# 4F3 - Predictive Control

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1 4F3 Predictive Control - Discrete-time systems p. 1/30 4F3 - Predictive Control Discrete-time State Space Control Theory For reference only Jan Maciejowski

2 4F3 Predictive Control - Discrete-time systems p. 2/30 Course Outline Introduction to predictive control Discrete-time state space control theory handout only Predictive control without constraints Predictive control with constraints Stability and feasibility in predictive control Setpoint tracking and offset-free control Industrial case study Dr Paul Austin Fri. 5 March Examples Class 2 Examples Papers Tue. 9 March

3 4F3 Predictive Control - Discrete-time systems p. 3/30 Outline of Lecture 2 Review of linear algebra Discrete time models and sampled data systems Stability of discrete time systems Reachability and Observability State feedback control design Observer design Output feedback and the separation principle

4 4F3 Predictive Control - Discrete-time systems p. 4/30 References Linear Algebra G. Strang. Introduction to Linear Algebra, Wellesley-Cambridge Press. 3 rd ed., D. C. Lay. Linear Algebra and Its Applications, Adison Wesley. 3 rd ed., Discrete Time Control G. F. Franklin et. al. Digital Control of Dynamic Systems, Adison Wesley. 3 rd ed., K. J. Åstrom and B. Wittenmark. Computer Controlled Systems, Prentice Hall. 3 rd ed., 2003.

5 4F3 Predictive Control - Discrete-time systems p. 5/30 Linear Independence and Rank Definition (Linear Independence) The vectors {v 1,...,v n } are linearly independent if and only if c 1 v c n v n = 0 only happens when c 1 = = c n v n = 0. The rank of a matrix A is defined as the number of linearly independent columns also known as the column rank. The column rank of A is the same as its row rank, i.e. ranka = ranka T

6 4F3 Predictive Control - Discrete-time systems p. 6/30 Solutions to Ax = b Given a set a linear inequalities Ax = b with A R r c Solution exists for a given b ranka = rank(a b) Solution exists for every b ranka = r. (Only possible when r c) If a solution exists, it is unique ranka = c (Only possible when r c) When A is square and invertible (i.e. full rank), the unique solution is x = A 1 b

7 4F3 Predictive Control - Discrete-time systems p. 7/30 Eigenvectors and Eigenvalues Suppose that A R n n and there exists x R n and λ R such that Ax = λx and x 0. Then: The vector x is an eigenvector of A. The scalar λ is an eigenvalue of A. The spectral radius of A is defined as: Some useful facts: ρ(a) := max { λ 1 (A),..., λ n (A) } Eigenvalues of A are the roots of its characteristic polynomial. Ax = λx det(λi A) = 0 Every eigenvalue has a non-zero eigenvector.

8 4F3 Predictive Control - Discrete-time systems p. 8/30 Digital (Sampled-Data) Control r(t) + r(kt) Digital Controller u(kt) DAC/ ZOH u(t) Plant y(t) Clock y(kt) ADC y(t) Sample period is T. Sample number is k. Time is t. s(t) continuous time signal. s(kt) discrete time signal. Will generally shorten s(kt) to s(k).

9 4F3 Predictive Control - Discrete-time systems p. 9/30 Linear Systems (Continuous Time) A linear continuous time (CT) state-space system: ẋ = Fx + Gu y = Cx + Du z = Hx x R n State vector. u R m Input vector {a.k.a. Manipulated Variables (MV)}. y R p Output vector { Measured Variables }. z R q Controlled variables (CV). In many cases, H = C, so that y = z.

10 4F3 Predictive Control - Discrete-time systems p. 10/30 Solution of Linear CT Systems Given: The CT linear system ẋ = Fx + Gu An initial state x(t 0 ) at time t 0 An input trajectory u( ) over the interval [t 0,t] the solution of the CT system at time t is given by: x(t) = e F(t t 0) x(t 0 ) + t t 0 e F(t τ) Gu(τ)dτ We can use this to get a DT model for our CT system.

11 4F3 Predictive Control - Discrete-time systems p. 11/30 DT models for Sampled-Data Systems With a ZOH at the the output of our digital controller: u(t) = u(kt) for the interval kt t k(t + 1) The DT state-space model x(kt + T) = Ax(kT) + Bu(kT) y(kt) = Cx(kT) + Du(kT) z(kt) = Hx(kT) is an exact representation of the sampled CT system if ( T ) A = e FT, B = e Fτ dτ G 0

12 4F3 Predictive Control - Discrete-time systems p. 12/30 DT models for Sampled-Data Systems With a ZOH at the the output of our digital controller, we can get an exact DT representation of our CT system if: the CT system is linear, or the CT system is linear with input saturation It is generally not possible to get an exact DT representation for a nonlinear CT system: ẋ = f(x,u) y = h(x,u) For a nonlinear model, we can use a DT approximation or use numerical integration to find x(kt).

13 4F3 Predictive Control - Discrete-time systems p. 13/30 Solution of Linear DT Systems Given: The DT linear system x(k + 1) = Ax(k) + Bu(k) An initial state x(k) at discrete time k An input sequence {u(k),...,u(k+n 1)} the solution of the DT system at time k + N is given by: x(k + N) = A N x(k) + N 1 i=0 A i Bu(k+N 1 i) Try to prove this to yourself we ll return to it in lecture 3.

14 4F3 Predictive Control - Discrete-time systems p. 14/30 Stability of Nonlinear DT Systems Consider a general nonlinear DT system with f : R n R n x(k + 1) = f(x(k)) Let x(k) denote the state of this system at time discrete time step k if the state at k = 0 is x(0). x(k) = f(f(...f(f(x(0)))...)) }{{} k times = f k (x(0)) Definition (DT Equilibrium Point) A state x is an equilibrium point of the DT system if x = f(x). Important: This is a different definition than in the CT case.

15 4F3 Predictive Control - Discrete-time systems p. 15/30 Stability of Nonlinear DT Systems Definition (Stability) An equilibrium point x is stable if for all ǫ > 0, there exists δ > 0 such that x(0) x δ x(k) x ǫ for all k 0 Definition (Asymptotic Stability) An equilibrium point x is locally asymptotically stable if it is stable and there exists an M > 0 such that x(0) x < M lim k x(k) = x The equilibrium point x is globally asymptotically stable if the above holds for all M > 0.

16 4F3 Predictive Control - Discrete-time systems p. 16/30 Stability of Linear DT Systems Consider a linear DT systems without an input: x(k + 1) = Ax(k) A state x is an equilibrium point if x = Ax: Any eigenvector with eigenvalue λ = 1 is an equilibrium point. The origin x = 0 is an equilibrium point. Some useful facts: The origin is globally asymptotically stable if and only if all of the eigenvalues of A are inside the unit disk (i.e. ρ(a) < 1). The matrix A is called stable if and only if ρ(a) < 1. For sampled data systems with ZOH, λ i (A) = e λ i(f)t CT system stable DT system stable.

17 4F3 Predictive Control - Discrete-time systems p. 17/30 Reachability Given the system x(k + 1) = Ax(k) + Bu(k) Definition (Reachability) The matrix pair (A, B) is reachable if it is possible to drive the above system to any state (from any other state) in finite time. Analogous to controllability of the CT system (F, G) Controllability/Reachability is not always preserved when the system is discretized, i.e. (F,G) controllable (A,B) reachable

18 4F3 Predictive Control - Discrete-time systems p. 18/30 Tests for Reachability Definition (Reachability Matrix) The reachability matrix of the pair (A,B) is defined as: ( ) C(A,B) := B AB... A n 1 B Proposition (Reachability Tests) The following are equivalent: a) The pair (A,B) is reachable. b) Rank of C(A,B) = n. c) The eigenvalues of (A + BK) can be freely assigned by suitable choice of K (requires complex eigenvalues to appear in conjugate pairs).

19 4F3 Predictive Control - Discrete-time systems p. 19/30 Observability Given the system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) Definition (Observability) The matrix pair (C, A) is observable if it is possible to determine the initial system state from a finite sequence of measurements. Analogous to observability of the CT system (C,F) Observability is not always preserved when the system is discretized, i.e. (C,F) observable (C,A) observable

20 4F3 Predictive Control - Discrete-time systems p. 20/30 Tests for Observability Definition (Observability Matrix) The observability matrix of the pair (C,A) is defined as: C CA O(C,A) :=. CA n 1 Proposition (Observability Tests) The following are equivalent: a) The pair (C,A) is observable. b) Rank of O(C,A) = n. c) The eigenvalues of (A + LC) can be freely assigned by suitable of choice of L (requires complex eigenvalues to appear in conjugate pairs). d) (A T,C T ) is reachable.

21 4F3 Predictive Control - Discrete-time systems p. 21/30 Stabilizability and Detectability Reachability and observability can be relaxed to deal with just the unstable part of the system: Definition (Stabilizability) The matrix pair (A, B) is stabilizable if there exists a matrix K such that (A + BK) is stable. Definition (Detectability) The matrix pair (C, A) is detectable if there exists a matrix L such that (A + LC) is stable. These are weaker definitions than those for reachability and observability. These definitions do not guarantee that the eigenvalues of (A + BK) or (A + LC) can be arbitrarily assigned.

22 4F3 Predictive Control - Discrete-time systems p. 22/30 Tests for Stabilizability and Detectability Let Λ be the set of eigenvalues on or outside the unit circle: Λ := {λ i (A) : λ i (A) 1} Proposition (Stabilizability) The pair (A, B) is stabilizable if and only if ( (A λi) B ) is full row rank for all λ Λ. Proposition (Detectability) The pair (C, A) is detectable if and only if ( ) A λi C is full column rank for all λ Λ.

23 4F3 Predictive Control - Discrete-time systems p. 23/30 State Feedback Control Design K u(k) DT System x(k) Given the DT system x(k + 1) = Ax(k) + Bu(k) want to design a state feedback control law u(k) = Kx(k) so that the origin of the closed loop system x(k + 1) = (A + BK)x(k) is globally asymptotically stable requires (A+BK) to be stable.

24 4F3 Predictive Control - Discrete-time systems p. 24/30 State Feedback and Pole Placement State feedback design procedure when (A, B) is reachable: 1) Specify desired eigenvalues {p 1,...,p n }, with each p i < 1 and complex values in conjugate pairs. 2) Compute coefficients of the polynomial p(λ) = (λ p 1 )(...)(λ p n ) 3) Compute characteristic polynomial det (λi (A + BK)) 4) Equate coefficients of the polynomials in (2) and (3). 5) Solve for the elements of the matrix K. Note that if (A,B) is only stabilizable, then the values {p 1,...,p N } can t be chosen arbitrarily.

25 4F3 Predictive Control - Discrete-time systems p. 25/30 Observer Design When all the states can not be measured (C I), then states can be estimated with an observer (or estimator). The DT system equations are: x(k + 1) = Ax(k) + Bu(k), y(k) = C(k)u(k) The equations for the observer are: ˆx(k k) = ˆx(k k 1) + L[ŷ(k k 1) y(k)] ˆx(k + 1 k) = Aˆx(k k) + Bu(k) ŷ(k k 1) = Cˆx(k k 1) Notation: ˆx(i j) denotes an estimate of x( ) at time i using all the measurements available through time j

26 4F3 Predictive Control - Discrete-time systems p. 26/30 Observer Design Define the state estimation error to be e(k) := x(k) ˆx(k k) Can eliminate variables in observer equations to get ˆx(k+1 k+1) = (A+ALC)ˆx(k k) + (B +LCB)u(k) Ly(k + 1) e(k + 1) = (A + LCA)e(k) When the matrix (A + LCA) is stable, then the observer is stable. the estimation error converges to zero.

27 4F3 Predictive Control - Discrete-time systems p. 27/30 Observer Design ˆx(k k) = ˆx(k k 1) + L[ŷ(k k 1) y(k)] ˆx(k + 1 k) = Aˆx(k k) + Bu(k) ŷ(k k 1) = Cˆx(k k 1)

28 4F3 Predictive Control - Discrete-time systems p. 28/30 Observer Design and Pole Placement We can place eigenvalues of (A + LCA) arbitrarily if (CA, A) is observable. If (C,A) is observable and λ i (A) 0 then (CA, A) observable. Observer design procedure when (CA, A) is observable: 1) Specify desired eigenvalues {p 1,...,p n }, with each p i < 1 and complex values in conjugate pairs. 2) Equate coefficients of polynomial p(λ) with those of det (λi (A + LCA)) 3) Solve for the elements of the matrix L. Note that if (CA,A) is only detectable, then the values {p 1,...,p N } can t be chosen arbitrarily.

29 4F3 Predictive Control - Discrete-time systems p. 29/30 Output Feedback Control K u(k) DT System y(k) ˆx(k k) Observer Controller The state feedback matrix K and observer L can be designed separately to form an output feedback control law.

30 4F3 Predictive Control - Discrete-time systems p. 30/30 The Separation Principle Define the state estimation error as before: e(k) := k(x) ˆx(k k) The closed loop system becomes: x(k + 1) = (A + BK)x(k) BKe(k) e(k + 1) = (A + LCA)e(k) The eigenvalues of the closed loop system are The eigenvalues of (A + BK) (from control design) The eigenvalues of (A + LCA) (from observer design) This is a special case of the separation principle.

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