Linear Systems Manfred Morari Melanie Zeilinger Institut für Automatik, ETH Zürich Institute for Dynamic Systems and Control, ETH Zürich Spring Semester 2016 Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification 4. State Estimation 4.1 Linear State Estimation 4.2 State Observer 4.3 Kalman Filter inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Systems Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification 4. State Estimation 4.1 Linear State Estimation 4.2 State Observer 4.3 Kalman Filter inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Systems Models of Dynamic Systems Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Systems Models of Dynamic Systems Models of Dynamic Systems Goal: Introduce mathematical models to be used in Model Predictive Control (MPC) describing the behavior of dynamic systems Model classification: state space/transfer function, linear/nonlinear, time-varying/time-invariant, continuous-time/discrete-time, deterministic/stochastic If not stated differently, we use deterministic models Models of physical systems derived from first principles are mainly: nonlinear, time-invariant, continuous-time, state space models (*) Target models for standard MPC are mainly: linear, time-invariant, discrete-time, state space models ( ) Focus of this section is on how to transform (*) to ( ) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-2
Linear Systems Models of Dynamic Systems Nonlinear, Time-Invariant, Continuous-Time, State Space Models (1/3) ẋ = g(x, u) y = h(x, u) x R n state vector g(x, u) : R n R m R n system dynamics u R m input vector h(x, u) : R n R m R p output function y R p output vector Very general class of models Higher order ODEs can be easily brought to this form (next slide) Analysis and control synthesis generally hard linearization to bring it to linear, time-invariant (LTI), continuous-time, state space form Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-3
Linear Systems Models of Dynamic Systems Nonlinear, Time-Invariant, Continuous-Time, State Space Models (2/3) Equivalence of one n-th order ODE and n 1-st order ODEs Define x (n) + g n (x, ẋ, ẍ,..., x (n 1) ) = 0 x i+1 = x (i), i = 0,..., n 1 Transformed system ẋ 1 = x 2 ẋ 2 = x 3.. ẋ n 1 = x n ẋ n = g n (x 1, x 2,..., x n ) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-4
Linear Systems Models of Dynamic Systems Nonlinear, Time-Invariant, Continuous-Time, State Space Models (3/3) Example: Pendulum Moment of inertia wrt. rotational axis m l 2 Torque caused by external force T c Torque caused by gravity m g l sin(θ) T c u l System equation m l 2 θ = Tc m g l sin(θ) Using x 1 θ, x 2 θ = ẋ 1 and u T c /m l 2 the system can be brought to standard form ( ) ( ) ẋ1 x ẋ = = 2 g l sin(x = g(x, u) 1) + u ẋ 2 Output equation depends on the measurement configuration, i.e. if θ is measured then y = h(x, u) = x 1. mg Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-5
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (1/6) ẋ = A c x + B c u y = Cx + Du x R n state vector A c R n n system matrix u R m input vector B c R n m input matrix y R p output vector C R p n output matrix D R p m throughput matrix Vast theory exists for the analysis and control synthesis of linear systems Exact solution (next slide) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-6
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (2/6) Solution to linear ODEs Consider the ODE (written with explicit time dependence) ẋ(t) = A c x(t) + B c u(t) with initial condition x 0 x(t 0 ), then its solution is given by t x(t) = e Ac (t t 0) x 0 + e A c (t τ) Bu(τ)dτ t 0 where e Act n=0 (A c t) n n! Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-7
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (3/6) Problem: Most physical systems are nonlinear but linear systems are much better understood Nonlinear systems can be well approximated by a linear system in a small neighborhood around a point in state space Idea: Control keeps the system around some operating point replace nonlinear by a linearized system around operating point First order Taylor expansion of f ( ) around x f (x) f ( x) + f x (x x), with f x= x x = f 1 x 1 f 1 x 2.... f n x 1 f n x 2... f 1 x n. f n x n Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-8
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (4/6) Linearization u s keeps the system around stationary operating point x s ẋ s = g(x s, u s ) = 0, y s = h(x s, u s ) ẋ x }{{} s =0 ẋ = g(x s, u s ) + g }{{} x =0 x=xs u=u s } {{ } =A c = ẋ = A c x + B c u y = h(x s, u s ) }{{} y s + y }{{} = C x + D u y y s h x x=xs u=u }{{} s =C (x x s ) + g }{{} u = x x=xs u=u s } {{ } =B c (x x s ) + h u x=xs u=u s } {{ } =D (u u s ) }{{} = u (u u s ) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-9
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (5/6) Linearization The linearized system is written in terms of deviation variables x, u, y Linearized system is only a good approximation for small x, u Subsequently, instead of x, u and y, x, u and y are used for brevity Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-10
Linear Systems Models of Dynamic Systems LTI Continuous-Time State Space Models (6/6) Example: Linearization of pendulum equations ẋ = ( ) ( ẋ1 = ẋ 2 y = x 1 = h(x, u) x 2 g l sin(x 1) + u ) = g(x, u) Want to keep the pendulum around x s = (π/4, 0) u s = g l sin(π/4) A c = g ( x = x=xs u=u s C = h x = (1 0), D = h x=xs u u=u s ) 0 1 g l cos(π/4) 0, B c = g u = 0 x=xs u=u s = x=xs u=u s ( 0 1 ) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-11
Linear Systems Models of Dynamic Systems LTI Discrete-Time State Space Models (1/4) Linear discrete-time systems are described by linear difference equations x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) Inputs and outputs of a discrete-time system are defined only at discrete time points, i.e. its inputs and outputs are sequences defined for k Z + Discrete time systems describe either 1 Inherently discrete systems, eg. bank savings account balance at the k-th month x(k + 1) = (1 + α)x(k) + u(k) 2 Transformed continuous-time system Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-12
Linear Systems Models of Dynamic Systems LTI Discrete-Time State Space Models (2/4) Vast majority of controlled systems not inherently discrete-time systems Controllers almost always implemented using microprocessors Finite computation time must be considered in the control system design discretize the continuous-time system Discretization is the procedure of obtaining an equivalent discrete-time system from a continuous-time system The discrete-time model describes the state of the continuous-time system only at particular instances t k, k Z + in time, where t k+1 = t k + T s and T s is called the sampling time Usually u(t) = u(t k ) t [t k, t k+1 ) is assumed (and implemented) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-13
Linear Systems Models of Dynamic Systems LTI Discrete-Time State Space Models (3/4) Discretization of LTI continuous-time state space models Recall the solution of the ODE x(t) = e Ac (t t 0) x 0 + t t 0 e Ac (t τ) Bu(τ)dτ Choose t 0 = t k (hence x 0 = x(t 0 ) = x(t k )), t = t k+1 and use t k+1 t k = T s and u(t) = u(t k ) t [t k, t k+1 ) tk+1 x(t k+1 ) = e Ac T s x(t k ) + t k Ts e A c (t k+1 τ) B c dτu(t k ) = e Ac T s }{{} x(t k ) + e Ac (T s τ ) B c dτ u(t k ) 0 }{{} A = Ax(t k ) + Bu(t k ) We found the exact discrete-time model predicting the state of the continuous-time system at time t k+1 given x(t k ), k Z + under the assumption of a constant u(t) during a sampling interval B = (A c ) 1 (A I )B c, if A c invertible B Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-14
Linear Systems Models of Dynamic Systems LTI Discrete-Time State Space Models (4/4) Example: Discretization of the linearized pendulum equations Using g/l = 10[s 2 ] the pendulum equations linearized about x s = (π/4, 0) are given by ( ẋ(t) = 0 1 10/ 2 0 ) ( 0 x(t) + 1 ) u(t) Discretizing the continuous-time system using the definitions of A and B, and T s = 0.1 s, we get the following discrete-time system ( ) ( ) 0.965 0.099 0.005 x(k + 1) = x(k) + u(k) 0.699 0.965 0.100 Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-15
Linear Systems Analysis of LTI Discrete-Time Systems Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Systems Analysis of LTI Discrete-Time Systems Analysis of LTI Discrete-Time Systems Goal: Introduce the concepts of stability, controllability and observability From this point on we consider only discrete-time LTI systems for the rest of the lecture Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-16
Linear Systems Analysis of LTI Discrete-Time Systems Coordinate Transformations (1/2) Consider again the system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) Input-output behavior, i.e. the sequence {y(k)} k=0,1,2... entirely defined by x(0) and {u(k)} k=0,1,2... Infinitely many choices of the state that yield the same input-output behavior Certain choices facilitate system analysis Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-17
Linear Systems Analysis of LTI Discrete-Time Systems Coordinate Transformations (2/2) Consider the linear transformation x = Tx with det(t) 0 (invertible) or T 1 x(k + 1) = AT 1 x(k) + Bu(k) y(k) = CT 1 x(k) + Du(k) Note: u(k) and y(k) are unchanged x(k + 1) = TAT }{{ 1 } x(k) + }{{} TB u(k) Ã B y(k) = CT }{{ 1 } x(k) + }{{} D u(k) C D Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-18
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Linear Systems (1/3) Theorem: Asymptotic Stability of Linear Systems The LTI system is globally asymptotically stable x(k + 1) = Ax(k) lim x(k) = 0, x(0) Rn k if and only if λ i < 1, i = 1,, n where λ i are the eigenvalues of A. inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-19
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Linear Systems (2/3) Proof of asymptotic stability condition Assume that A has n linearly independent eigenvectors e 1,, e n then the coordinate transformation x = [e 1,, e n ] 1 x = Tx transforms an LTI discrete-time system to λ 1 0 0 x(k + 1) = TAT 1 x(k) = 0 λ 2 0... 0.. x(k) = Λ x(k) 0 0 0 λ n The state x(k) can be explicitly formulated as a function of x(0) = Tx(0) λ k 1 0 0 x(k) = Λ k x(0) = 0 λ k 2 0... 0.. 0 x(0) 0 0 λ k n Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-20
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Linear Systems (3/3) Proof of asymptotic stability condition We thus have that x(k) = Λ k x(0) x(k) = Λ k x(0) (component-wise) x(k) = Λ k x(0) x i (k) = λ k i x i (0) = λ i k x i (0) If any λ i 1 then lim k x(k) 0 for x(0) 0. On the other hand if λ i < 1 i 1,, n then lim k x(k) = 0 and we have asymptotic stability If the system does not have n linearly independent eigenvectors it can not be brought into diagonal form and Jordan matrices have to be used for the proof but the assertions still hold Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-21
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Nonlinear Systems (1/5) For nonlinear systems there are many definitions of stability. Informally, we define a system to be stable in the sense of Lyapunov, if it stays in any arbitrarily small neighborhood of the origin when it is disturbed slightly. In the following we always mean stability in the sense of Lyapunov. We consider first the stability of a nonlinear, time-invariant, discrete-time system x k+1 = g(x k ) (1) with an equilibrium point at 0, i.e. g(0) = 0. Note that system (1) encompasses any open- or closed-loop autonomous system. We will then derive simpler stability conditions for the specific case of LTI systems. Note that always stability is a property of an equilibrium point of a system. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-22
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Nonlinear Systems (2/5) Definitions Formally, the equilibrium point x = 0 of a system (1) is stable if for every ɛ > 0 there exists a δ(ɛ) such that x 0 < δ(ɛ) x k < ɛ, k 0 unstable otherwise. An equilibrium point x = 0 of system (1) is asymptotically stable in Ω R n if it is Lyapunov stable and lim k x k = 0, x 0 Ω globally asymptotically stable if it is asymptotically stable and Ω = R n Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-23
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Nonlinear Systems (3/5) Lyapunov functions We can show stability by constructing a Lyapunov function Idea: A mechanical system is asymptotically stable when the total mechanical energy is decreasing over time (friction losses). A Lyapunov function is a system theoretic generalization of energy Definition: Lyapunov function Consider the equilibrium point x = 0 of system (1). Let Ω R n be a closed and bounded set containing the origin. A function V : R n R, continuous at the origin, finite for every x Ω, and such that V (0) = 0 and V (x) > 0, x Ω \ {0} V (g(x k )) V (x k ) α(x k ) x k Ω \ {0} where α : R n R is continuous positive definite, is called a Lyapunov function. inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-24
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Nonlinear Systems (4/5) Lyapunov theorem Theorem: Lyapunov stability (asymptotic stability) If a system (1) admits a Lyapunov function V (x), then x = 0 is asymptotically stable in Ω. Theorem: Lyapunov stability (global asymptotic stability) If a system (1) admits a Lyapunov function V (x) that additionally satisfies x V (x), then x = 0 is globally asymptotically stable. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-25
Linear Systems Analysis of LTI Discrete-Time Systems Stability of Nonlinear Systems (5/5) Remarks Note that the Lyapunov theorems only provide sufficient conditions Lyapunov theory is a powerful concept for proving stability of a control system, but for general nonlinear systems it is usually difficult to find a Lyapunov function Lyapunov functions can sometimes be derived from physical considerations One common approach: Decide on form of Lyapunov function (e.g., quadratic) Search for parameter values e.g. via optimization so that the required properties hold For linear systems there exist constructive theoretical results on the existence of a quadratic Lyapunov function Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-26
Linear Systems Analysis of LTI Discrete-Time Systems Global Lyapunov Stability of Linear Systems (1/3) Consider the linear system x(k + 1) = Ax(k) (2) Take V (x) = x Px with P > 0 (positive definite) as a candidate Lyapunov function. It satisfies V (0) = 0, V (x) > 0 and x V (x). Check energy decrease condition V (Ax(k)) V (x(k)) = x (k)a PAx(k) x (k)px(k) = x (k)(a PA P)x(k) α(x(k)) We can choose α(x(k)) = x (k)qx(k), Q > 0. Hence, the condition can be satisfied if a P > 0 can be found that solves the discrete-time Lyapunov equation A PA P = Q, Q > 0. (3) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-27
Linear Systems Analysis of LTI Discrete-Time Systems Global Lyapunov Stability of Linear Systems (2/3) Theorem: Existence of solution to the DT Lyapunov equation The discrete-time Lyapunov equation (3) has a unique solution P > 0 if and only if A has all eigenvalues inside the unit circle, i.e. if the system x(k + 1) = Ax(k) is stable. Therefore, for LTI systems global asymptotic Lyapunov stability is not only sufficient but also necessary, and it agrees with the notion of stability based on eigenvalue location. Note that stability is always global for linear systems. inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-28
Linear Systems Analysis of LTI Discrete-Time Systems Global Lyapunov Stability of Linear Systems (3/3) Property of P The matrix P can also be used to determine the infinite horizon cost-to-go for an asymptotically stable autonomous system x(k + 1) = Ax(k) with a quadratic cost function determined by Q. Proof More precisely, defining Ψ(x(0)) as we have that Ψ(x(0)) = x(k) Qx(k) = x(0) (A k ) QA k x(0) (4) k=0 k=0 Ψ(x(0)) = x(0) Px(0). (5) Define H k (A k ) QA k and P k=0 H k (limit of the sum exists because the system is assumed asymptotically stable). We have that A H k A = (A k+1 ) QA k+1 = H k+1. Thus A PA = k=0 A H k A = k=0 H k+1 = k=1 H k = P H 0 = P Q. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-29
Linear Systems Analysis of LTI Discrete-Time Systems Controllability (1/3) Definition: A system x(k + 1) = Ax(k) + Bu(k) is controllable 1 if for any pair of states x(0), x there exists a finite time N and a control sequence {u(0),, u(n 1)} such that x(n ) = x,i.e. x = x(n ) = A N x(0) + ( B AB A N 1 B ) u(n 1) u(n 2). u(0) It follows from the Cayley-Hamilton theorem that A k can be expressed as linear combinations of A i, i 0, 1,, n 1 for k n. Hence for all N n range ( B AB A N 1 B ) = range ( B AB A n 1 B ) 1 Often referred to as reachable for discrete time systems. inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-30
Linear Systems Analysis of LTI Discrete-Time Systems Controllability (2/3) If the system cannot be controlled to x in n steps, then it cannot in an arbitrary number of steps Define the controllability matrix C = ( B AB A n 1 B ) The system is controllable if u(n 1) u(n 2) C. = x A n x(0) u(0) has a solution for all right-hand sides (RHS) From linear algebra: solution exists for all RHS iff n columns of C are linearly independent Necessary and and sufficient condition for controllability is rank(c) = n Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-31
Linear Systems Analysis of LTI Discrete-Time Systems Controllability (3/3) Remarks Another related concept is stabilizability A system is called stabilizable if it there exists an input sequence that returns the state to the origin asymptotically, starting from an arbitrary initial state A system is stabilizable iff all of its uncontrollable modes are stable Stabilizability can be checked using the following condition if rank ([λ i I A B]) = n λ i Λ + A (A, B) is stabilizable where Λ + A is the set of all eigenvalues of A lying on or outside the unit circle. Controllability implies stabilizability Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-32
Linear Systems Analysis of LTI Discrete-Time Systems Observability (1/3) Consider the following system with zero input x(k + 1) = Ax(k) y(k) = Cx(k) Definition: A system is said to be observable if there exists a finite N such that for every x(0) the measurements y(0), y(1), y(n 1) uniquely distinguish the initial state x(0) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-33
Linear Systems Analysis of LTI Discrete-Time Systems Observability (2/3) Question of uniqueness of the linear equations y(0) C y(1) CA. y(n 1) =. CA N 1 x(0) As previously we can replace N by n wlog. (Cayley-Hamilton) Define O = ( C (CA) (CA n 1 ) ) From linear algebra: solution is unique iff the n columns of O are linearly independent Necessary and sufficient condition for observability of system (A, C) is rank(o) = n Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-34
Linear Systems Analysis of LTI Discrete-Time Systems Observability (3/3) Remarks Another related concept is detectability A system is called detectable if it possible to construct from the measurement sequence a sequence of state estimates that converges to the true state asymptotically, starting from an arbitrary initial estimate A system is detectable iff all of its unobservable modes are stable Detectability can be checked using the following condition if rank ([A λ i I C ]) = n λ i Λ + A (A, C) is detectable where Λ + A is the set of all eigenvalues of A lying on or outside the unit circle. Observability implies detectability Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 1-35
Linear Quadratic Optimal Control Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification 4. State Estimation 4.1 Linear State Estimation 4.2 State Observer 4.3 Kalman Filter inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Quadratic Optimal Control Optimal Control Table of Contents 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Quadratic Optimal Control Optimal Control Optimal Control Introduction (1/2) Discrete-time optimal control is concerned with choosing an optimal input sequence U 0 N [u 0, u 1,...] (as measured by some objective function), over a finite or infinite time horizon, in order to apply it to a system with a given initial state x(0). The objective, or cost, function is often defined as a sum of stage costs q(x k, u k ) and, when the horizon has finite length N, a terminal cost p(x N ): N 1 J 0 N (x 0, U 0 N ) p(x N ) + q(x k, u k ) k=0 The states {x k } N k=0 must satisfy the system dynamics x k+1 = g(x k, u k ), k = 0,..., N 1 x 0 = x(0) and there may be state and/or input constraints h(x k, u k ) 0, k = 0,..., N 1. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-36
Linear Quadratic Optimal Control Optimal Control Optimal Control Introduction (2/2) In the finite horizon case, there may also be a constraint that the final state x N lies in a set X f x N X f A general finite horizon optimal control formulation for discrete-time systems is therefore J 0 N(x(0)) min U 0 N J 0 N (x(0), U 0 N ) subject to x k+1 = g(x k, u k ), k = 0,..., N 1 h(x k, u k ) 0, k = 0,..., N 1 x N X f x 0 = x(0) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-37
Linear Quadratic Optimal Control Optimal Control Linear Quadratic Optimal Control In this section, only linear discrete-time time-invariant systems and quadratic cost functions x(k + 1) = Ax(k) + Bu(k) N 1 J 0 (x 0, U 0 ) x NPx N + [x kqx k + u kru k ] (6) are considered, and we consider only the problem of regulating the state to the origin, without state or input constraints. The two most common solution approaches will be described here k=0 1 Batch Approach, which yields a series of numerical values for the input 2 Recursive Approach, which uses Dynamic Programming to compute control policies or laws, i.e. functions that describe how the control decisions depend on the system states. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-38
Linear Quadratic Optimal Control Optimal Control Unconstrained Finite Horizon Control Problem Goal: Find a sequence of inputs U 0 [u 0,..., u N 1 ] that minimizes the objective function N 1 J0 (x(0)) min x NPx N + [x kqx k + u kru k ] U 0 k=0 subject to x k+1 = Ax k + Bu k, k = 0,..., N 1 x 0 = x(0) P 0, with P = P, is the terminal weight Q 0, with Q = Q, is the state weight R 0, with R = R, is the input weight N is the horizon length Note that x(0) is the current state, whereas x 0,..., x N and u 0,..., u N 1 are optimization variables that are constrained to obey the system dynamics and the initial condition. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-39
Linear Quadratic Optimal Control Batch Approach Table of Contents 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Quadratic Optimal Control Batch Approach Solution approach 1: Batch Approach (1/4) The batch solution explicitly represents all future states x k in terms of initial condition x 0 and inputs u 0,..., u N 1. Starting with x 0 = x(0), we have x 1 = Ax(0) + Bu 0, and x 2 = Ax 1 + Bu 1 = A 2 x(0) + ABu 0 + Bu 1, by substitution for x 1, and so on. Continuing up to x N we obtain: x 0 x 1.. x N = I A.. A N x(0) + The equation above can be represented as 0 0 B 0 0 AB B 0....... 0 A N 1 B AB B u 0 u 1.. u N 1 X S x x(0) + S u U 0. (7) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-40
Linear Quadratic Optimal Control Batch Approach Solution approach 1: Batch Approach (2/4) Define Q blockdiag(q,..., Q, P) and R blockdiag(r,..., R) Then the finite horizon cost function (6) can be written as J 0 (x(0), U 0 ) = X QX + U 0 RU 0. (8) Eliminating X by substituting from (7), equation (8) can be expressed as: J 0 (x(0), U 0 ) = (S x x(0) + S u U 0 ) Q(S x x(0) + S u U 0 ) + U 0 RU 0 = U 0 HU 0 + 2x(0) FU 0 + x(0) S x QS x x(0) where H S u QS u + R and F S x QS u. Note that H 0, since R 0 and S u QS u 0. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-41
Linear Quadratic Optimal Control Batch Approach Solution approach 1: Batch Approach (3/4) Since the problem is unconstrained and J 0 (x(0), U 0 ) is a positive definite quadratic function of U 0 we can solve for the optimal input U 0 by setting the gradient with respect to U 0 to zero: U0 J 0 (x(0), U 0 ) = 2HU 0 + 2F x(0) = 0 U 0 (x(0)) = H 1 F x(0) = (S u QS u + R) 1 S u QS x x(0), which is a linear function of the initial state x(0). Note H 1 always exists, since H 0 and therefore has full rank. The optimal cost can be shown (by back-substitution) to be J 0 (x(0)) = x(0) FHF x(0) + x(0) S x QS x x(0) = x(0) (S x QS x S x QS u (S u QS u + R) 1 S u QS x )x(0), Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-42
Linear Quadratic Optimal Control Batch Approach Solution approach 1: Batch Approach (4/4) Summary The Batch Approach expresses the cost function in terms of the initial state x(0) and input sequence U 0 by eliminating the states x k. Because the cost J 0 (x(0), U 0 ) is a strictly convex quadratic function of U 0, its minimizer U0 is unique and can be found by setting U0 J 0 (x(0), U 0 ) = 0. This gives the optimal input sequence U0 as a linear function of the intial state x(0): U 0 (x(0)) = (S u QS u + R) 1 S u QS x x(0) The optimal cost is a quadratic function of the initial state x(0) J 0 (x(0)) = x(0) (S x QS x S x QS u (S u QS u + R) 1 S u QS x )x(0) If there are state or input constraints, solving this problem by matrix inversion is not guaranteed to result in a feasible input sequence Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-43
Linear Quadratic Optimal Control Recursive Approach Table of Contents 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (1/8) Alternatively, we can use dynamic programming to solve the same problem in a recursive manner. Define the j-step optimal cost-to-go as the optimal cost attainable for the step j problem: J j (x(j)) N 1 min x u j,...,u N 1 NPx N + [x kqx k + u kru k ] k=j subject to x k+1 = Ax k + Bu k, k = j,..., N 1 x j = x(j) This is the minimum cost attainable for the remainder of the horizon after step j Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-44
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (2/8) Consider the 1-step problem (solved at time N 1) J N 1(x N 1 ) = min u N 1 {x N 1Qx N 1 + u N 1Ru N 1 + x NP N x N } (9) subject to x N = Ax N 1 + Bu N 1 (10) P N = P where we introduced the notation P j to express the optimal cost-to-go x jp j x j. In particular, P N = P. Substituting (10) into (9) J N 1(x N 1 ) = min u N 1 {x N 1(A P N A + Q)x N 1 + u N 1(B P N B + R)u N 1 + 2x N 1A P N Bu N 1 } Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-45
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (3/8) Solving again by setting the gradient to zero leads to the following optimality condition for u N 1 Optimal 1-step input: 1-step cost-to-go: where 2(B P N B + R)u N 1 + 2B P N Ax N 1 = 0 u N 1 = (B P N B + R) 1 B P N Ax N 1 F N 1 x N 1 J N 1(x N 1 ) = x N 1P N 1 x N 1, P N 1 = A P N A + Q A P N B(B P N B + R) 1 B P N A. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-46
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (4/8) The recursive solution method used from here relies on Bellman s Principle of Optimality For any solution for steps 0 to N to be optimal, any solution for steps j to N with j 0, taken from the 0 to N solution, must itself be optimal for the j-to-n problem Therefore we have, for any j = 0,..., N J j (x j ) = min u j {J j+1(x j+1 ) + x jqx j + u jru j } s.t. x j+1 = Ax j + Bu j Suppose that the fastest route from Los Angeles to Boston passes through Chicago. Then the principle of optimality formalizes the obvious fact that the Chicago to Boston portion of the route is also the fastest route for a trip that starts from Chicago and ends in Boston. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-47
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (5/8) Now consider the 2-step problem, posed at time N 2 { N 1 } JN 2(x N 2 ) = min x kqx k + u kru k + x NPx N u N 1,u N 2 k=n 2 s.t. x k+1 = Ax k + Bu k, k = N 2, N 1 From the Principle of Optimality, the cost function is equivalent to J N 2(x N 2 ) = min u N 2 {J N 1(x N 1 ) + x N 2Qx N 2 + u N 2Ru N 2 } = min u N 2 {x N 1P N 1 x N 1 + x N 2 Qx N 2 + u N 2Ru N 2 } Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-48
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (6/8) As with 1-step solution, solve by setting the gradient with respect to u N 2 to zero Optimal 2-step input 2-step cost-to-go where u N 2 = (B P N 1 B + R) 1 B P N 1 Ax N 2 F N 2 x N 2 J N 2(x N 2 ) = x N 2P N 2 x N 2, P N 2 = A P N 1 A + Q A P N 1 B(B P N 1 B + R) 1 B P N 1 A We now recognize the recursion for P j and u j, j = N 1,, 0. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-49
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (7/8) We can obtain the solution for any given time step k in the horizon u (k) = (B P k+1 B + R) 1 B P k+1 Ax(k) F k x(k) for k = 1,..., N where we can find any P k by recursive evaluation from P N = P, using P k = A P k+1 A + Q A P k+1 B(B P k+1 B + R) 1 B P k+1 A (11) This is called the Discrete Time Riccati Equation or Riccati Difference Equation (RDE). Evaluating down to P 0, we obtain the N -step cost-to-go J 0 (x(0)) = x(0) P 0 x(0) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-50
Linear Quadratic Optimal Control Recursive Approach Solution approach 2: Recursive Approach (8/8) Summary From the Principle of Optimality, the optimal control policy for any step j is then given by u (k) = (B P k+1 B + R) 1 B P k+1 Ax(k) = F k x(k) and the optimal cost-to-go is J k (x(k)) = x kp k x(k) Each P k is related to P k+1 by the Riccati Difference Equation P k = A P k+1 A + Q A P k+1 B(B P k+1 B + R) 1 B P k+1 A, which can be initialized with P N = P, the given terminal weight Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-51
Linear Quadratic Optimal Control Recursive Approach Comparison of Batch and Recursive Approaches (1/2) Fundamental difference: Batch optimization returns a sequence U 0 (x(0)) of numeric values depending only on the initial state x(0), while dynamic programming yields feedback policies u (k) = F k x(k), k = 0,..., N 1 depending on each x(k). If the state evolves exactly as modelled, then the sequences of control actions obtained from the two approaches are identical. The recursive solution should be more robust to disturbances and model errors, because if the future states later deviate from their predicted values, the exact optimal input can still be computed. The Recursive Approach is computationally more attractive because it breaks the problem down into single-step problems. For large horizon length, the Hessian H in the Batch Approach, which must be inverted, becomes very large. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-52
Linear Quadratic Optimal Control Recursive Approach Comparison of Batch and Recursive Approaches (2/2) Without any modification, both solution methods will break down when inequality constraints on x k or u k are added. The Batch Approach is far easier to adapt than the Recursive Approach when constraints are present: just perform a constrained minimization for the current state. Doing this at every time step within the time available, and then using only the first input from the resulting sequence, amounts to receding horizon control. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-53
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Table of Contents 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Infinite Horizon Control Problem: Optimal Sol n (1/2) In some cases we may want to solve the same problem with an infinite horizon: { } J (x(0)) = min [x kqx k + u kru k ] u( ) k=0 subject to x k+1 = Ax k + Bu k, k = 0, 1, 2,...,, x 0 = x(0) As with the Dynamic Programming approach, the optimal input is of the form u (k) = (B P B + R) 1 B P Ax(k) F x(k) and the infinite-horizon cost-to-go is J (x(k)) = x(k) P x(k). Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-54
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Infinite Horizon Control Problem: Optimal Sol n (2/2) The matrix P comes from an infinite recursion of the RDE, from a notional point infinitely far into the future. Assuming the RDE does converge to some constant matrix P, it must satisfy the following (from (11), with P k = P k+1 = P ) P = A P A + Q A P B(B P B + R) 1 B P A, which is called the Algebraic Riccati Equation (ARE). The constant feedback matrix F is referred to as the asymptotic form of the Linear Quadratic Regulator (LQR). In fact, if (A, B) is stabilizable and (Q 1/2, A) is detectable, then the RDE (initialized with Q at k = and solved for k 0) converges to the unique positive definite solution P of the ARE. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-55
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Stability of Infinite-Horizon LQR In addition, the closed-loop system with u(k) = F x(k) is guaranteed to be asymptotically stable, under the stabilizability and detectability assumptions of the previous slide. The latter statement can be proven by substituting the control law u(k) = F x(k) into x(k + 1) = Ax(k) + Bu(k), and then examining the properties of the system x(k + 1) = (A + BF )x(k). (12) The asymptotic stability of (12) can be proven by showing that the infinite horizon cost J (x(k)) = x(k) P x(k) is actually a Lyapunov function for the system, i.e. J (x(k)) > 0, k 0, J (0) = 0, and J (x(k + 1)) < J (x(k)), for any x(k). This implies that lim x(k) = 0. k Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-56
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Choices of Terminal Weight P in Finite Horizon Control (1/2) 1 The terminal cost P of the finite horizon problem can in fact trivially be chosen so that its solution matches the infinite horizon solution To do this, make P equal to the optimal cost from N to (i.e. the cost with the optimal controller choice). This can be computed from the ARE: P = A PA + Q A PB(B PB + R) 1 B PA This approach rests on the assumption that no constraints will be active after the end of the horizon. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-57
Linear Quadratic Optimal Control Infinite Horizon Optimal Control Choices of Terminal Weight P in Finite Horizon Control (2/2) 2 Choose P assuming no control action after the end of the horizon, so that x(k + 1) = Ax(k), k = N,..., This P can be determined from solving the Lyapunov equation APA + Q = P. This approach only makes sense if the system is asymptotically stable (or no positive definite solution P will exist). 3 Assume we want the state and input both to be zero after the end of the finite horizon. In this case no P but an extra constraint is needed x k+n = 0 Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 2-58
Uncertainty Modeling Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification 4. State Estimation 4.1 Linear State Estimation 4.2 State Observer 4.3 Kalman Filter inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Uncertainty Modeling Objective Statement, Stochastic Processes Table of Contents 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Uncertainty Modeling Objective Statement, Stochastic Processes Objective Statement One of the main reasons for control is to suppress the effect of disturbances on key process outputs. A model is needed to predict the disturbances influence on the outputs on the basis of measured signals. For unmeasured disturbances, stochastic models are used. Objective In this part we introduce stochastic models for disturbances and show how to integrate them into deterministic system models for estimation and control. We discuss how to construct models of the form: x(k + 1) = Ax(k) + Bu(k) + Fw(k) y(k) = Cx(k) + Gw(k) (13) where w(k) is a disturbance signal. inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-59
Uncertainty Modeling Objective Statement, Stochastic Processes Stochastic processes (1/2) Stochastic processes are the mathematical tool used to model uncertain signals. A discrete-time stochastic process is a sequence of random variables {w(0), w(1), w(2),...} The realization of the process is uncertain. We can model a stochastic process via its probability distribution In general, one must specify the joint probability distribution function (pdf) for the entire time sequence P(w(0), w(1),...) inear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-60
Uncertainty Modeling Objective Statement, Stochastic Processes Stochastic processes (2/2) Stochastic processes are modeled using data: Estimating the joint pdf usually is intractable Thus the normal distribution assumption is often made: only models of the mean and the covariance are needed Further distinction of stochastic processes: Stationary stochastic processes Nonstationary stochastic processes Informally, a stationary process with normal distribution has mean and variance that do not vary over a shifting time window. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-61
Uncertainty Modeling Objective Statement, Stochastic Processes Normal stochastic process Joint pdf is a normal distribution Completely defined by its mean and covariance function µ w (k) E{w(k)} R w (k, τ) E{w(k + τ)w (k)} µ w (k + τ)µ w (k) Stationary if µ w (k) = µ w and R w (k, τ) = R w (τ) Typically data are used to estimate µ w (k) and R w (k, τ) Special case: Normal white noise stochastic process ε(k) µ ε = 0 { R ε if τ = 0 R ε (k, τ) = 0 otherwise Since jointly normally distributed and uncorrelated over time, ε(k) is independent of time Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-62
Uncertainty Modeling Modeling using State Space Descriptions Table of Contents 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Uncertainty Modeling Modeling using State Space Descriptions Stationary Case (1/2) We model the stochastic process as the output w(k) of a linear system driven by normal white noise ε(k). It will turn out that w(k) is a normal stochastic process and µ w (k), R w (k, τ) can be chosen through A w,b w,c w. In the stationary case and using a state space description we have: x w (k + 1) = A w x w (k) + B w ε(k) w(k) = C w x w (k) + ε(k) (14) where: x w is an additional state introduced to model the linear system s response to white noise all the eigenvalues of A w lie strictly inside the unit circle. (14) is the standard form for many filter and control design tools. We will show how to determine A w,b w,c w in practical situations. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-63
Uncertainty Modeling Modeling using State Space Descriptions Stationary Case (2/2) The output w(k) of system (14) with white noise ε(k) and stable A w has the following properties: E{x w (k)} = A w E{x w (k 1)} = A k we{x(0)} E{x w (k)x w(k)} = A w E{x w (k 1)x w(k 1)}A w + B w R ε B w E{x w (k + τ)x w(k)} = A τ we{x w (k)x w(k)} (15) From this one can deduce that w = lim k E{w(k)} = 0 R w (τ) = lim k E{w(k + τ)w (k)} = C w A τ P w w C w + C w A τ 1 w B w R ε (16) where P w = A w Pw A w + B w R ε B w, (17) i.e. Pw is a positive semi-definite solution to a Lyapunov equation. These relations can be used in order to determine A w, B w, C w matching a certain covariance R w (τ). Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-64
Uncertainty Modeling Modeling using State Space Descriptions Nonstationary Case (1/2) If a disturbance signal has persistent characteristics (exhibiting shifts in the mean), it is not appropriate to model it with a stationary stochastic process. For example, controller design based on stationary stochastic processes will generally lead to offset. In this case one can superimpose the output of a linear system driven by integrated white noise ε int (k) to the stationary signal: The state space description is then: ε int (k + 1) = ε int (k) + ε(k) (18) x w (k + 1) = A w x w (k) + B w ε int (k) w(k) = C w x w (k) + ε int (k) (19) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-65
Uncertainty Modeling Modeling using State Space Descriptions Nonstationary Case (2/2) The state space description (19) can be rewritten, using differenced variables, as: x w (k + 1) = A w x w (k) + B w ε(k 1) (20) w(k) = C w x w (k) + ε(k 1) where w(k) = w(k) w(k 1) and ε(k) is a zero-mean stationary process. Since ε(k) is a white noise signal, (20) is equivalent to: x w (k + 1) = A w x w (k) + B w ε(k) w(k) = C w x w (k) + ε(k) (21) Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-66
Uncertainty Modeling Obtaining Models from First Principles Table of Contents 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Uncertainty Modeling Obtaining Models from First Principles Obtaining Models from First Principles From first principles, after linearization, one obtains an ODE of the form: ẋ p = A c px p + Bpu c + Fp c w y = C p x + G p w (22) which can be discretized, leading to: x p (k + 1) = A p x(k) + B p u(k) + F p w(k) y(k) = C p x p (k) + G p w(k) (23) Remarks: Subscript p is used here to distinguish the process model matrices from the disturbance model matrices introduced before. If the physical disturbance variables cannot be identified, one can express the overall effect of the disturbances as a signal directly added to the output output disturbance, i.e. G p = I and F p = 0. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-67
Uncertainty Modeling Obtaining Models from First Principles Stationary Case We can combine the model (14) with (23) to get: [ ] xp (k + 1) x w (k + 1) = [ ] [ Ap F p C w xp (k) 0 A w x w (k) y(k) = [ C p G p C w ] [ x p (k) x w (k) ] + ] + G p ε(k) [ ] [ ] Bp Fp u(k) + ε(k) 0 B w (24) With some appropriate re-definition of system matrices, the above is in the standard state-space form of: x(k + 1) = Ax(k) + Bu(k) + Fε(k) }{{} ε 1(k) y(k) = Cx(k) + Gε(k) }{{} ε 2(k) (25) Notice that the state is now expanded to include both the original system state x p and the disturbance state x w. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-68
Uncertainty Modeling Obtaining Models from First Principles Nonstationary Case (1/2) We can combine (21) with a differenced version of (23) to obtain: [ ] [ ] [ ] [ ] [ ] xp (k + 1) Ap F = p C w xp (k) Bp Fp + u(k) + ε(k) (26) x w (k + 1) 0 A w x w (k) 0 B w }{{}}{{}}{{} A B F y(k) = [ [ ] ] xp (k) C p G p C w + G }{{} x w (k) p ε(k) (27) }{{} C G where x p (k) x p (k) x p (k 1), and x w (k), u(k) are defined similarly. For estimation and control, it is further desired that the model output be y rather than y. This requires yet another augmentation of the state... Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-69
Uncertainty Modeling Obtaining Models from First Principles Nonstationary Case (2/2) The augmented system is: [ ] x(k + 1) = y(k + 1) [ ] [ ] A 0 x(k) C I y(k) y(k) = [ 0 I ] [ ] x(k) y(k) [ ] [ B F + u(k) + ε(k) (28) 0 G] It can be brought into the standard state-space form after re-definition of the system matrices: (29) x(k + 1) = Ā x(k) + B u(k) + Fε(k) y(k) = C x(k) (30) except that now the system input is u rather than u. System (30) has n y integrators to express the effects of the white noise disturbances and the system input u on the output. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-70
Uncertainty Modeling Obtaining Models from System Identification Table of Contents 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification inear Systems M. Morari, M. Zeilinger - Spring Semester 2016
Uncertainty Modeling Obtaining Models from System Identification Stationary Case Input output models obtained from an identification have the typical structure of: y(z) = H 1 (z)u(z) + H 2 (z)ε(z) (31) where H 1 (z) and H 2 (z) are stable transfer matrices. This can be brought into the form of (25) by finding a state-space realization of [ H1 (z) H 2 (z) ] : x(k + 1) = A(k) + Bu(k) + Fε(k) y(k) = Cx(k) + Du(k) + Gε(k) (32) Remarks: H 1 (z) has relative degree of at least one We may assume without loss of generality that H 2 (0) = I, D = 0 and G = I. Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-71
Uncertainty Modeling Obtaining Models from System Identification Nonstationary Case In this case, the driving noise should be integrated white noise. 1 y(z) = H 1 (z)u(z) + H 2 (z) ε(z) 1 z 1 }{{} ε int(z) (33) Using the fact (1 z 1 )y(z) = ( y)(z), we can rewrite the above as: ( y)(z) = H 1 (z) u(z) + H 2 (z)ε(z) (34) Denoting the realization of [ H 1 (z) H 2 (z) ] as x(k + 1) = Ax(k) + B u(k) + Fε(k) y(k) = Cx(k) + ε(k) (35) The state can be augmented with y as before to bring it into the form of (30). Linear Systems M. Morari, M. Zeilinger - Spring Semester 2016 3-72
State Estimation Table of Contents 1. Linear Systems 1.1 Models of Dynamic Systems 1.2 Analysis of LTI Discrete-Time Systems 2. Linear Quadratic Optimal Control 2.1 Optimal Control 2.2 Batch Approach 2.3 Recursive Approach 2.4 Infinite Horizon Optimal Control 3. Uncertainty Modeling 3.1 Objective Statement, Stochastic Processes 3.2 Modeling using State Space Descriptions 3.3 Obtaining Models from First Principles 3.4 Obtaining Models from System Identification 4. State Estimation 4.1 Linear State Estimation 4.2 State Observer 4.3 Kalman Filter inear Systems M. Morari, M. Zeilinger - Spring Semester 2016