = k 1 1 (x 1 )+x 2 ẋ 2 = k 2 2 (x 1 ), (13) 1 (e 1 )=µ 1 e 1 1 2 sign (e 1 )+µ 2 e 1 q sign (e 1 ),µ 1,µ 2 0, q 1, 2 (e 1 )= 0 1 (x 1 ) 1 (x 1 )= = µ2 1 2 sign (e 1)+ q + 1 µ 1 µ 2 e 1 q 1 2 sign (e 1 )+ 2 + qµ 2 2 e 1 2q 1 sign (e 1 ), Standard STA: µ 1 = 1, µ 2 = 0 Linear Algorithm: µ 1 = 0, µ 2 > 0, q = 1. GSTA: µ 1 = 1, µ 2 > 0, q > 1. Jaime A. Moreno Nonlinear Observers 333 / 363

Quadratic-like Lyapunov Functions System can be written as: apple = 0 1 (x 1 ) A, = Family of strong Lyapunov Functions: Time derivative of Lyapunov Function: apple 1 (x 1 ) k1 1, A = x 2 k 2 0. V (x) = T P,P = P T > 0. V (x) = 0 1 (x 1 ) T A T P + PA = 0 1 (x 1 ) T Q Algebraic Lyapunov Equation (ALE): A T P + PA = Q Jaime A. Moreno Nonlinear Observers 334 / 363

Figure : The Lyapunov function. Jaime A. Moreno Nonlinear Observers 335 / 363