The Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
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1 The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture -4: The choice of controller generally corresponds to finding Q(s), to get desirable properties of the map from w to z: z P zw (s) P zu (s)q(s)p yw (s) w Most of this lecture is based on source material from Boyd, Vandenberghe and coauthors. See Once Q(s) is determined, a corresponding controller is derived. Example: Spring-mass System Lecture 3: Synthesis by Convex Optimization. Position of the first mass, d d d.8.6 u M m.4. b The step response is not within its upper and lower bounds. Lecture 3: Synthesis by Convex Optimization Lecture 3: Synthesis by Convex Optimization 6 Control Signal, u(t) Sensitivity Function S Frequency (rad/s) The step input stays within its amplitude bound u(t) 6. Lecture 3: Synthesis by Convex Optimization The sensitivity does not satisfy the magnitude bound S.3 Least-squares Introduction to convex optimization 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 (A R n ); less if structured a mature technology using least-squares Most of this lecture is based on source material from Boyd, Vandenberghe and coauthors. See least-squares problems are easy to recognize a few standard techniques increase flexibility (e.g., including weights, adding regularization terms) Introduction 5
2 Linear program (LP) Linear programming minimize c T x d subject to Gx h convex problem with affine objective and constraint functions feasible set is a polyhedron P x c 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 trics used to convert problems into linear programs (e.g., problems involving l - or l -norms, piecewise-linear functions) Convex optimization problems 4 7 Introduction 6 Convex optimization problem 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 solving convex optimization problems no analytical solution reliable and efficient algorithms computation time (roughly) proportional to max{n 3, 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 trics for transforming problems into convex form surprisingly many problems can be solved via convex optimization Introduction 7 Introduction 8 Brief history of convex optimization theory (convex analysis): ca9 97 algorithms 947: simplex algorithm for linear programming (Dantzig) 96s: early interior-point methods (Fiacco & McCormic, Diin,... ) 97s: ellipsoid method and other subgradient methods 98s: polynomial-time interior-point methods for linear programming (Karmarar 984) late 98s now: polynomial-time interior-point methods for nonlinear convex optimization (Nesterov & Nemirovsi 994) 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) Examples 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 Introduction 5 Convex functions 3 3 Examples 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 x examples on R m n (m n matrices) affine function m n f(x) = tr(a T X) b = A ij X ij b i= j= Convex optimization problem standard form convex optimization problem minimize f (x) subject to f i(x), i =,..., m a T i x = bi, i =,..., p f, f,..., f m are convex; equality constraints are affine problem is quasiconvex if f is quasiconvex (and f,..., f m convex) spectral (maximum singular value) norm f(x) = X = σ max (X) = (λ max (X T X)) / Convex functions 3 4
3 Quadratic program (QP) minimize (/)x T P x q T x r subject to Gx h P S n, so objective is convex quadratic minimize a convex quadratic function over a polyhedron f(x ) Second-order cone programming minimize f T x subject to A i x b i c T i x d i, i =,..., m F x = g (A i R ni n, F R p n ) x P Convex optimization problems 4 Newton s method with F i, G S Semidefinite program (SDP) minimize c T x subject to x F x F x n F n G inequality constraint is called linear matrix inequality (LMI) 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. 3. Line search. Choose step size t by bactracing line search. 4. Update. x := x t x nt. 5 x () x () f(x () ) p Barrier method for constrained minimization Outline 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 an equality constrained problem for t >, (/t) log( u) is a smooth approximation of I approximation improves as t u Controller optimization using Youla parametrization Interior-point methods 4 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 φ (s) 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) = Pulse response parameterization We will use an intuitively simple parametrization of Q(s) where each parameter Q represents a point on the corresponding impulse response in time domain Q 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. But, what specifications give a convex design problem?. 5 3
4 Mini-problem Lower bound on step response Which specifications are convex constraints on Q?. Stability of the closed loop system. Lower bound on step response from w i to z j at time t i.5 3. Upper bound on step response from w i to z j at time t i 4. Lower bound on Bode amplitude from w i to z j at frequency ω i 5. Upper bound on Bode amplitude from w i to z j at frequency ω i 6. Interval bound on Bode phase from w i to z j at frequency ω i The step response depends linearly on Q, so every time t with a lower bound gives a linear constraint. Upper bound on step response Upper bound on Bode amplitude Bode Magnitude Diagram.5 - G a (iω) G b (iω).5 - Frequency (rad/sec) An amplitude bound G(iω i ) <cis a quadratic constraint. Every time t with an upper bound also gives a linear constraint. Lower bound on Bode amplitude Synthesis by convex optimization Bode Magnitude Diagram G a (iω) - - Frequency (rad/sec) G b (iω) An lower bound G(iω i ) is a non-convex quadratic constraint. This should be avoided in optimization. A general control synthesis problem can be stated as a convex optimization problem in the variables Q,..., Q m. The problem has a quadratic objective, with linear and quadratic constraints: Minimize subject to Q(iω) {{ Pzw(iω) Pzu(iω) Q φ (iω) P yw(iω) dω quadratc objective step response w i z j is smaller than f ij at time t linear constraints step response w i z j is bigger than ij at time t Bode magnitude w i z j is smaller than h ij at ω quadratic constraints Once the variables Q,..., Q m have been optimized, the controller is obtained as C(s)= [ I Q(s)P yu (s) ] Q(s) Outline Example DC-motor w z C(s) s(s) z w Examples revisited The transfer matrix from(w, w ) to(z, z ) is ] G zw (s)= [ P PC PC PC PC C PC with P(s)= s(s). We will choose C(s) to minimize trace subject time-domain bounds. G zw (iω)g zw (iω) dω 4
5 DC-servo with time domain bounds DC-servo with time domain bounds Input step disturbance Reference step Input step disturbance Reference step What if we remove the upper bound on the response to input disturbances? The integral action in the controller is lost, just as in lecture! Summary 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 parametrization allows us to use these algorithms for control synthesis Resulting controllers have high order. Order reduction will be studied in the next lecture. 5
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