6.245: Problem Set 8 Solutions

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 8 Solutions 1 Problem 8.1 For the standard LTI feedback design setup defined by equations ⎡ � ⎡ � x(t) ẋ(t) ẋ(t) = ax(t) + u(t) + w1 (t), z(t) = , y= , u(t) x(t) + w2 (t) where a � R is a parameter, find matrices T0 , T1 , T2 defining a valid Qparameterization of all closed loop transfer matrices T : w � z which can be achieved while using a finite order stabilizing dynamic feedback u = Ky. This state space model has matrix coefficients ⎡ � ⎡ � ⎥ ⎦ 1 a A = a, B1 = 1 0 , B2 = 1, C1 = , C2 = , 0 1 ⎡ � ⎡ � ⎡ � ⎡ � 0 0 0 1 0 1 D11 = , D12 = , D21 = , D22 = . 0 0 1 0 1 0 Since D22 →= 0, consider the case when K is strictly proper. Then the set of all achievable closed loop transfer matrices will not change when D22 is replaced by zero. Indeed, for a strictly proper K = K(s), u = K(C2 x + D21 w + D22 u) 1 Version of April 27, 2004 2 is equivalent to where ˆ 2 x + D21 w), u = K(C ˆ = (I − KD22 )−1 K, K = K(I ˆ + KD ˆ 22 )−1 , K ˆ is strictly proper whenever K is strictly proper. and K To apply the Q-parameterization theorem of Lecture 10, take F, L such that A + B 2 F and A + LC2 are Hurwitz matrices, for example, F = −1 − a, L = [−1 − 1], which yields A + B2 F = A + LC2 = −1. Then the explicit formulae for the transfer matrices T0 , T1 , T2 in Twz = T0 + T1 QT2 , Q − stable, proper yield T0 (s) = � 1 s+1 − 1+a s+1 1+a − (s+1) 2 − 1+a + s+1 (1+a)2 (s+1)2 � , T1 (s) = ⎡ 1 1 s+1 − 1+a s+1 � , T2 (s) = ⎡ a 1 − s+1 s 0 s+1 � . Problem 8.2 For the standard discrete time LTI feedback design setup defined by equa­ tions ⎡ � ax[k] x[k + 1] = −x[k] + u[k] + w1 [k], z[k] = , y[k] = x[k] + w2 [k], u[k] where a > 0 is a parameter, find the H2 optimal feedbacl law by using a Tustin transformation to an equivalent continuous time problem. Also give explicit expressions for the equivalent CT setup, and for the cor­ responing CT H2 optimal feedback. As in most discrete time formulations, this H2 optimization setuo has at least two different interpretations: the one in which only strictly causal controllers are allowed (i.e. u[k] is allowed to depend on y[k − 1], y[k − 1], . . . ), and the one with an arbitrary causal controller (u[k] depends on y[k], y[k − 1], . . . ). Below, a solution for the second case is presented. 3 First, since there is an open loop pole at z = −1, we introduce the new control variable v[k] = u[k] − y[k] = u[k] − x[k] − w2 [k]. The new system equations will have the form � ⎡ ax[k] , y[k] = x[k] + w2 [k]. x[k + 1] = w1 [k] + w2 [k] + v[k], z[k] = x[k] + w2 [k] + v[k] Then, in order to reduce the problem to that of desining a strictly causal controller, introduce the new sensor output ȳ[k] = y[k + 1] = x[k + 1] + w2 [k + 1] = w1 [k] + w2 [k] + v[k] + w2 [k + 1]. Since the equations now depend on both w2 [k] and w2 [k + 1], we introduce an additional system state x2 [k] = w2 [k], and a modified noise vector ⎡ � ⎡ � f1 [k] w1 [k] f [k] = = . f2 [k] w2 [k + 1] Now system equations have the form ⎡ � ⎡ � ⎡ � x1 [k + 1] x2 [k] + f1 [k] + v[k] ax1 [k] = , z[k] = , ȳ[k] = x2 [k]+f1 [k]+f2 [k]+v[k], x2 [k + 1] f2 [k] x1 [k] + x2 [k] + v[k] where x1 [k] = x[k]. Here D11 = 0. To make sure that the transfer matrix from control to sensor is zero at z = −1, use D22 = 0, which means introducing a new sensor variable y� [k] = ȳ[k] − v[k] = x2 [k] + f1 [k] + f2 [k]. Now the open loop plant transfer matrix is ⎤ a Applyin the Tustin transform P DT (z) = � z= z 1 z 1 a z2 z+1 z2 z+1 z a z z+1 z 0 1+s z−1 , s= , 1−s z+1 � ⎣. 4 yields a continuous time plant ⎤ � (1−s)2 1−s a a a 1−s 2 (1+s) 1+s � 1+s ⎢ 1−s 2 P (s) = � 1−s . 2 2 1+s (1+s) 1+s ⎣ 2 1 0 1+s After dividing Pf z (s) by 1 − s and multiplying Pvy (s) by 1 − s, we get ⎤ � a(1−s) 1−s a a (1+s)2 1+s � 1+s ⎢ 1 2 2 . P̂ (s) = � 1+s (1+s)2 1+s ⎣ 2 0 1 1+s A state space model of this CT plant is given by ⎡ � ⎡ � ⎡ � ẋ1 (t) −x1 (t) + 2x2 (t) + f1 (t) + 2v(t) a(x1 (t) − x2 (t) − v(t)) = , z(t) = , y(t) = f1 (t)+2x2 (t). ẋ2 (t) −x2 (t) + f2 (t) x1 (t) To solve the corresponding standard CT H2 optimization problem, consider the asso­ ciated full information abstract H2 optimization: ⎡ � ⎡ � � � ẋ1 −x1 + 2x2 + 2u = , {|x1 |2 + a2 |u + x2 − x1 |2 }dt � min . ẋ2 −x2 0 With a modified control vector ũ = u + x2 − x1 , the problem can be re-written in an equivalent one-dimensional form � � ẋ1 = x1 + 2˜ u, {|x1 |2 + a2 |˜ u|2 }dt � min . 0 The optimal controller is given by u = −gf x1 − x2 , where gf = and � 1 1 1 2pf + 2 − =1− 2 , 4 a 2 a a2 pf = + 4 � a2 a4 + 4 16 5 is the stabilizing solution of the associated Riccati equation p2 − a2 a2 p− = 0. 2 4 Similarly, consider the associated state estimation abstract H2 optimization: ⎡ � ⎡ � � � �˙ 1 −�1 , = {|�1 + q|2 + |�2 |2 }dt � min . ˙ 2� − � + 2q �2 1 2 0 With a modified control vector q̃ = �1 + q, the problem can be re-written in an equivalent one-dimensional form � � ˙ {|�2 |2 + |q̃|2 }dt � min . �2 = −�2 + 2q̃, 0 The optimal controller is given by q = −�1 − ge �2 , where ge = and � 1 1 + 1 − = 2pe , 4 2 � 1 1 1 pe = − + + 4 4 16 is the stabilizing solution of the associated Riccati equation 1 1 p2 + p − = 0. 2 4 The optimal CT controller K CT has state space model ⎡ � ⎡ � ⎡ � d xˆ1 −ˆ x2 + 2u 1 x1 + 2ˆ = − x2 − y), u = −gf xˆ1 − xˆ2 , (2ˆ x2 −ˆ ge dt xˆ2 which corresponds to controller transfer function K CT (s) = − gf (s + 1) + ge (s + 1 + 2gf ) . (s + 1 + 2gf )(s + 1 + 2ge ) 6 ˆ The inverse Tustin transform K(z) of (1 − s)K CT (s) is given by ˆ K(z) =− 2zgf + ge (2z + gf (z + 1)) . (2z + gf (z + 1))(2z + ge (z + 1)) Due to the changes of control and sensor variables introduced in the original setup, the true optimal controller (with input y[k] and output u[k]) is given by K(z) = 1 + which yields K(z) = ˆ zK(z) , ˆ 1 + K(z) 4z − ge gf . (2 + ge )(2 + gf )z + ge gf Problem 8.3 Consider a system described by the hyperbolic partial differential equation vt = vxx + rv, v(0, t) = 0, y(t) = v(1, t) + w(t), u(t) = vx (1, t), where v = v(x, t), for fixed time, is a function of the spatial parameter x � [0, 1], v t denotes the time derivative of v, vxx denotes the double spatial derivative of v, and r > 0 is a given parameter. The control action is the Dirichlet boundary condition u(t) = v x (1, t), while a noisy measurement of y(t) = v(1, t) + w(t) is used as the sensor signal. (a) Find an analytical expression for the transfer function P = Pr (s) from u to y. (b) For r = 1, find a good low order rational approximation P̂1 of P1 , such that � = P1 − P̂1 is stable, together with an upper bound ���� < �. (c) Using the results from (b), small gain theorem, and H-Infinity optimization, design a finite order stabilizing feedback u = Ky for the original system, while trying to provide an upper bound for the closed loop H-Infinity norm �Twu � which is as small as possible. Note that this will only be possible when � is small enough.