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1 NONLINEAR CONTROL - AN OVERVIEW Fernando Lobo Pereira, fl[email protected] Porto University, Portugal Nonlinear Control, 2008, FEUP References 1 M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cli®s, NJ, 1978. 2 A. Isidori, Nonlinear Control Systems, Springer-Verlag, 199X. 3 E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, 1998. 4 C. T. Chen, Introduction to Linear Systems Theory, Van Nostrand Reinhold Company, New York, NY, 1970. 5 C. A. Desoer, Notes for a Second Course on Linear Systems, Holt Rhinehart, Winston, 1970. 6 K. Ogata, State Space Analysis of Control Systems, Prentice-Hall, Englewood Cli®s, NJ, 1965. 7 J. M. Carvalho, Dynamical Systems and Automatic Control, Prentice-Hall, New York, NY, 1994. Fernando Lobo Pereira, fl[email protected] 1 Nonlinear Control, 2008, FEUP Introduction Problem description - system components, objectives and control issues. Example 1: telescope mirrors control PICTURE Most of real systems involve nonlinearities in one way or another. However, many concepts for linear systems play an important role since some of the techniques to deal with nonlinear systems are: ² Change of variable in phase space so that the resulting system is linear. ² Nonlinearities may be cancelled by picking up appropriate feedback control laws. ² Pervasive analogy ... Fernando Lobo Pereira, fl[email protected] 2 Nonlinear Control, 2008, FEUP Introduction Example 2: Pendulum Modeling PICTURE Dynamics (Newton's law for rotating objects): mθÄ(t) + mgsin(θ(t)) = u(t) State variables: θ; θ_ Control/Input variable: u (external applied torque) Let us assume m = 1; l = 1; g = 1. Fernando Lobo Pereira, fl[email protected] 3 Nonlinear Control, 2008, FEUP Introduction Control Design ½ (0; 0) stable equilibrium Stationary positions: (1) (¼; 0) unstable equilibrium. Consider the local control around the later. Objective: Apply u so that, for small (θ(0); θ_(0)), (θ(t); θ_(t)) ! (¼; 0) as t ! 1. Step 1 - Analysis - Linearization For θ close to ¼: sin(θ) = ¡(θ ¡ ¼) + o(θ ¡ ¼). Let ' = θ ¡ ¼, then: 'Ä(t) ¡ '(t) = u(t) Fernando Lobo Pereira, fl[email protected] 4 Nonlinear Control, 2008, FEUP Introduction Step 2 - Control Synthesis Take ' > 0 and pick u(t) = ¡®'(t) with ® > 0. Then 'Ä(t) + (® ¡ 1)'(t) = 0: 8 < ® > 1 oscillatory behavior. ® = 1 only stable point:' _(0) = 0 (2) : p ® < 1 stable points:' _(0) = ¡'(0) 1 ¡ ® Conclusion: \proportional control" does not su±ce. Some anticipative control action is required. This achieved by \proportional and derivative control", i.e., u(t) = ¡®'(t) ¡ ¯'_(t) with ® > 1 and ¯ > 0. Then 'Ä(t) + ¯'_(t) + (® ¡ 1)'(t) = 0: p ¡¯§ ¯2¡4(®¡1) The roots of the characteristic polynomial are 2 . Fernando Lobo Pereira, fl[email protected] 5 Nonlinear Control, 2008, FEUP Introduction Linearization Principle Linear designs apply to linearized nonlinear systems operating locally. Extension: controller scheduling Organize the phase space into regions and \patch" together local linear designs. Fernando Lobo Pereira, fl[email protected] 6 Nonlinear Control, 2008, FEUP Introduction Issues in control system design: 1 - Study the system of interest and decide on sensors and actuators 2 - Model the resulting system to be controlled 3 - Simplify the model so that it becomes tractable 4 - Analyze the system to determine its properties 5 - De¯ne performance speci¯cations 6 - Design the controller to meet the speci¯cations 7 - Evaluate the design system via, say, simulation 8 - If not happy go to the beginning; otherwise 9 - Proceed to implementation 10 - Being the case, tune the controller on-line Fernando Lobo Pereira, fl[email protected] 7 Nonlinear Control, 2008, FEUP Introduction Strategy of the course: ² Easy introduction to the main concepts of nonlinear control by making as much use as possible of linear systems theory and with as little as possible Mathematics. ² Example-oriented introduction to concepts. ² Matlab hands-on exercises Fernando Lobo Pereira, fl[email protected] 8 Nonlinear Control, 2008, FEUP Introduction Structure of the course: ² Selected topics of linear systems ² Issues and speci¯c background topics for nonlinear systems ² Main issues Lyapunov Stability ² Examples of feedback linearization design ² Basic issues in optimal control ² Introduction to adaptive control Fernando Lobo Pereira, fl[email protected] 9 Nonlinear Control, 2008, FEUP Selected topics of linear systems State-space representation of Linear Dynamic Systems ½ x_(t) = A(t)x(t) + B(t)u(t)[t0; tf ] ¡ a:e:; x(t0) = x0 y(t) = C(t)x(t) + D(t)u(t)[t0; tf ] ¡ a:e: where ² x 2 <n, y 2 <q, and u 2 <m ² A 2 <n£n, B 2 <n£m, C 2 <q£n, and D 2 <q£m System's state trajectory given by a closed form solution Z t x(t) = Á(t; t0)x0 + Á(t; ¿)B(¿)u(¿)d¿ t0 where Á(t; s) is the State Transition Matrix from s to t. Fernando Lobo Pereira, fl[email protected] 10 Nonlinear Control, 2008, FEUP Selected topics of linear systems De¯nition of State Transition Matrix Fundamental matrix X(¢) of the pair (A(¢);X0) is the solution to X_ (t) = A(t)X(t);X(t0) = X0. The State Transition Matrix Á(t; t0) is given by X(t) when X0 = I. Thus, it satis¯es the n £ n matrix di®erential equation @ Á(t; t ) = A(t)Á(t; t ): @t 0 0 Main Properties 1. Á(t; t0) is uniquely de¯ned 2. The solution tox _(t) = A(t)x(t); x(t0) = x0, is given by Á(t; t0)x0 3. For all t; t0; t1, Á(t; t0) = Á(t; t1)Á(t1; t0) ¡1 4. Á(t1; t0) is nonsingular and [Á(t; t0)] = Á(t0; t) ¡1 5. Á(t; t0) = X(t; t1)[X(t0)] R R R t t σ1 6. Á(t; t0) = I + A(σ1)dσ1 + A(σ1) A(σ2)dσ2dσ1 + R R t0 R t0 t0 t σ1 σ2 A(σ1) A(σ2) A(σ3)dσ3dσ2dσ1 + ::: t0 t0 t0 Fernando Lobo Pereira, fl[email protected] 11 Nonlinear Control, 2008, FEUP Selected topics of linear systems \Variations-by-parts" formula Exercise: Check \variations-by-parts" formula by noting that Z t _ _ x_(t) = Á(t; t0)x0 + Á(t; t0)B(t)u(t) + Á(t; s)B(s)u(s)ds t0 Heuristic derivation Let u(σ) = u(s) when σ 2 [s; s + ds] where t0 < s < t. Then ² x(s) = Á(s; t0)x0 ² x(s + ds) ' x(s) + [A(s)x(s) + B(s)u(s)] ds = [Á(s; t0) + A(s)Á(s; t0)ds] x0 + B(s)u(s)ds ' Á(s + ds; t0)x0 + B(s)u(s)ds ² x(t) = Á(t; s + ds)x(s + ds) ' Á(t; t0)x0 + Á(t; s + ds)B(s)u(s)ds ² Summing the input contributions for all s: Z t x(t) = Á(t; t0)x0 + Á(t; s)B(s)u(s)ds t0 Fernando Lobo Pereira, fl[email protected] 12 Nonlinear Control, 2008, FEUP Selected topics of linear systems Computation of the State Transition Matrix Z Z t X1 1 t Á(t; t ) = exp( A(s)ds) = ( A(s)ds)i 0 i! t0 i=0 t0 XN Z t i = ®i(t; t0)( A(s)ds) (3) i=0 t0 for some N · n ¡ 1. The last equality holds due to the Cayley Hamilton Theorem: p(A) = 0. Here p is the characteristic polynomial of A. Algorithm R ² Compute eigenvalues and eigenvectors of t A(s)ds t0 ² Compute coe±cients ®i(t; t0) by solving a system of linear functional equations obtained by noting that A and its eigenvalues satisfy (3) ² Plug the ®i(t; t0)'s in (3) in order to obtain Á(t; t0) Fernando Lobo Pereira, fl[email protected] 13 Nonlinear Control, 2008, FEUP Selected topics of linear systems Observations: ² The above algorithm is more generic: it enables the computation of any su±ciently regular function of the matrix A. ² When all the eigenvalues are distinct, N = n ¡ 1 and one equation is written for each eigenvalue. Otherwise, N is the degree of the Minimal polynomial. ² When an eigenvalue has multiplicity d > 1 and its greatest Jordan block of its eigenspace has dimension m, then q equations associated with this eigenvalue have to be added by taking the successive derivatives (from 0 to q ¡ 1) w.r.t. the eigenvalue of both sides. ² Geometric interpretation plays a key role. Def. The minimal polynomial à is the one of least degree s. t. Ã(A) = 0. Let A have σ distinct eigenvalues, ¸i whose multiplicity is di and let mi be the dimension of the associated greatest invariant subspace. Then: σ mi σ di Ã(s) = ¦i=1(¸ ¡ ¸i) and p(s) = ¦i=1(¸ ¡ ¸i) : Fernando Lobo Pereira, fl[email protected] 14 Nonlinear Control, 2008, FEUP Selected topics of linear systems Jordan Representation m Let @k = @[(A ¡ ¸kI) k] and Bk a basis for @k. Then: ² dim(@k) = dk n L L L ² < = @1 @2 ::: @σ Sσ ² A is represented by diag(Ai) in B = 1 Bi where Bi is a basis for @i σ σ dk ² det(A ¡ ¸I) = ¦i=1det(Ai ¡ ¸Ii) = ¦k=1®k(¸ ¡ ¸k) Function of a Matrix P P σ mk¡1 (l) f(A) = k=1 l=0 f (¸k)pkl(A) where l (¸¡¸k) Ák(¸) ² pkl(¸) = l! , l = 0; :::; mk ¡ 1 Ã(¸) ² Ák(¸) = nk(¸) m , and (¸¡¸k) k 1 ² nk(¸) are the coe±cients of a partial fraction expansion of , i.e., P Ã(¸) 1 σ nk(¸) = m Ã(¸) k=1 (¸¡¸k) k Fernando Lobo Pereira, fl[email protected] 15 Nonlinear Control, 2008, FEUP Selected topics of linear systems Geometric Interpretation of Eigenvalues and Eigenvectors Assume that the eigenvalues of A, ¸i, i = 1; :::; n are distinct and denote by vi the associated eigenvectors.P P n n ¸it Let x(0) = i=1 ®ivi.
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