Under assumptions (A1), (A2), (A3), the closed-loop system in figure() is stable if:

Output multiplicative uncertainty Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, s( jω) G( jω) K( jω) Ip G( jω) K( jω) + <, (0.39) or, equivalently, ( ω) ( ω)( p ( ω) ( ω)) δm ( ω) G j K j I + G j K j <. (0.40) Under assumption (A4), the closed-loop system in figure() is stable if and only if: ω \, ( ω) ( ω)( p ( ω) ( ω)) δm ( ω) G j K j I + G j K j <. (0.4) Figure 26: Equivalent M interconnection for output multiplicative uncertainty This last condition can be written as: m T ( jω) < δ ( ω), where y Ty(): s = G() s K() s I p + G() s K() s (0.42) is the output complementary sensitivity matrix. We can see from the robust stability condition for a multiplicative uncertainty, that T y reflects the capacity of the system to tolerate uncertainty of the output multiplicative form. ROBSTAB - 5/7

5REXVW VWDELOLW\ ZLWK LQYHUVH PXOWLSOLFDWLYH XQFHUWDLQW\ Input inverse multiplicative uncertainty Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, ( jω) [ I + K( jω) G( jω) ] <, (0.43) s m I + K( jω) G( jω) < δ ( ω). (0.44) or, equivalently, [ ] m m Under assumption (A4), the closed-loop system in figure() is stable if and only if: ω \, [ I + K( jω) G( jω) ] < δ ( ω). (0.45) m m Figure 27: Equivalent M interconnection for input inverse multiplicative uncertainty Output inverse multiplicative uncertainty Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, s( jω) Ip G( jω) K( jω) + <, (0.46) or, equivalently, m Ip + G( jω) K( jω) < δ ( ω). (0.47) Under assumption (A4), the closed-loop system in figure() is stable if and only if: ω \, m Ip + G( jω) K( jω) < δ ( ω). (0.48) ROBSTAB - 52/7

Figure 28: Equivalent M interconnection for output inverse multiplicative uncertainty 5REXVW VWDELOLW\ ZLWK IHHGEDFN XQFHUWDLQW\ Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, f ( jω) I p + G( jω) K( jω) G( jω) <, (0.49) or, equivalently, f Ip + G( jω) K( jω) G( jω) < δ ( ω). (0.50) Under assumption (A4), the closed-loop system in figure() is stable if and only if: ω \, f Ip + G( jω) K( jω) G( jω) < δ ( ω). (0.5) f ( s) + K( s) + G( s) Figure 29: Interconnection for feedback uncertainty ROBSTAB - 53/7

5REXVW VWDELOLW\ ZLWK OLQHDU IUDFWLRQDO XQFHUWDLQW\ Theorem Under assumptions (A), (A2), (A3), the closed-loop system in figure() is stable if: ω \, ( jω) [ P( jω), K( jω) ] <, (0.52) or, equivalently, [ P j K j ] s L l ( ω), ( ω) < δ ( ω). (0.53) L Under assumption (A4), the closed-loop system in figure() is stable if and only if: ω \, [ P( jω), K( jω) ] < δ ( ω). (0.54) L The figure below shows a feedback controlled upper LFT with the perturbation. This is the most general representation as all other robust stability conditions can obtained from this one. Typically, the generalized plant Ps () would embed all of the weighting functions used to specify performance and also weighting functions that shape the uncertainty. l s ( s) z 2 P( s) w 2 K( s) Figure 30: LFT for linear fractional uncertainty ROBSTAB - 54/7

The first condition given in each theorem above are just sufficient conditions for robust stability. But the second condition given based on the small-gain theorem are necessary and sufficient, and these are the most useful to control engineers. As previously mentioned, the fact that () s must be stable in this case is not too restrictive. The uncertainty bound δω ( ) is usually specified by a scalar stable rational weigthing function W() s as follows. Let the perturbation be defined as (): s = W() s () s where () s is a normalized perturbation in * such that () <. Then s ( jω) < W( jω) = δ( ω), ω W j j j ( ω) ( ω) = ( ω) <, ω () s < \ \ (0.55) Thus, the upper bound () < on the -norm of the normalized perturbation can be used in s the theorems, along with the weighting functions. These conditions are listed in the table below. ROBSTAB - 55/7

Table : Robust stability conditions Uncertainty form Assumptions W, x * x Necessary and sufficient condition for robust stability Additive = W, a a a < Input multiplicative = W, a m m m < Output multiplicative = W, m m m m < Input inverse multiplicative = W, m i i i < Output inverse multiplicative = W, i i i i < Feedback = W, i f f f < LFT = W, f l l l < l KS W = K I + GK W ( ) y a p a [ ] TW = KG I + KG W u m m m y m p m TW = GK I + GK W [ ] SW = KG I + KG W u i m i y i p i SW = I + GK W y f p f SGW = I + GK GW L [ PK, ] W l absorbed in P ROBSTAB - 56/7

These results indicate that we have to minimize the -norm of certain closed-loop transfer matrices in order to increase robustness margins, i.e., the level of uncertainty that can be tolerated without destabilizing the system. Example: Robust stability of room heating process control with uncertain sensor dynamics The temperature sensor with first-order dynamics was assumed to have 0% uncertainty in its frequency response, i.e., l ( jω ) < 0., ω, so that Wm () s = 0.. This uncertainty was modeled with an output multiplicative model, which was then transformed into an LFT model. The controller Ks ( ) = 000 is chosen as a pure gain. sensor uncertainty m ( s) Wm ( s) Q 0.0 000 s + T 0 s + + + T m nominal room dynamics nominal sensor 000 + 0 controller Note that the controller stabilizes the nominal loop gain. The condition for robust stability with output multiplicative uncertainty is TW y m = GK Ip + GK Wm. Let's compute the norm of this closed-loop transfer function (from the output of the weighted uncertainty to its input): 0(0.) 0 GK I p + GK Wm = (000 )(0 ) + s s (000s )(0s ) + + + + = = s+ s+ + s + s+ 0.09 = (80.5s+ )(.3s+ ) 2 (000 )(0 ) 0 0000 00 (0.56) The absolute value of the DC gain of this transfer function is 0.09, which is also the maximum magnitude of its frequency response. Therefore, ROBSTAB - 57/7

0.09 GK I p + GK Wm = (80.5s+ )(.3 s+ ) 0.09 = sup = 0.09 < ω \ (80.5 jω + )(.3 jω + ) (0.57) and hence the closed-loop system has robust stability. It also means that the closed-loop system could tolerate up to (=/0.9) times more uncertainty in the sensor before the possible onset of instability. The Bode plot of GK I p + GK Wm is shown below. 0 0 Log Magnitude 0-2 0-4 0-6 0-5 0-4 0-3 0-2 0-0 0 0 Frequency (radians/sec) 0 Phase (degrees) -50-00 -50-200 0-5 0-4 0-3 0-2 0-0 0 0 Frequency (radians/sec) We will now check that the robustness condition based on the LFT representation will give us the same PK, conclusion. This condition is L [ ] a previous example (but Wl () s must be added to it):, where the generalized plant Ps () was obtained in P () s = G(), s P () s = I, P () s = 0, P () s = G() s W () s (0.58) 22 2 2 l ROBSTAB - 58/7

l ( s) P( s) T m 0 0.00 (000s + )(0s + ) 0.0 (000s + )(0s + ) Q 000 [ ] = + [ ] L Ps ( ), Ks ( ) P( s) P( sks ) ( ) I P ( sks ) ( ) P( s) 2 22 2 0 = + (000s+ )(0s+ ) (000s )(0s ) + + = = s+ s+ + s + s+ 2 (000 )(0 ) 0 0000 00 (0.59) As we expected, we get the same closed-loop transfer function, and therefore we get robust stability since [ ] L PK, = 0.09<. (0.60) ROBSTAB - 59/7

6WUXFWXUHG XQFHUWDLQW\ Sometimes the uncertainty in a model will come from different sources: uncertain time delay, neglected hig-frequency dynamics, actuator uncertainty, etc. One way to model this uncertainty is to try to "cover" all of these possible variations with a single full complex block at each frequency (i.e., an unstructured uncertainty block.) This approach often turns out to be conservative in the sense that the chosen unstructured uncertainty will have a large size and hence may generate a much larger set of perturbed plant models than what is required. Another approach is to model each source of uncertainty separately with its own perturbation. To each perturbation will correspond a delta block in the block diagram of the perturbed plant model. The best framework to use in this case is the LFT form with a so-called structured perturbation. We proceed with an example to show how such a structured perturbation can be constructed. Example: Structured uncertainty for the room heating process Suppose that there is uncertainty not only in the sensor, but also in the process model itself, particularly in the time constant. Assume that there is 20% uncertainty in this time constant. The additional perturbation can be "pulled out" as follows: 0.0 = 0.0 000s +, δ \, δ <. (0.6) (000 + 200 δ) s+ 200δs + 000 s + This equation is just a feedback interconnection with the perturbation in the feedback path, as shown below: ROBSTAB - 60/7

200s 0.000s + δ Q 0.0 + 000 s + T 200s Notice that the weighting function was taken to be instead of 200s as (0.6) suggests. 0.000s + This is because because all weighting functions must be in *, and 200s isn't. The addition of the high-frequency pole should not affect the design at frequencies of interest. Plugging the above diagram into the overall perturbed plant block diagram, we get: sensor uncertainty Q 0.0 + 200s 0.000s + 000s + δ T m ( s ) 0. + 0s + + T m nominal room dynamics 000 nominal sensor + 0 controller Define the structured perturbation δ 0 s(): s =. (0.62) 0 m ( s) The block diagram above can be rearranged into an LFT form. A good technique to find the block entries of the generalized plant transfer matrix is to follow the signal paths on the block diagram from and to the perturbations, and from and to the controller. ROBSTAB - 6/7

δ m ( s) P( s) K( s) We have ROBSTAB - 62/7

P P2 P3 Ps () = P2 P22 P 23 P3 P32 P 33 200s P() s = (000s+ )(0.000s+ ) P () s = 0 2 0.0 P3 () s = 000s + 200s P2() s = (000s+ )(0s+ )(0.000s+ ) P P 22 23 () s = 0 0.0 () s = (000s+ )(0s+ ) 200s P3() s = (000s+ )(0s+ )(0.000s + ) P 32 ( s) = 0. 0.0 P33 () s = (000s+ )(0s+ ) With the structured perturbation defined above, the LFT can be rearranged into another simpler LFT as shown below, where P' P' P P P P'( s) =, P', P', P' [ P P ], P' P P P = = = = P P 2 2 3 2 2 3 32 22 33 ' 2 ' 22 2 22 P 23 s ( s) P'( s) 000 ROBSTAB - 63/7

5REXVW FORVHGORRS VWDELOLW\ ZLWK VWUXFWXUHG XQFHUWDLQW\ µ $QDO\VLV Robust stability analysis with structured uncertainty was developed by Doyle (Doyle, 982). He introduced the structured singular value of a matrix as a tool of linear algebra that can be used to analyze stability robustness with structured uncertainty in the frequency domain. This seminal work has developed into a theory called µ -analysis. The robust control design counterpart of this theory is called µ -synthesis and will be studied in the next chapter. 7KH VWUXFWXUHG VLQJXODU YDOXH Basic concept The structured singular value is a generalization of the maximum singular value of constant complex matrices. Let us consider again the robust stability problem of the following standard feedback interconnection with stable M() s and () s. ( s) M ( s) The closed-loop poles are given by det [ I M( s) ( s) ] 0 unstable if det [ I M( s) ( s) ] = 0 for some s ^ + (closed RHP). Let 0 =, and the feedback system becomes β > a sufficiently small < β. number such that the closed-loop system is stable for all stable perturbation bounded by Next, increase β up to β max such that the closed-loop system becomes unstable for some < β max. So β max is the robust stability margin. By the small-gain theorem, β max s ^ + ( ()) ( ( )) = M : = sup σ M s = supσ M jω ω (0.63) if () s is unstructured. In fact, we can tell how large the bound on () s can get in terms of the -norm of M() s such that stability of the closed-loop system will be maintained. Namely, the feedback system can tolerate uncertainty with -norm up to M. Note that for any fixed complex number s 0 + can be written as: ^, the maximum singular value (the norm) σ M() s ROBSTAB - 64/7 ( )

( 0) σ ( M( s0) ) M s = = p p { σ ( ) ( I M( s0 ) ) = ^ } min : det 0, (0.64) The reciprocal of the largest singular value of M( s 0 ) is a measure of the smallest unstructured p p ^ that can cause instability of the feedback system. To quantify the smallest destabilizing structured complex s, the concept of maximum singular value needs to be generalized. We loosely define (we'll give a more formal definition later) µ ( M( s0 )) = { σ ( s) ( I M( s0 ) s) = s } min : det 0, is structured as the largest structured singular value of M( s 0 ) with respect to the structured complex perturbation s. Then, the robust stability margin β max of the feedback system with structured complex uncertainty s is given by: = sup µ ( M() s ) = sup µ ( M( jω) ). s ^ ω \ β max + Definition of the structured singular value µ We consider complex matrices nxn M ^. In the definition of ( M ) µ, there is an underlying structure (set of structured, block-diagonal matrices) on which everything in the sequel deps. For each problem, this structure is, in general, different; it deps on the uncertainty and performance objectives of the problem. Defining the structure involves specifying three things: the type of each block, the total number of blocks, and their dimensions. There are two types of blocks: (possibly-repeated) scalar blocks and full blocks. Two nonnegative integers, S and F, represent the number of (possibly-repeated) scalar blocks and the number of full blocks, respectively. We consider the general case of a S -scalar-block, F -full-block structured perturbation, where the n n scalar blocks may be repeated. Let M ^, and r,!, rq, m,!, ml be positive integers satisfying S F ri + mj = n (this is just for bookkeeping of the dimensions of the uncertainty blocks). i= j= We define the block structure n n Γ ^ as m m { s { δ I } } r δqirq l δi j j j Γ= : = diag,!,,,!, : ^, ^. (0.65) ROBSTAB - 65/7

Definition: structured singular value For a complex matrix M ^ n n n n ^, the function µ Γ : + \ is defined as { s s s } µ Γ ( M) : = min : Γ, det( I M ) = 0 (0.66) unless no Γ makes det( I M ) singular, in which case µ Γ ( M ) = 0. s s Remarks This definition is for complex matrices, not for transfer matrices. µ Γ ( M ) is called the structured singular value because, as previously explained, if the structure is simplified to a single-block complex uncertainty, then µ Γ ( M) = σ ( M), the maximum singular value of the matrix. Hence, µ Γ ( M ) is a generalization of the maximum singular value (or spectral norm) of matrix M. In words, µ Γ ( M ) is the inverse of the size of the smallest structured perturbation that makes I M singular. Thus, this definition is directly related to the small-gain theorem. s The full uncertainty blocks do not have to be square. We assume that they are for ease of presentation. Computation of the structured singular value is usually performed frequency-by-frequency on the frequency response of a system. Suppose M n n ^ and consider the loop shown in the figure below. s M Here, µ Γ ( M ) is the reciprocal of the norm of the smallest structured perturbation, s Γ, that causes "instability" of the feedback loop above, in the sense of singularity of the matrix I M s, or well-posedness of the corresponding closed-loop equations. Define the " unit open ball" of complex structured perturbations: { } Γ= : Γ: <. (0.67) ROBSTAB - 66/7

Then, an alternative expression for µ Γ ( M ) follows from the definition: ( M) = max ( M ) µ ρ Γ s Γ nxn Since the spectral radius is a continuous function, the structured singular value µ Γ : ^ \ + is also a continuous function. However, µ Γ () is not a norm, since it doesn t satisfy the triangle inequality. In the special cases of only a single scalar block, and only a single full block, µ Γ ( M ) reduces to the following. If Γ= { δi : δ ^ } ( S =, F = 0, r = n) which is the spectral radius of M. nxn If Γ= ^ ( S = 0, F =, m = n), then ( M) ( M) µ ρ s Γ =, (0.68), then ( M) ( M) µ σ Γ =. (0.69) Obviously, for a general uncertainty structure Γ as defined in the beginning, we must have nxn { δ : δ ^} Γ ^. (0.70) I n The bigger the uncertainty set, the more possible it is for a small perturbation to make singular. We conclude that I M s The gap between ρ ( M ) and ( M ) ( M) Γ ( M) ( M) ρ µ σ. (0.7) σ can be large, which makes the evaluation of the structured singular value using these bounds difficult. However, the bounds can be refined by considering transformations on M that do not affect µ M, but do affect ρ( M)and σ ( M). To do this, define the following two subsets of ^nxn : ( ) * { U UU I } : = Γ : = n which is the set of structured unitary matrices, and -, (0.72) ROBSTAB - 67/7

: { D Ds dim df Im I m } diag,...,,,...,, : F F = rxr i i * Di ^, Di = Di > 0, d j \, d j > 0. (0.73) which is the set of matrices that commute with any s Γ, U, D, - we have the following properties: s Γ. Mathematically, for any Consequently, we have: U For all -, : ( ) ( ) ( ) * U, U s Γ, su Γ, σ U s = σ su = σ s -, (0.74) D = D. (0.75) D s s ( MU) ( UM) ( M) ( DMD ) µ = µ = µ = µ (0.76) Γ Γ Γ Γ Therefore, the lower and upper bounds on µ in (0.7) can be tightened to: ( s ) Γ ( ) ( ) max ρ( UM ) max ρ M = µ M inf σ DMD U - s Γ D (0.77) Bounds The lower bound of the expression obtained above is always an equality: ( ) ( ) max ρ ( UM ) = µ M inf σ DMD U - Γ D (0.78) Unfortunately, the function ρ ( UM ) can have multiple local maxima which are not global. Thus local search cannot be guaranteed to obtain µ Γ ( M ), but can only yield a lower bound. The upper bound can be reformulated as a convex optimization problem, so the global minimum can, in principle, be found. Unfortunately, the upper bound is not always equal to µ Γ ( M ). For block structures Γ with a number of scalar blocks and full blocks satisfying 2S+ F 3, the upper bound is always equal to µ Γ M, and for block structures with 2S F 3 µ Γ M is less ( ) + >, there exist matrices for which ( ) than the infimum. The following table gives the cases for which we have equality between the upper inf σ DMD µ Γ M. bound ( ) D ( ) ( ) and the value of ( ) µ M = inf σ DMD if 2S+ F 3 Γ D ROBSTAB - 68/7

Table 2: Number of full and scalar blocks for which the upper bound equals mu F= 0 2 3 4 S= 0 Yes yes Yes no Yes Yes no No no 2 No No no No no Where especially, if: S=0, F= : µ ( M) σ( M). Γ = S=, F=0 : µ Γ ( M) ρ( M) σ ( DMD ) Recap = = inf. D The reciprocal of the largest singular value of matrix M is a measure of the smallest unstructured p p ^ that can cause non-invertibility ("instability") of an M feedback interconnection. The structured singular value µ Γ ( M ) is the inverse of the size of the smallest structured perturbation that can cause non-invertibility ("instability") of an M s feedback interconnection (i.e., makes I M singular.) s The structured singular value µ Γ ( M ) can't be computed directly in general, but the lower and max ρ ( UM ) µ M inf σ DMD are usually very close, if not equal. upper bounds Γ ( ) ( ) U - D :HOOSRVHGQHVV DQG WKH PDLQ ORRS WKHRUHP We now present a technical result that is used in the proof of the robust performance theorem. Let M be a complex matrix partitioned as M M M M M 2 = 2 22 (0.79) ROBSTAB - 69/7

and suppose there are two defined block structures, Γ and Γ 2, which are compatible in size with M and M 22, respectively. Define a third structure Γ as 0 Γ= : : Γ, Γ2. 0 2 2 (0.80) We may compute µ with respect to three structures as follows: µ (). is with respect to Γ, 2 (). with respect to Γ 2, and µ Γ (). is with respect to Γ. By definition, the notations, ( M) µ ( M ) and µ Γ ( M ) all make sense. 2 22 Definition: Well-posedness of LFT Γ, the LFT L ( M, ) Let 2 2 Theorem: I M 2 is well-posed if 22 The linear fractional transformation L ( M, 2 ) µ ( M ). 2 22 2 is invertible. µ is µ, is well-posed for all Γ if and only if 2 2 Main Loop Theorem: The following are equivalent: µ Γ ( M ) ( M ), and µ ( L ( M 2) ) µ 2 22 sup,. Γ2 (0.8) M M M 2 M 2 22 2 ROBSTAB - 70/7

5REXVW VWDELOLW\ ZLWK VWUXFWXUHG XQFHUWDLQW\ Note that so far we have defined the structured singular value and gave a number of results for complex matrices. Now, we come back to transfer matrices to give a theorem for robust closed-loop stability with structured uncertainty. We use the general framework of LFT's. P() s P2() s Suppose Ps () = P2() s P22() s is a generalized plant with a structured uncertainty and a controller connected to it. Let Γ be an uncertainty structure, as defined in (0.65), and assume that the dimensions of the perturbations in Γ are compatible with P () s. Now define the set of real-rational, proper, stable structured perturbations: { s s s s s } + : = ( ) * : ( ) Γ, closed RHP (0.82) 0 0 and consider the perturbed feedback loop in the figure below where s () s +. The LFT is simply reduced to an M() s s () s interconnection and we can use a generalization of the small gain theorem to assess robust stability. s ( s) s ( s) P( s) K( s) L [ Ps ( ), Ks ( )] We have the following result, which is the so-called small- µ theorem: Theorem: Robust stability with structured perturbation (small- µ theorem) Assume controller Ks () is stabilizing for the nominal plant Ps. () Then given β > 0, the closed-loop system in the above figure is well-posed and internally stable for all +, < if and only if s s β ROBSTAB - 7/7

, { L [ P j K j ]} sup µ ( ω), ( ω) ω \ Γ. (0.83) β In particular, if the structured perturbation is normalized, i.e., +, < the condition becomes: Remark: { L [ P j K j ]} sup µ ( ω), ( ω). (0.84) ω \ Γ How do we check Condition (0.83) in practice? Well, we select a wide grid of frequency points covering the frequencies of interest, and we plot µ Γ { L [ P( jω), K( jω) ]} across these frequencies. If this plot is below β at all frequencies, then the closed-loop system with structured uncertainty is robustly stable. If at least one point on this plot exceeds β, then closed-loop system with structured uncertainty is not robustly stable. s s Example: Robust stability with structured uncertainty for the room heating process Recall that in this example, the sensor dynamics and the plant's time constant were uncertain. Now the real parametric uncertainty in the time constant δ \, δ < can be covered with a complex parameter δ ^, δ < at each frequency. Therefore we will model this uncertainty as. δ() s *, δ < First we have to define a structured uncertainty set: δ 0 Γ= : : δ, m. 0 m Then we compute the LFT L [ P'( s), K( s) ] ^ ^ (0.85) with the generalized plant and controller already given. This is tedious to do by hand, but the "sysic" and "starp" functions in Matlab facilitate the calculations, see the m-file muroomheating.m. ROBSTAB - 72/7

s ( s) P'( s) 000 After the LFT is built, its frequency response is computed and µ { L [ P( jω), K( jω) ]} Γ is computed, or more specifically, its upper and lower bounds are computed. Even though we have two scalar blocks, these bounds are equal to each other and hence to µ Γ { L [ P( jω), K( jω) ]} in this case. The "mu-plot" is shown below, and it can be seen that µ Γ L [ P( jω), K( jω) ] is well below { }. Therefore the closed-loop system is robustly stable to both sensor uncertainty and uncertainty in the process time constant. 0.24 0.22 { L [ P( j ), K( j )]} µ ω ω Γ 0.2 0.8 0.6 0.4 0.2 0. 0.08 0-5 0-4 0-3 0-2 0-0 0 0 ω (log) Note that mu-theory was recently enhance as the so-called "mixed-mu" theory. In this theory, real and complex parametric uncertainty (not just complex) can be analysed at the same time. This means that ROBSTAB - 73/7

we can leave the parametric uncertainty in the time constant as a real perturbation δ \, δ <, instead of covering it with a complex parameter δ ^, δ < at each frequency, which is more conservative. The "mu" command in Matlab can also compute bounds on mixed- µ Γ, it suffices to specify that the first perturbation is real in the argument of the command (see muroomheating.m). The mixed- µ Γ plot is shown below. As we expected, the mixed- µ Γ bounds lie below the mu-plot (also shown on this plot) as the uncertainty set is more restricted, and hence the closed-loop system is harder to destabilize. It is of course robustly stable since the peak is below. 0.25 0.2 { L [ P( j ), K( j )]} µ ω ω Γ 0.5 0. 0.05 bounds on mixed { L [ P( j ), K( j )]} µ ω ω Γ 0 0-5 0-4 0-3 0-2 0-0 0 0 ω (log) 5REXVW SHUIRUPDQFH ZLWK VWUXFWXUHG XQFHUWDLQW\ Often, stability is not the only property of a closed-loop system that must be robust to perturbations. Typically, there are exogenous disturbances acting on the system (wind gusts, sensor noise) which result in tracking and regulating errors. Under dynamic perturbations, the effect that these disturbances have on error signals can greatly increase, and therefore closed-loop performance can degrade. Recall that nominal performance specifications can be written in terms of the -norms of selected closed-loop transfer matrices. Usually the system is rearranged in LFT form so that nominal performance is specified by a bound on the -norms of the LFT shown below. ROBSTAB - 74/7

z w Ps ( ) e K( s) u Figure 3: Standard LFT diagram for -optimal control design That is, with given weighting function embedded in the generalized plant Ps, () the performance spec is written as follows: where Tzw L [ P(), s K() s ] =. T <, (0.86) zw In reality there is always some uncertainty in the model, and thus a more approriate block diagram would be the one shown below with structured uncertainty. Then the robust performance specification can be stated as follows: { [ Ps s] Ks} L U s (), (), () <, () s +, () s < s s (0.87) s ( s) z 2 P( s) d K( s) We define the robust performance criterion to be met in our design as guarantee of closed loop performance under to the perturbation acting on the system. In fact it indicates the worst-case level of performance associated with a given unstructured or structured uncertainty set. ROBPERF - 75/7

The following result provides a means to check for robust performance, rather than just robust stability. 2 Consider the perturbed closed-loop system in the figure below, where () nz nd z2 t \ and dt () \, and define the augmented block structure 0 d z2 : s :, n n Ω= sa = s Γ p 0 ^ (0.88) p The uncertainty block p is a fictitious uncertainty linking the controlled output z to the exogenous input w. This fictitious uncertainty block allows us to transform the robust performance problem in an equivalent robust stability problem with respect to the augmented structured perturbation sa, as shown in the figure below. sa ( s) s () s 0 0 p ( s) z2 P( s) d K( s) Intuitively, if the closed-loop system shown above is robustly stable to all () <, then this must mean that 2 (for all admissible () s ) by virtue of the small gain theorem. This T z d intuition is right, as shown by the following robust performance theorem. Note that the generalized plant P P2 is 2x2: Ps () = P2 P, with its first input signal being the output of () s sa, and its first output 22 signal being the input to () s. sa s p s ROBPERF - 76/7

Theorem: Robust performance with structured perturbation Assume controller Ks () is stabilizing for the nominal plant Ps. () Then for the closed-loop system in the above figure, for all s +, s < β the closed-loop system is well-posed, internally stable and T z 2 d if and only if β { L [ P j K j ]} sup µ ( ω), ( ω) ω \ Ω, (0.89) β where T z2d is the transfer matrix from the input d to the output z 2. Remarks The theorem is used mostly with β = and normalized perturbations () s The proof of this theorem uses the main-loop theorem stated above. s. This theorem allows us to check robust performance by computing µ { L [ P( jω), K( jω) ]} Ω at "all" frequencies and making sure it is less than. In practice, this is done for a reasonable frequency grid of, say, 200 points. Computation of the structured singular value is readily performed in Matlab with the "mu" command. The condition µ Ω { L [ P jω K jω ]} sup ( ), ( ) ω \ implies robust stability and nominal performance (as it should, as these are weaker specs than robust performance) Note that this result doesn't tell us how to design a controller that will achieve robust performance. It will be the subject of the next chapter. ROBPERF - 77/7

as Example: Robust performance of room temperature control system We consider a performance spec for temperature setpoint tracking. sensor uncertainty Q 0.0 + 200s 0.000s + 000 s + δ T m ( s ) 0. + 0s + + T m nominal room dynamics 000 nominal sensor e + T des controller e We ( s) The LFT obtained from the above block diagram has the desired temperature Tdes as an input and the weighted room temperature error e: = We( s)( Tdes Tm) the output to be minimized. Right away, we can define the augmented structured perturbation with a fictitious block for performance: and define the augmented block structure: () s 0 m sa(): s =, (0.90) 0 p ( s) s 0 Ω= : sa = : s Γ, p 0 ^ (0.9) p The block diagram above can be rearranged into an LFT form for the equivalent robust stability problem. ROBPERF - 78/7

sa ( s) s () s 0 0 p ( s) e P( s) T des K( s) Note that we must lump all the input and output signals going to/coming from the augmented P P2 perturbation together, so that the generalized plant is 2x2: Ps () = P2 P. We won't do this by 22 hand, but rather use Matlab to build the LFT and compute the robust performance condition (see muroomheating.m). Selecting the performance weighting function as { L [ P( j ), K( j )]} µ ω ω Ω 0 We () s =, a plot of 2500s + is shown below indicating that robust performance was attained. Note that it often takes a few iterations in which the designer has to relax the performance weighting function (less gain, lower bandwidth) before robust performance can be achieved. Sometimes the uncertainty is just too large and robust performance can't be achieved. This is the robustness/performance tradeoff at play..2. { L [ P( j ), K( j )]} µ ω ω Ω 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0-5 0-4 0-3 0-2 0-0 0 0 ROBPERF - 79/7

5REXVW + &RQWURO 'HVLJQ We have seen that robustness of a control system to stable perturbations of its dynamics is related to the -norms of certain closed-loop transfer matrices. Therefore we can use -optimal control theory to design robust control systems by minimizing -norms. 2EMHFWLYH Recall that the small-gain theorem says that a feedback interconnection of a nominal closed-loop transfer matrix with a stable perturbation () s * is stable if the norm of each is less than. This theorem is widely used for robust controller design because such an interconnection would be stable for all perturbations of norm less than in the figure below: ( s) H( s) As we have seen in the previous chapter, the closed-loop transfer matrix H() s is determined as the system "seen" by the perturbation. The transfer matrix is relatively straightforward to obtain by solving the loop equations. As previously mentioned any type of uncertainty model can be transformed into an upper LFT. In the case of H() s being a lower LFT of a generalized plant and a controller, we have the following block diagram: ( s) z w y P( s) K( s) u HINF - 80/7

The objective of controller design for robust stability with this setup is to design a finite-dimensional, linear time-invariant controller Ks () such that the nominal closed-loop system [ ] H(): s = L P(), s K() s is stable, and the perturbed closed-loop system U H(), s () s is stable for all () <. s The necessary and sufficient condition to achieve robust stability with normalized linear fractional uncertainty () s is [ ] L Ps (), Ks (). (0.92) Therefore, the controller design problem for robust stability is just an -norm minimization problem: min ( ), ( ). (0.93) K L [ Ps Ks] Example: Room temperature design for robust stability We want to maximize the stability robustness of our room temperature control system with respect to the uncertainty in sensor dynamics. The temperature sensor with first-order dynamics was assumed to have 0% uncertainty in its frequency response, i.e., l ( jω) < 0., ω, so that Wm () s = 0.. This uncertainty was modeled with an output multiplicative model, which was then transformed into an LFT model. sensor uncertainty m ( s) Wm ( s) Q 0.0 000 s + T 0 s + + + T m nominal room dynamics nominal sensor K( s) + 0 controller HINF - 8/7

The corresponding LFT is shown below, where the generalized plant Ps () is P = 0, P = G, P = W, P = G (0.94) 2 2 m 22 m ( s) Ps ( ) T m 0 0. 0.0 (000s + )(0s + ) 0.0 (000s + )(0s + ) Q K( s) The controller is designed to minimize the -norm of the lower LFT: min ( ), ( ). (0.95) K L [ Ps Ks] It turns out that the optimally-robust controller is no controller at all, i.e., Ks () = 0! In this case, the lower LFT is simply 0 which is stable, and L[ Ps (), Ks ()] = L[ Ps (),0] = 0 = 0 is minimized. Then, any stable perturbation of any size can be tolerated since the resulting open-loop system will always be stable. To avoid this trivial solution, we must add a performance specification for 0 temperature tracking with We () s =. To make the problem well-posed, we also add a 000s + weighting function on the control signal. Let's add the weighting function Wu ( s ) = 0.00. HINF - 82/7

u e sensor uncertainty m ( s) Wm ( s) Q 0.0 000 s + T 0 s + + + T m nominal room dynamics nominal sensor K( s) + T des Wu ( s) controller We ( s) The corresponding LFT is shown below, with the generalized plant Ps. () m ( s) Ps ( ) u e T m 0.0 (000s )(0s ) + + 0 0 0.0W e 0.We W e (000s )(0s ) + + 0 0 Wu 0.0 [ 0. ] (000s + )(0s + ) T des Q K( s) This is done in the m-file Hinfroomheating.m. All of the assumptions on the generalized plant previously explained earlier for -optimal control design must be checked first. Here is the output produced by the hinfsyn function, indicating that the -norm of the lower LFT L [ Ps (), Ks ()] ("gamma value achieved") is almost equal to. This implies that the -norms of sub-blocks of HINF - 83/7

[ (), ()] L Ps Ks are all less than or equal to, some of them being the closed-loop transfer matrices that we wanted to minimize to get robust stability, and nominal performance. This technique is actually called the mixed-sensitivity problem where the -norm of a bigger transfer matrix than what the original robustness or performance spec called for is minimized. Test bounds: 0.0000 < gamma <= 2.0000 gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f 2.000 8.7e-003 2.0e-003.0e-003-3.0e-02 0.0000 p.000 2.5e-06# *******.0e-003 -.3e-042 ****** f.500 7.5e-003 2.0e-003.0e-003.5e-042 0.0000 p.250 6.0e-003 2.0e-003.0e-003 -.8e-02 0.0000 p.25 4.6e-003 2.0e-003.0e-003 -.0e-04 0.0000 p.063 3.4e-003 2.0e-003.0e-003 -.e-02 0.0000 p.03 2.5e-003 2.0e-003.0e-003-9.5e-022 0.0000 p.06.8e-003 2.0e-003.0e-003 4.2e-042 0.0000 p.008.2e-003 2.0e-003.0e-003 6.e-042 0.0000 p.004 8.9e-004 2.0e-003.0e-003 5.8e-04 0.0000 p.002 6.3e-004 2.0e-003.0e-003-2.5e-04 0.0000 p.00 4.4e-004 2.0e-003.0e-003-5.7e-022 0.0000 p Gamma value achieved:.000 The robust controller obtained is of third order. 0L[HGVHQVLWLYLW\ UREXVW + FRQWUROOHU GHVLJQ The last example dealt with two concurrent objectives: maximize the robust stability margin, and minimize the sensitivity. These two objectives lead to the minimization of the -norms of two different transfer matrices. But control can only deal with one objective at a time, e.g., a single full uncertainty block, or a single performance specification. One approach to satisfy two (or more) objectives at the same time is to simply add inputs and outputs corresponding to all of these objectives to the generalized plant, and then minimize the -norm of the big closed-loop transfer matrix (lower LFT) relating all of the inputs to all of the outputs. This is what we did in the previous room temperature control example. It is called a mixed-sensitivity design. If this -norm is less than or equal to, then all of the objectives are attained as a result (it is a sufficient condition). Consider for example the block diagram below. HINF - 84/7

l ( s) z w z 2 P( s) w 2 K( s) If we can somehow manage to design a controller that will give then as a result we will get w z = L [ Ps (), Ks ()] w 6 z, (0.96) 2 2 T zw (robust stability), and T zw 2 2 However, this is a conservative approach, because [ (), ()] (nominal performance). L Ps Ks contains other transfer matrices that are minimized as well, putting more constraints than called for. The µ -synthesis approach for controller design is often a better solution. HINF - 85/7

&RQWUROOHU 'HVLJQ 9LD 0X6\QWKHVLV We now present the µ -synthesis technique to design controllers for robust performance. But first, let's review the main result of µ -analysis. Consider the perturbed closed-loop system in the figure below, 2 where () nz 2 z2 t \ and () nw w2 t \, and define the augmented block structure 0 w2 z2 : s :, n n Ω= sa = s Γ p 0 ^ (0.97) p The uncertainty block input w 2. p is a fictitious uncertainty linking the controlled output z 2 to the exogenous sa ( s) s () s 0 0 p ( s) z 2 P( s) w 2 K( s) Recall the theorem on robust performance with structured perturbation Assume controller Ks () is stabilizing for the nominal plant Ps. () Then for the above block diagram, for all s +, s < the closed-loop system is well-posed, internally stable and T z 2 d if sup µ P ( j L ω), K ( j ω). and only if Ω { [ ]} ω \ MUSYN - 86/7

A µ -synthesis is the solution to the following optimization problem: K( s) stabilizing ω \ { L [ P j K j ]} min sup µ ( ω), ( ω) Ω (0.98) This is still an open problem. Recall that an upper bound on the structured singular value µ Ω { L [ P( jω), K( jω) ]} is given by inf σ ( DMD ) D { D Ds dim df Im Im } diag,...,,,...,, : F F : =. (0.99) rxr i i * Di ^, Di = Di > 0, d j \, d j > 0 which is the set of matrices that commute with any s Γ. However, a reasonable approach is to minimize the upper bound on [ ( ω), ( ω) ] solve ( ) min inf Ds ( ) ( ) L Ps ( ), Ks ( ) D ( s) K s D( s), D ( s) * L P j K j, i.e., to (0.200) by iteratively solving for the controller Ks () and the D-scale Ds (). This is the so-called D-K Iteration. The stable and minimum phase scaling matrix () Ds is chosen such that D() s () s = () s D() s (note that D(s) is not necessarily belong to D since D(s) is not necessarily Hermitian). For a fixed scaling transfer matrix Ds (), K( s) ( ) min D L G, K D (0.20) is a standard H optimization problem. For a given stabilizing controller Ks (), ( ) inf, D G K D (0.202) DD, L is a standard convex optimization problem and it can be solved pointwise in the frequency domain: ( )( ω) sup inf σ DF ω l GK, j D ω Dω ω (0.203) MUSYN - 87/7

Indeed, DD, ( ) σ ( )( ω) inf D, sup inf, L G K D = D G K j D. (0.204) * ω L ω ω Dω This last equation follows intuitively from the following arguments: the left-hand side is always no smaller than the right-hand side, and, on the other hand, given the minimizing D ω from the right hand side across the frequency, there is always a rational function Ds () uniformly approximating the magnitude frequency response D ω. Note that when S=0, (no scalar blocks, only full bocks) ω ω ( F ) D = diag d I,..., d I, I, (0.205) ω which is a block-diagonal scaling matrix applied pointwise across frequency to the frequency response L GK, jω. ( )( ) ( ) D s P( s) D ( s) ( ) K s Figure 32: µ -synthesis via scaling D-K Iterations proceed by performing this two-parameter minimization in sequential fashion: first minimizing over Ks () with Ds () fixed, then minimizing pointwise over D ω with Ks () fixed, then again over Ks (), and again over D ω, etc. Details of this process are summarized in the following steps:. Fix an initial estimate of the scaling matrix D ω pointwise across frequency. 2. Find scalar transfer functions di() s, di () s * for i =,, F d ( j ) d ω i i! such that ω. This interpolation process is currently done mostly by graphical matching using lower order transfer functions. MUSYN - 88/7

{ F } 3. Let () () () D s = diag d s I,..., d s I, I. Construct a state space model for the system in the above figure: P l () s () () D s 0 D s 0 = P() s. (0.206) 0 I 0 I ( ) 4. Solve an -optimization problem to minimize l L PK, over all stabilizing controllers. Note that this optimization problem uses the scaled version of P() s. Let its minimizing controller be denoted by l K. ( ) 5. Minimize σ D, l ω L P K D ω over D ω, pointwise across frequency. Note that this evaluation uses the minimizing K l from the last step, but that P() s is unscaled. The minimization itself produces a new scaling function. Let this new function be denoted by l Dω. 6. Compare l Dω with the previous estimate D ω. Stop if they are close, but, otherwise, replace D ω with l Dω and return to step (2). With either Ks () or Ds () fixed, the global optimum in the other variable may be found using the µ and solutions. Although the joint optimization over Ks () and Ds () is not convex and the global convergence is not guaranteed, many designs have shown that this approach works very well. In fact, this is probably the most effective design methodology available today for dealing with such complicated multivariable problems. We will solve two examples to show how powerful the µ -synthesis technique is. The dkit is used to implement the DK-iteration. MUSYN - 89/7

as Example: µ -synthesis for robust performance of room temperature control system We consider a performance spec for temperature setpoint tracking. sensor uncertainty Q 0.0 + 200s 0.000s + 000 s + δ T m ( s ) 0. + 0s + + T m nominal room dynamics K( s) nominal sensor e + T des controller e We ( s) The LFT obtained from the above block diagram has the desired temperature Tdes as an input and the weighted room temperature error e: = We( s)( Tdes Tm) the output to be minimized. Recall that we defined the augmented structured perturbation with a fictitious block for performance: () s 0 m sa(): s =, and we also defined the augmented block structure: 0 p ( s) s 0 Ω= : sa = : s Γ, p 0 ^. (0.207) p The block diagram above can be rearranged into an LFT form for the equivalent robust stability problem. MUSYN - 90/7

sa ( s) s () s 0 0 p ( s) e P( s) T des K( s) Note that we must lump all the input and output signals going to/coming from the augmented P P2 perturbation together, so that the generalized plant is 2x2: Ps () = P2 P. We use Matlab to 22 build the LFT and implement a D-K iteration. (see musynroomheating.m). 0 The performance weighting function is We () s =. A plot of µ { [ ] } Ω L P( jω), K( jω) 2500s + is shown below indicating that robust performance was attained. Note that there is room to modify the performance weighting function (more gain, higher bandwidth) for better robust performance, since µ Ω L [ P( jω), K( jω) ] is well below. { } 0.38 0.36 0.34 CLOSED-LOOP MU: CONTROLLER #2 { L [ P( j ), K( j )]} µ ω ω Ω 0.32 0.3 MU 0.28 0.26 0.24 0.22 0.2 0.8 0-3 0-2 0-0 0 0 0 2 0 3 FREQUENCY (rad/s) MUSYN - 9/7

The robust controller obtained is of the fourth order. Its state-space matrices as given by Matlab are shown below: A matrix.0e+003 * -0.0243-0.0082 0.006 0.0002 0.0703 0.0236-0.0046-0.0006 0.0044 0.005-0.002-0.0763 0.0000 0.0033-0.079-9.8977 B matrix 0.0374 0.005-0.0098 0.000 C matrix.0e+004 * 9.902 3.0789-0.6032-0.0723 D matrix 0 Example: µ -synthesis for robust performance of mixing tank process control system Recall that the nominal state-space representation of the mixing tank process is: x = Ax+ Bu y = Cx+ Du (0.208) where 0 0 A = 0.0298 0.00 B 0 30 2.38 0 = 7 C = [ 0 ] D = 02 2. (0.209) We consider an uncertainty in the parameter T in, i.e., the assumed operating liquid temperature at the inlet valve. Variations in T in are evaluated to be within ± 5% of the nominal temperature. In fact we MUSYN - 92/7

can write T in as : T in = 293± 5. This variation in the temperature in will lead to variation in the matrices representing the state-space model. If we take the extreme values of T in, we can compute the extreme values of the state space matrices: T For T in = 293+ 5 For T = 293 5 in 0 0 A = 0.048 0.00 0 0 A = 0.0448 0.00 B B 0 5 2.38 0 = 7 0 45 2.38 0 = 7 C = [ 0 ] D = 02 2. C = [ 0 ] D = 02 2. In the framework of the uncertainty representation the perturbed matrices can be written as: A p 0 0 δ 0 0 0 0.0298 0.00 0.05 0 = + 0 δ (0.20) and Bp 0 0 0 = 7 + δ 2 30 2.38 0 5 0 (0.2) such that < δ, δ2 <. We can now represent the uncertain transfer matrix of the process as given in the figure below: u u 2 z 3 5 2 B w 3 w w 2 0 0 z z 2 A ( si A) C y Where 0 0 A : = 0.05 0. MUSYN - 93/7

In order to convert the design problem for this uncertain plant into LFT representation, we need to find the generalized plant Ps () that includes the outputs of the uncertainty parameters w, w 2, w 3 as exogenous inputs and the inputs of the parameters uncertainty as variables to control z, z2, z 3. This is tedious to do by hand, but is easy to do in Matlab. See the m-file musynmixing.m given in appix. MUSYN - 94/7

LTI: IMC: $FURQ\PV Linear time-invariant Internal model control MIMO: Multi-input multi-output SISO: LFT: Single-input single-output Linear fractional transformation APP - 95/7

5HIHUHQFHV [] K. Zhou, J.C. Doyle, K. Glover "Robust and Optimal Control", Prentice Hall, 996, 596 pp. [2] K. Zhou, J.C. Doyle "Essentials of Robust Control", Prentice Hall, 998, 4 pp. [3] S. Skogestad, I. Postlethwaite "Multivariable Feedback Control", Wiley, 996,559 pp. [4] M. Morari, E. Zafiriou, "Robust Process Control", Prentice Hall, 989. APP - 96/7

$SSHQGL[ 0DWODE PILOHV H2mixing.m %% (c) Coronado Systems 2000 % Author: Benoit Boulet % This m-file was used to compute and simulate the optimal H-infinity controller % for the mixing tank process % 2-input, -output plant state-space matrices A=[0 0; 0.0298-0.00]; B=[ 0; -30 2.388e-7]; C=[0 ]; % only the temperature sensor is used D=zeros(,2); G=pck(A,B,C,D); % weighting functions numwo=[0.0 0]; denwo=[2000 ]; Wo=nd2sys(numWo,denWo); numwi=[0.00]; denwi=[]; Wi=nd2sys(numWi,denWi); Wi=daug(Wi,Wi); numwe=[0]; denwe=[2000 ]; We=nd2sys(numWe,denWe); numwu=[0]; denwu=[]; numwu2=[0.000]; denwu2=[]; Wu=nd2sys(numWu,denWu); Wu2=nd2sys(numWu2,denWu2); Wu=daug(Wu,Wu2); % Build the system interconnection to obtain the generalized plant P(s) % using the "sysic" command systemnames=' Wo Wi Wu We G '; inputvar='[ w; w2{2}; u{2} ]'; outputvar='[ Wu; We; Wo - G ]'; input_to_wo='[ w ]'; input_to_wi='[ w2 ]'; input_to_wu='[ u ]'; input_to_we='[ Wo - G ]'; input_to_g='[ u + Wi ]'; sysoutname='p'; APP - 97/7

cleanupsysic='yes'; sysic; % Extract partitioned state-space matrices [AP,BP,CP,DP]=unpck(P); BP=BP(:,:3); BP2=BP(:,4:5); CP=CP(:3,:); CP2=CP(4,:); DP=DP(:3,:3); DP2=DP(:3,4:5); DP2=DP(4,:3); DP22=DP(4,4:5); % Stabilizability of (AP,BP2) and (AP,BP) % PBH test: [-AP+sI BPi] must not lose rank at closed RHP eigvals of AP rhpeigs=[]; [V,EIGS] = eig(ap); eigs = diag(eigs); for i=:length(ap) if real(eigs(i)) >= 0 rhpeigs=[rhpeigs;eigs(i)]; p = length(rhpeigs); r = length(ap); rkb=[]; rkb2=[]; % Rank of [si-ap] for s not eigval(ap) % ranks at rhp eigenvalues % ranks at rhp eigenvalue for j=:p rkb=[rkb ; rank([(rhpeigs(j)*eye(length(ap)))-ap BP])]; rkb2=[rkb2 ; rank([(rhpeigs(j)*eye(length(ap)))-ap BP2])]; if min(rkb) < length(ap) sprintf('problem: (AP,BP) IS NOT STABILIZABLE') stop if min(rkb2) < length(ap) sprintf('problem: (AP,BP2) IS NOT STABILIZABLE') stop % Detectability of (CP2,AP) and (CP,AP) % PHB test: [(-AP+sI)' CP2']' must not lose rank at closed RHP eigvals of AP rkc=[]; rkc2=[]; % ranks at rhp eigenvalues % ranks at rhp eigenvalues for j=:p rkc=[rkc ; rank([(rhpeigs(j)*eye(length(ap)))-ap; CP])]; APP - 98/7

rkc2=[rkc2 ; rank([(rhpeigs(j)*eye(length(ap)))-ap; CP2])]; if min(rkc) < length(ap) sprintf('problem: (CP,AP) IS NOT DETECTABLE') stop if min(rkc2) < length(ap) sprintf('problem: (CP2,AP) IS NOT DETECTABLE') stop % There are two more condition that should be checked. The matrices % [AP-jwI BP2; CP DP2] and [AP-jwI BP; CP2 DP2] must have full column % and row rank respectively for all w. ww = logspace(-4,2,200); for i=:200 rk = rank([sqrt(-)*ww(i)*eye(length(ap))-ap BP2; CP DP2]); if rk < (length(ap)+length(bp2(,:))) sprintf('problem: FIRST MATRIX IS RANK DEFICIENT AT FREQ. = %g',ww(i)) stop for i=:200 rk = rank([sqrt(-)*ww(i)*eye(length(ap))-ap BP; CP2 DP2]); if rk < (length(ap)+length(cp2(:,))) sprintf('problem: SECOND MATRIX IS RANK DEFICIENT AT FREQ. = %g',ww(i)) stop % compute the H2 optimal controller nmeas=; % number of measurements to controller ncon=2; % number of control signals from controller ricmethod=2; [K,Tzw,norms,Kfi,Gfi,hamx,hamy]=h2syn(P,nmeas,ncon,ricmethod) % H2-norm achieved h2norm(tzw) % closed-loop sensitivity from output temperature disturbance to process output temp. % i.e. S(s)=(I+GK)^(-) S=starp(abv(sbs(0,),sbs(,mmult(-,G,K))),); % closed-loop system from output temperature disturbance to process inputs % i.e. SK(s)=-(I-GK)^(-)K APP - 99/7

KS=starp(abv(sbs([0; 0],eye(2)),sbs(,G)),mmult(K,-)); % simulate the output response to a change in setpoint for deltat tfinal=0000; % final time for simulation intstep=5; % integration step step_input=-; % step input: -K disturbance in temperature % simulate plant response first plant_output_resp=trsp(mmult(s,wo,0.0),step_input,tfinal,intstep); K_output_resp=trsp(mmult(KS,Wo,0.0),step_input,tfinal,intstep); % plot the output and actuator responses figure() subplot(3) vplot(plant_output_resp); subplot(32) vplot(sel(k_output_resp,,)); subplot(33) vplot(sel(k_output_resp,2,)); APP - 00/7

bodeh2imc.m %% (c) Coronado systems 2000 % This m-file was used to generate the Bode plot of % sensitivity with the SISO IMC H2 controller design % frequency grid ww=logspace(-5,,200); % Output sensitivity transfer function numtyd=[.007 339.3 0.67]; dentyd=conv([007 ],[0.00 ]); Tyd=nd2sys(numTyd,denTyd); % Frequency response and Bode plot Tydjw=frsp(Tyd,ww); figure() vplot('bode_g',tydjw) % Response to step disturbance Tydstep=trsp(Tyd,5,5000,,0); % Plot the response figure(2) vplot(tydstep) APP - 0/7

Hinfmixing.m %% (c) Coronado Systems 2000 % Author: Benoit Boulet % This m-file was used to compute and simulate the optimal H-infinity controller % for the mixing tank process % 2-input, -output plant state-space matrices A=[0 0; 0.0298-0.00]; B=[ 0; -30 2.388e-7]; C=[0 ]; % only the temperature sensor is used D=zeros(,2); G=pck(A,B,C,D); % weighting functions numwo=[0.0 0]; denwo=[2000 ]; Wo=nd2sys(numWo,denWo); numwi=[0.00]; denwi=[]; Wi=nd2sys(numWi,denWi); Wi=daug(Wi,Wi); numwe=[0]; denwe=[2500 ]; We=nd2sys(numWe,denWe); numwu=[0]; denwu=[]; numwu2=[0.000]; denwu2=[]; Wu=nd2sys(numWu,denWu); Wu2=nd2sys(numWu2,denWu2); Wu=daug(Wu,Wu2); % Build the system interconnection to obtain the generalized plant P(s) % using the "sysic" command systemnames=' Wo Wi Wu We G '; inputvar='[ w; w2{2}; u{2} ]'; outputvar='[ Wu; We; Wo - G ]'; input_to_wo='[ w ]'; input_to_wi='[ w2 ]'; input_to_wu='[ u ]'; input_to_we='[ Wo - G ]'; input_to_g='[ u + Wi ]'; sysoutname='p'; cleanupsysic='yes'; sysic; % Extract partitioned state-space matrices [AP,BP,CP,DP]=unpck(P); BP=BP(:,:3); APP - 02/7