Multivariable Control Systems Ali Karimpour Assistant Professor Ferdowsi University of Mashhad <<<1.1>>> ###Control System Design### {{{Control, Design}}}

Topics to be covered include: Chapter 8 Multivariable Control System Design: LQG Method Topics to be covered include: LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

LQG Control LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

LQG Control In traditional LQG Control, it is assumed that the plant dynamics are linear and known and that the measurement noise and disturbance signals (process noise) are stochastic with known statistical properties. That is, wd and wn are white noise processes with covariances The problem is then to devise a feedback-control law which minimizes the ‘cost’

LQG Control The solution to the LQG problem is prescribed by the separation theorem, which states that the optimal result is achieved by adopting the following procedure. First, obtain an optimal estimate of the state x Optimal in the sense that is minimized Then use this estimate as if it were an exact measurement of the state to solve the deterministic linear quadratic control problem.

LQG Control: Optimal state feedback The optimal solution for any initial state is where Where X=XT ≥ 0 is the unique positive-semidefinite solution of the algebraic Riccati equation

LQG Control: Kalman filter The Kalman filter has the structure of an ordinary state-estimator or observer, as Where Y=YT ≥ 0 is the unique positive-semidefinite solution of the algebraic Riccati equation

LQG Control: Combined optimal state estimation and optimal state feedback KLQG(s) Exercise 1: Proof the relation of KLQG(s) according to above figure.

Robustness Properties LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

Robustness Properties For an LQR-controlled system (i.e. assuming all the states are available and no stochastic inputs) it is well known (Kalman, 1994; Safonov and Athans, 1997) that, if the weight R is chosen to be diagonal, the sensitivity function satisfies Nyquist plot in MIMO case From this it can be shown that the system will have a gain margin equal to infinity, a gain reduction margin (lower gain margin) equal to 0.5 and a (minimum) phase margin of 60˚ in each plant input control channel. -1 -1

Robustness Properties

Robustness Properties Example 8-1: LQR design of a first order process. Consider a first order process For a non-zero initial state the cost function to be minimized is The algebraic Riccati equation becomes

Robustness Properties Example 8-1: LQR design of a first order process. Consider a first order process

Robustness Properties So, an LQR-controlled system has good stability margins at the plant inputs, Arguments dual to those employed for the LQR-controlled system can then be used to show that, if the power spectral density matrix V is chosen to be diagonal, then at the input to the Kalman gain matrix Kf there will be an infinite gain margin, a gain reduction margin of 0.5 and a minimum-phase margin of 60˚.

Robustness Properties So, an LQR-controlled system has good stability margins at the plant inputs, And Kalman filter has good stability margins at the inputs to Kf For an LQG-controlled system with a combined Kalman filter and LQR control law are there any guaranteed stability margins? Unfortunately there are no guaranteed stability margins. This was brought starkly to the attention of the control community by Doyle (1978 ) (in a paper entitled “Guaranteed Margins for LQR Regulators” with a very compact abstract which simply states “There are none”). Doyle showed, by an example, that there exist LQG combinations with arbitrarily small gain margins.

Robustness Properties Why there are no guaranteed stability margins in LQG controller. (Regulator transfer function) guaranteed stability margins L2 L3 (Kalman Filter transfer function) guaranteed stability margins L4 The most important loop but no guaranteed stability margins

Loop Transfer Recovery LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

Loop transfer recovery (LTR) procedures Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs. The LQG loop transfer function Guaranteed stability margins If Kr in the LQR problem is designed to be large using the sensitivity recovery procedure of Kwakernaak (1969). The LQG loop transfer function Guaranteed stability margins If Kf in the Kalman filter to be large using the robustness recovery procedure of Doyle and Stein (1979).

Loop transfer recovery (LTR) procedures Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs. The LQG loop transfer function L2 Recovering robustness at the plant output The LQG loop transfer function L1 Recovering robustness at the plant input

Loop transfer recovery (LTR) procedures Recovering robustness at the plant input Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs. L1 Step I: First, design the linear quadratic problem whose transfer function KrΦ(s)B is desirable. This is done, in an iterative fashion, by manipulate the matrices Q and R, emphasis of The design is on aspects such as gains, possibly ‘balancing’ the principal gains, and adjusting the low frequency behavior. Step II: When the singular values of KrΦ(s)B are thought to be satisfactory, LTR is achieved by designing Kf in the Kalman filter by setting Г=B, W=I and V= ρI ,where ρ is a scalar. As ρ tends to zero

and R= ρI, where ρ is a scalar. As ρ tends to zero Loop transfer recovery (LTR) procedures Recovering robustness at the plant output Assume that the plant model G(s) is minimum-phase and that it has at least as many inputs as outputs. L2 Step I: First, we design a Kalman filter whose transfer function CΦ(s)Kf is desirable. By choosing the power spectral density matrices W and V so that the minimum singular value of CΦ(s)Kf is large enough at low frequencies for good performance and its maximum singular value is small enough at high frequencies for robust stability. Step II: When the singular values of CΦ(s)Kf are thought to be satisfactory, loop transfer recovery is achieved by designing Kr in an LQR problem with M=C, Q=I and R= ρI, where ρ is a scalar. As ρ tends to zero

Loop transfer recovery (LTR) procedures LTR procedure guaranteed to work only with minimum-phase plants. Since it relies on the ‘cancellation’ of some of the plant dynamics by the filter Dynamics) If RHP zeros exist in the plant the procedure may still work, particularly if these zeros lie beyond the operation bandwidth of the system as finally designed.

Loop transfer recovery (LTR) procedures Proof: Recovering robustness at the plant input ? Г=B, W=I and V= ρI ,As ρ tends to zero The LQG loop transfer function at the plant input is: By matrix-inversion lemma we have Now the algebraic Riccati equation is: Exercise: Derive equation I .

Loop transfer recovery (LTR) procedures Proof: Recovering robustness at the plant input Г=B, W=I and V= ρI ,As ρ tends to zero Now the algebraic Riccati equation is: It can be shown (Kwakernaak and Sivan, 1973) that, if C(sI-A)-1ГW1/2 has no RHP zero and if it has at least as many outputs as rank(Σ), then

Loop transfer recovery (LTR) procedures Proof: Recovering robustness at the plant input Г=B, W=I and V= ρI ,As ρ tends to zero It can be shown (Kwakernaak and Sivan, 1973) that, if C(sI-A)-1ГW1/2 has no RHP zero and if it has at least as many outputs as rank(Σ), then In particular if we choose and provided C(sI-A)-1B has no zeros in LHP, then Substituting this in the LQG loop transfer function at the plant input leads to:

Loop transfer recovery (LTR) procedures Proof: Recovering robustness at the plant input Г=B, W=I and V= ρI ,As ρ tends to zero By push-through rule: Finally we have:

Shaping the Principal Gains (Singular Values) LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

Shaping the principal gains (singular values) How to modify W and V in order to bring about desirable changes in C(sI-A)-1Kf In order to exploit LTR technique, we must to know: How to modify Q and R in order to bring about desirable changes in Kr(sI-A)-1B In order to obtain an intuitive grasp of this, consider the Kalman filter. (let u=0) This is now looks like a feedback system which is to: track (in a sense) the ‘reference input’ z, while rejecting the measurement errors v.

Shaping the principal gains (singular values) To shape the principal-gain plots we can do one of two things: Modify the plant model by augmenting it with additional dynamics For example adding integrator in each loop. Modify the matrices ГWГT and V in a more sophisticated way, We can use ГWГT to increase the smallest principal gain of the sensitivity matrix, or decrease the largest one near some particular frequency.

Shaping the principal gains (singular values) Modify the matrices W and V Let Ff as the return difference of Kalman filter, and define Then we can show that Suppose we choose V=I. Then from which it follows that Exercise I : Derive equation I . (Hint Maciejowski 1989 pp. 227-231) Exercise II: Derive equation II .

Shaping the principal gains (singular values) Modify the matrices W and V So we can reduce by increasing , etc. But the point is not merely to reduce all the singular values of , but to reduce the largest one, relative to smallest. One way is: Suppose we need adjustment at ω1 Now let So so the jth singular value has been changed by a factor (1+α), while all the other singular values have been left unchanged.

Shaping the principal gains (singular values) Modify the matrices W and V Example: Let Singular value of GfW1/2 is: We want to change 7.6 to 3*7.6 so we change W1/2 by

Shaping the principal gains (singular values) Modify the matrices W and V Now let So so the jth singular value has been changed by a factor (1+α), while all the other singular values have been left unchanged. The problem with this approach is that uj is usually a complex vector, whereas we wish to keep W1/2 real. Once again we are faced with the problem of approximating a complex matrix by a real matrix, and as before we can employ the align algorithm. In this case other algorithms may be more appropriate, however, since we really want to approximate uj rather than align it. In particular, Re{uj} is sometimes an adequate approximation. A further possibility is to approximate uj by the output direction of the matrix [Re{uj} Im{uj}] which corresponds to its largest singular value.

Some Practical Consideration LQG Control Robustness Properties Loop transfer recovery (LTR) procedures - Recovering robustness at the plant output - Recovering robustness at the plant input - Shaping the principal gains (singular values) Some practical consideration

Some practical consideration LTR procedures are limited in their applicability. Their main limitation is to minimum phase plants. This is because the recovery procedures work by canceling the plant zeros, and a cancelled non-minimum phase zero would lead to instability. The cancellation of lightly damped zeros is also of concern because of undesirable oscillations at these modes during transients. A further disadvantage is that the limiting process For full recovery Introduces high gains which may cause problems with unmodelled dynamics. The recovery procedures are not usually taken to their limits. The result is a somewhat ad-hoc design procedure.

Loop transfer recovery (LTR) procedures Design example Consider the aircraft model AIRC described in the following state-space model. the model has three inputs, three outputs and five states.

Loop transfer recovery (LTR) procedures Design example THE SPECIFICATION We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop. Integral action in each loop, little interaction between outputs. Good damping of step responses and zero steady-state error in the face of step demands or disturbances. PROPERTIES OF THE PLANT The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very severe interaction between outputs. The poles of the plant (eigenvalues of A) are so the system is stable (but not asymptotically stable). Thus this plant has no finite zeros, and we do not expect any limitations on performance to be imposed by zeros.

Loop transfer recovery (LTR) procedures Design example Recovering robustness at the plant output The LQG loop transfer function Recovering robustness at the plant input The LQG loop transfer function Here we use Recovering robustness at the plant output

Loop transfer recovery (LTR) procedures Design example Kalman filter design We need to choose the matrices Г, W, V, which appear in and obtain the Kalman-filter gain Kf from We shall write It is generally advisable to start with simple choices of Г, W, V, inspect L4 Then adjust Г, W, V accordingly, and so gradually improve L4 One of the simplest possible choices is Г=B, W=I3 and V=I3.

Loop transfer recovery (LTR) procedures Design example Kalman filter design So try with The loop transfer function is:

Loop transfer recovery (LTR) procedures Design example Kalman filter design BW around 1 rad/sec Constant gain at low frequencies Decreasing with 20 db/dec at low frequencies

Loop transfer recovery (LTR) procedures Design example Kalman filter design The first thing to do is to insert integral action, by augmenting the plant model. Placing poles of the augmented model at the origin leads to problems in the recovery step later, so in this case we place them at -0.001, which is virtually at the origin, when compared to the required bandwidth 10 rad/sec We could also have chosen Cw more carefully, with the aim of adjusting the low frequency gains. The augmented model is

Loop transfer recovery (LTR) procedures Design example Kalman filter design Now we have

Loop transfer recovery (LTR) procedures Design example Kalman filter design The loop transfer function is: The gain was added around 60 db at low frequency according to integrator Kf2 Kf1 We want to increase the gain at low frequency. 60 db By tuning W one can manipulate gains.

Loop transfer recovery (LTR) procedures Design example Kalman filter design By tuning W one can manipulate gains. But how? In a case of diagonal system every diagonal element of W corresponds to a singular value. But in non-diagonal system we must use singular value decomposition. Which gives

Loop transfer recovery (LTR) procedures Design example Kalman filter design By tuning W one can manipulate gains. But how? Now let α=9 (α+1=10) so we have (better value for α+1 is 4637/651=7.12) so

Loop transfer recovery (LTR) procedures Design example Kalman filter design The loop transfer function is: Kf2 Kf1 Kf3 Band width problem? We need at least 7 rad/sec. W3 must increase.

Loop transfer recovery (LTR) procedures Design example Kalman filter design W3 must increase. Maximum singular value of W3 and 10W3 and 100W3 are considered Kfx Kf3 Kf4 Which one is ok?

Loop transfer recovery (LTR) procedures Design example Kalman filter design The loop transfer function is: We need to find closed loop transfer functions

Loop transfer recovery (LTR) procedures Design example Kalman filter design We need to find closed loop transfer functions -3 We could therefore terminate the Kalman filter design and move on to the recovery step. However we shall suppose that we wish to improve the sensitivity further.

Loop transfer recovery (LTR) procedures Design example Kalman filter design However we shall suppose that we wish to improve the sensitivity further. W51/2 can be shape as follows:

Loop transfer recovery (LTR) procedures Design example Kalman filter design So we have

Loop transfer recovery (LTR) procedures Design example Kalman filter design The loop transfer function is: The principle gains have clearly been “squeezed together” near 10 rad/sec, and at higher frequency.

Loop transfer recovery (LTR) procedures Design example Kalman filter design We need to find closed loop transfer functions

Loop transfer recovery (LTR) procedures Design example Kalman filter design The characteristic loci of is: All the loci remain outside or on the boundary of the unit circle as predicted by Kalman filter theory, from which we know that:

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Since the system has no transmission zero in the RHP so arbitrary good recovery should be possible. To obtain LTR, we solve the following Riccati equation With M=Ca, Q=I and R=ρI We shall write When Kr has been find the controller realization is: Now we need

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Now we need ρ=10-2 L2 L4 ρ=10-4 L2 L4

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Now we need ρ=10-6 L2 L4 ρ=10-8 L2 L4

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output We have for ρ=10-8

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Characteristic loci at the output of compensated plant, for ρ=10-6 and ρ=10-8 ρ=10-6 ρ=10-8

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Principle gains of S and T, for ρ=10-6 and ρ=10-8 ρ=10-6 S T ρ=10-8 S T

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output Closed-loop step responses to step responses to different inputs.

Loop transfer recovery (LTR) procedures Design example Recovery at the plant output The closed-loop poles are located at