Lecture Notes on Q-Parameterization

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Q-Parameterization1 This lecture introduces the so-called “Q-Parameterization” (sometimes referred to as the “Youla Parameterization”). 10.1 Introduction Not every stable transfer matrix w � z can be obtained by closing a stabilizing LTI feedback loop u = K(s)y in canonical feedback design setup shown on Figure 10.1. In w z P (s) y u K(s) � Figure 10.1: LTI feedback this lecture, we investigate the limits of LTI feedback’s ability to change the closed loop transfer matrix. More precisely, it will be shown that the set of all closed loop transfer matrices is affine. In addition, the role of unstable poles and zeros of P in limiting LTI feedback designs will be investigated. 1 Version of April 2, 2004 2 Q-parameterization gives a very simple affine description of the set of all achievable closed loop transfer matrices as function of the so-called “Q” parameter Q = Q(s), which is an arbitrary stable proper transfer matrix of size m × k, where m is the total number of actuator inputs, and k is the total number of sensors in the system. For design purposes, the result is extremely important: instead of thinking in terms of the controller transfer matrix K = K(s), one will be much better off by designing Q(s). 10.2 The Q-parameterization theorem We consider LTI systems P from Figure 10.1 (where u and y are vectors of size m and k respectively) given in the state space format � � B1 B2 A (10.1) P (s) := ⎞ C1 D11 D12 � . C2 D21 0 An LTI feedback controller u = −Kx, where ⎠ ⎛ Af Bf K := C f Df is said to stabilize the feedback system if all eigenvalues of the matrix ⎝ � A + B 2 Df C 2 B 2 C f Acl = Bf C2 Af (10.2) (10.3) have negative real part. Let T = T (s) be the transfer matrix of the resulting closed loop system with input w and output z: � � B1 + B2 Df D21 A + B 2 Df C 2 B 2 C f �. Bf D21 Bf C2 Af (10.4) T := ⎞ C1 + D12 Df C2 D12 Cf D11 + D12 Df D21 Note that the description of T given in (10.4) is very inconvenient for design purposes, because the design parameters Af , Bf , Cf , Df enter the transfer matrix defined by (10.4) in a very non-linear way, and are themselves constrained nonlinearly by the condition that Acl in (10.3) must be a Hurwitz matrix. The following result gives a much more useful parameterization of all T = T (s). 3 Theorem 10.1 Let F, L be constant gains such that matrices A + B2 F and A + LC2 are Hurwitz. Then a given transfer matrix T = T (s) can be made equal to the transfer matrix of the closed loop system (10.4) by an appropriate selection of a stabilizing feedback controller (10.2) if and only if T (s) = T0 (s) + T1 (s)Q(s)T2 (s) (10.5) for some stable proper rational transfer matrix Q = Q(s), where T0 , T1 , T2 are the transfer matrices of systems � � B1 A B2 F −LD21 � , (10.6) T0 = ⎞ −LC2 A + B2 F + LC2 C1 D12 F D11 ⎠ ⎛ A + B2 F B2 T1 = , (10.7) C1 + D12 F D12 ⎠ ⎛ A + LC2 B1 + LD21 T2 = . (10.8) C2 D21 Moreover, T (s) can be achieved by using a strictly proper stabilizing controller if and only if representation (10.5) with a stable strictly proper rational Q is possible. Theorem 10.1 is a remarkable result, which, in particular, ensures that any convex feedback optimization problem formulated in terms of the closed loop transfer matrices can be solved relatively easily. In (10.5), T0 represents some admissible closed loop design, and the structure of T1 , T2 imposes limitations on the closed loop transfer matrix. 10.3 Derivation of Youla parameterization Let F, L be defined as in Theorem 10.1, i.e. A + B2 F and A + LC2 are Hurwitz matrices. For any u(·) let x̂ be defined by xˆ˙ = Aˆ x + B2 u + L(C2 xˆ − y). Then for e = x − x̂ we have ė = (A + LC2 )e + (B1 + LD21 )w. Let � = y − C2 x̂ = C2 e + D21 w. 4 Note that � is the output of system T2 with input w. Now the equations for x̂, xf , u can be re-written in the form x̂˙ = (A + B2 Df C2 )x̂ + B2 Cf xf + (B2 Df − L)�, ẋf = Bf C2 x̂ + Af xf + Bf �, u − F x̂ = (Df C2 − K)x̂ + Cf xf + Df �. Hence, for a stabilizing controller (10.2), u − F x̂ is the output of the stable system � � ⎝ � B 2 Df − L Acl � Bf Q=⎞ � � Df C 2 − F C f Df with input �, where Acl is defined by (10.3). Since, in addition, the equations for x, u − F x̂, w, e, z can be written as ẋ = (A + B2 F )x + B2 (u − F x̂) + (B1 w − B2 F e) z = (C1 + D12 F )x + D12 (u − F x̂) + (D11 w − D12 F e), z equals the output of T1 with input u − F x̂ plus the output of T0 with input w. This proves that the closed loop transfer matrix from w to z has the form (10.5) with some stable Q. Conversely, if Q is stable, using u which is a sum of F x̂ and the output of Q with input � defines a stabilizin feedback controller u = Ky. Figure 10.2 gives a simple interpretation of Q-parameterization: in order to describe a general form of a stabilizing LTI controller, one has to find a special stabilizing controller first, then use a copy of the stabilized system as an estimator of the sensor output, and feed a stable LTI transformation Q = Q(s) of the difference between the actual and the estimated sensor values back into the control input. 10.4 Open loop zeros The restrictions on the closed loop transfer function imposed by the Youla parameteriza­ tion can be understood in terms of the so-called open loop zeros. To define zeros of MIMO systems, consider state space models ẋ1 = ax1 + bu1 , y1 = cx1 + du1 , where the dimensions of x1 , u1 and y1 are n, m and k respectively. Consider the complex matrix ⎝ � a − sI b M (s) = c d 5 w z P � � K0 � − � � Q 0 P � � K0 � Figure 10.2: Youla parameterization where s � C is a scalar complex variable. The system is said to have a right zero at a point s if ker M (s) = � {0}, i.e. if M (s) is not left invertible, or, equivalently, if there exists a non-zero pair (X1 , U1 ) of complex vectors such that sX1 = aX1 + bU1 , 0 = cX1 + dU1 . Similarly, the system is said to have a left zero at s if the range of M (s) is not the whole vector space Cn+k , i.e. if M (s) is not right invertible, or, equivalently, if there exists a non-zero pair (p1 , q1 ) of complex vectors such that sp1 = p1 a + q1 c, 0 = p1 b + q1 d. As it can be seen from the formulae for T1 and T2 , the restrictions on the closed loop transfer matrix are caused by the unstable right zeros of the system ẋ = Ax + B1 w, y = C2 x + D21 w, (10.9) and by the unstable left zeros of the system ẋ = Ax + B2 u, z = C1 x + D12 u. (10.10) It can be seen immediately that the right zeros of (10.9) cause problems by obstructing the observation process, while the left zeros of (10.10) describe problems which a control action will expecrience even in the case of a complete knowledge of w and x. 6 10.5 Time-Domain Q-Parameterization The following statement re-states Theorem 10.1 in a time-domain format. Theorem 10.2 A matrix-valued function Z = Z(t) is a closed-loop impulse response from w to z generated in the closed loop system (10.4) by an appropriate selection of a stabilizing feedback controller (10.2) if and only if there exists a function Q = Q(t), elements of which are finite linear combinations of Dirac deltas � = �(t) and generalized one-sided exponents tk est u(t) with Re(s) < 0, such that Z(t) = (C + D12 F )X(t) + D12 V (t) + D11 �(t), where V (t) = �(t)(B1 + LD21 ) + Q(t)D21 , ˙ �(t) = �(t)(A + LC2 ) + Q(t)C2 , Ẋ(t) = (A + B2 F )X(t) + B2 V (t) + B1 �(t), and X(·), �(·) are zero for t < 0.