Course Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
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1 Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization: Analytic approach L-L Controller optimization: Numerical approach. Youla parameterization, internal model control. Synthesis by convex optimization. Controller simplification Lecture Outline General idea for Lectures. controlled variables z Plant distubances w.. measurements y Controller control inputs u. The choice of controller corresponds to designing a transfer matrix Q(s), to get desirable properties of the following map from w to z: Parts of this lecture is based on material from Boyd, Vandenberghe and coauthors. See also lecture notes and links on course homepage. z P zw (s) P zu (s)q(s)p yw (s) w Once Q(s) has been designed, the corresponding controller can be found. Lecture Outline Example (Doyle Stein, 979) Given the process ẋ= x u 6 v y= xv where v and v are independent unit-intensity white noise processes, find a controller that minimizes { } E 8 x T xu while satisfying the robustness constraint M s Example (Doyle Stein, 979) Example DC-motor LQG design gives a controller that does not satisfy the constraint on S (see Lecture ):..... S - - Frequency (rad/s) C(s) w z P(s) Assume we want to optimize the closed-loop transfer matrix from (w, w ) to(z, z ), P PC G zw (s)= PC PC C PC PC when P(s)= s(s). z w
2 Example DC-motor Example DC-motor Minimizing G zw (iω) dω is equivalent to solving the LQG problem with (see Lecture ) A=, B=N=, C= Q = C T C, Q = R = R = Step responses of gang of four: PC/(PC).. C/(PC) P/(PC) /(PC) Example DC-motor Lecture Outline Suppose we want to add some time-domain constraints: C/(PC) P/(PC) Control signal u. for unit output disturbance (or setpoint change) Output signal y. for t for unit load disturbance Least-squares Linear program (LP) minimize Ax b solving least-squares problems analytical solution: x = (A T A) A T b reliable and efficient algorithms and software computation time proportional to n k (A R k n ); less if structured a mature technology minimize c T x d subject to Gx h Ax = b convex problem with affine objective and constraint functions feasible set is a polyhedron using least-squares least-squares problems are easy to recognize P x c a few standard techniques increase flexibility (e.g., including weights, adding regularization terms) Introduction Convex optimization problems 7 Linear programming Convex optimization problem solving linear programs minimize c T x subject to a T i x b i, no analytical formula for solution reliable and efficient algorithms and software i =,..., m computation time proportional to n m if m n; less with structure a mature technology using linear programming not as easy to recognize as least-squares problems a few standard tricks used to convert problems into linear programs (e.g., problems involving l - or l -norms, piecewise-linear functions) minimize f (x) subject to f i (x) b i, i =,..., m objective and constraint functions are convex: f i (αx βy) αf i (x) βf i (y) if α β =, α, β includes least-squares problems and linear programs as special cases Introduction 6 Introduction 7
3 solving convex optimization problems no analytical solution reliable and efficient algorithms computation time (roughly) proportional to max{n, n m, F }, where F is cost of evaluating f i s and their first and second derivatives almost a technology using convex optimization often difficult to recognize many tricks for transforming problems into convex form surprisingly many problems can be solved via convex optimization Brief history of convex optimization theory (convex analysis): ca9 97 algorithms 97: simplex algorithm for linear programming (Dantzig) 96s: early interior-point methods (Fiacco & McCormick, Dikin,... ) 97s: ellipsoid method and other subgradient methods 98s: polynomial-time interior-point methods for linear programming (Karmarkar 98) late 98s now: polynomial-time interior-point methods for nonlinear convex optimization (Nesterov & Nemirovski 99) applications before 99: mostly in operations research; few in engineering since 99: many new applications in engineering (control, signal processing, communications, circuit design,... ); new problem classes (semidefinite and second-order cone programming, robust optimization) Introduction 8 Introduction on R convex: affine: ax b on R, for any a, b R exponential: e ax, for any a R powers: x α on R, for α or α powers of absolute value: x p on R, for p negative entropy: x log x on R concave: affine: ax b on R, for any a, b R powers: x α on R, for α logarithm: log x on R on R n and R m n affine functions are convex and concave; all norms are convex examples on R n affine function f(x) = a T x b norms: x p = ( n i= x i p ) /p for p ; x = max k x k examples on R m n (m n matrices) affine function f(x) = tr(a T X) b = spectral (maximum singular value) norm m i= j= n A ij X ij b f(x) = X = σ max (X) = (λ max (X T X)) / Convex functions Convex functions Convex optimization problem standard form convex optimization problem minimize f (x) subject to f i (x), i =,..., m a T i x = b i, i =,..., p f, f,..., f m are convex; equality constraints are affine problem is quasiconvex if f is quasiconvex (and f,..., f m convex) Quadratic program (QP) minimize (/)x T P x q T x r subject to Gx h Ax = b P S n, so objective is convex quadratic minimize a convex quadratic function over a polyhedron f (x ) x P Convex optimization problems Second-order cone programming Semidefinite program (SDP) minimize f T x subject to A i x b i c T i x d i, F x = g (A i R ni n, F R p n ) i =,..., m with F i, G S k minimize c T x subject to x F x F x n F n G Ax = b inequality constraint is called linear matrix inequality (LMI)
4 Newton s method Barrier method for constrained minimization given a starting point x dom f, tolerance ǫ >. repeat. Compute the Newton step and decrement. x nt := f(x) f(x); λ := f(x) T f(x) f(x).. Stopping criterion. quit if λ / ǫ.. Line search. Choose step size t by backtracking line search.. Update. x := x t x nt. minimize f (x) subject to f i (x) =,..., m Ax=b approximation via logarithmic barrier minimize f (x) (/t) m i= log( f i(x)) subject to Ax = b x () x () f(x (k) ) p k an equality constrained problem for t >, (/t) log( u) is a smooth approximation of I approximation improves as t u Interior-point methods Lecture Outline Scheme for numerical optimization of Q Given some fixed set of basis function φ (s),...,φ N (s), we will search numerically for matrices Q,..., Q N such that the closed-loop transfer matrix G zw (s) satisfies given specifications when G zw (s)= P zw (s) P zu (s)q(s)p yw (s) and Q(s)= N Q k φ k (s) k= Once Q(s) has been determined, we will recover the desired controller from the formula C(s)= [ I Q(s)P yu (s) ] Q(s) It is possible to choose the sequence φ (s),φ (s),φ (s),... such that every stable Q can be approximated arbitrarily well. Hence, in principle, every convex control design problem can be solved this way. Choice of basis functions Specifications that lead to convex constraints Many possibilities. Common choices: Laguerre basis polynomials, φ k (s)= where a should be wisely selected (s/a) k (rule of thumb: close to bandwidth of closed-loop system) Pulse response parameterization (discrete time approximation). Stability of the closed-loop system. Lower bound on step response from w i to z j at time t i. Upper bound on step response from w i to z j at time t i. Upper bound on Bode amplitude from w i to z j at frequency ω i Q k. Interval bound on Bode phase from w i to z j at frequency ω i Lower bound on step response Upper bound on step response.... The step response depends linearly on Q k, so every time t k with a lower bound gives a linear constraint. Every time t k with an upper bound also gives a linear constraint.
5 Upper bound on Bode amplitude Lower bound on Bode amplitude Bode Magnitude Diagram Bode Magnitude Diagram G a (iω) G a (iω) G b (iω) G b (iω) Frequency (rad/sec) Frequency (rad/sec) An amplitude bound G(iω i ) <c is a quadratic constraint. An lower bound G(iω i ) is a non-convex quadratic constraint. This should be avoided in optimization. Synthesis by convex optimization Software for convex optimization Minimize Quite general control synthesis problems can be stated as convex optimization problems in the variable Q(s). The problem could have a quadratic objective, with linear/quadratic constraints, e.g.: subj. to Q(iω) { }}{ } P zw(iω) P zu (iω) Q k φ k (iω) P yw (iω) dω quadratic objective k } step response w i z j is smaller than f ijk at time t k linear constraints step response w i z j is bigger than ijk at time t k } Bode magnitude w i z j is smaller than h ijk at ω k quadratic constraints Here Q(s)= k Q kφ k (s), where φ,...,φ m are some fixed basis functions, and Q,..., Q m are optimization variables. Once Q(s) has been determined, the controller is obtained as C(s)= [ I Q(s)P yu (s) ] Q(s) CVX Matlab software for disciplined convex programming, developed at Stanford by Stephen Boyd and co-workers Easily integrated with Python, Julia CVXGEN C code generation YALMIP Matlab toolbox for convex and nonconvex optimization problems SeDuMi software for optimiztion over symmetric cones SDPT Matlab software for semidefinite programming... Lecture Outline Example (Doyle Stein, 979) Reformulate LQG problem as extended plant model to be optimized: z z z Q R P Q R w w w y u C Minimize P zw (iω) P zu (iω) q k φ k (iω)p yw (iω) dω k with q k scalar and φ k (s)= (s/a) k Example (Doyle Stein, 979) Example DC-servo Green: Optimization-based design with constraint on S : Introduce stabilizing controller C and reformulate for optimization:.... S z z y [ P ] P P P C ũ w w y [ Pc P c P c P c P c P c P c P c P c ] ũ C (s) C (s). (Controller order: ) - - Frequency (rad/s) G zw (s)= [ ] [ ] Pc P c Pc ] Q [P P c P c P c P c c
6 Example DC-servo Example DC-servo Green: Optimization with control signal limitation: Green: Also adding the limit on y, t :.8 C/(PC) P/(PC) (Controller order: ) (Controller order: ) Example DC-servo Summary Final controller: Phase (deg) Bode Diagram There are efficient algorithms for convex optimization, e.g. Linear programming (LP) Quadratic programming (QP) Second order cone programming (SOCP) Semi-definite programming (SDP) The Youla parameterization allows us to use these algorithms for control synthesis Resulting controllers have high order. Order reduction will be studied in the next lecture Frequency (rad/s) Is it any good? With optimization, you get what you ask for! 6
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