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Schematic representation of the structural decompositions. This decomposition has several impor- minreal sys returns an orthogonal tant properties as stated in the following theorem. In discrete time, 4. Statement 4 of Theorem The pair A, C is detectable if it is algebraically Note.
Any observable equivalent to a system in the standard form for unobservable systems How- ever, there are alternative tests that avoid this intermediate step. These tests can be deduced by duality from the stabilizability tests. The discrete-time LTI system AC-LTI is detectable if and only if every eigenvec- tor of A corresponding to an eigenvalue with magnitude larger than or equal to 1 is not in the kernel of C.
When only the output y can be measured, the control law In Lecture 15, we saw that the state of an observable system can be reconstructed from the its input and output over an interval [t0 , t1 ] using the observability or con- structibility Gramians. However, the formulas derived provide only the value of the state at a particular instant of time, instead of the continuous estimate required to implement The simplest state estimator consists of a copy of the original system, Notation.
Therefore, when A is a stability matrix, the open-loop state estimator When the matrix A is not a stability matrix, it is still possible to construct an asymptotically correct state estimate, but to achieve this we need a closed-loop es- timator of the form Note.
Consider the closed-loop state estimator Assume that the pair A, C is observable. The condition in Theorem The following theorem results from the triangular structure of this matrix. The closed loop of the process LTI with the output feed- back controller This decomposition places the unobservable modes on top of the observable ones, opposite to what happens in Hint: Take a look at Exercise Minimal Realizations 2. Markov Parameters 3.
Similarity of Minimal Realizations 4. The size n of the state-space vector x is called the order of the realization. We saw in Exercise 4. Every minimal realization must be both controllable and observable. This theorem can be easily proved by contradiction. To prove this, one needs to introduce the so-called Markov parameters. From In addition, they must have the same impulse response, and we conclude from We are now ready to prove one of the key results regarding minimal realizations, which completely characterizes minimality in terms of controllability and observabil- ity.
A realization is minimal if and only if it is both controllable and ob- servable. We have already shown in Theorem We prove the converse by contradiction. Denoting by C and O the controllability and observability matrices of As these columns of C Moreover, since Markov parameter cf. All minimal realizations of a transfer function are algebraically equiv- alent. To prove this theorem, let M. From Theorem We show next that this matrix T provides a similarity transformation between the two realizations.
Right-multiplying Left-multiplying In this case, the Notation. Two right-hand side of The roots of d s are called the poles of the transfer function, and the no common roots. Minimal- ity then results from Theorem To proceed, consider then the realization in the controllable canonical form derived in Exercise 4.
We show that it is observable using the eigenvector test. It is straightfor- ward to show that this is so for the matrix A in the minimal realization More- over, since by Theorem We shall see in This result has the following immediate consequence. Lecture 19 that this Corollary See Corollary When sys is in state-space form, msys is a state-space system from which all uncontrollable and unobservable modes were removed.
When sys is in transfer function form, msys is a transfer function from which all common poles and zeros have been canceled. The function pinv M computes the pseudoinverse of the matrix M.
Write the controllability and observability matrices for this system. Justify your answer. Polynomial Matrices: Smith Form 3. McMillan Degree, Poles, and Zeros 5. Transmission-Blocking Property of Transmission Zeros 6. When the polynomials n s and d s in Two the roots of n s and the poles are the roots of d s. We recall that the number polynomial are coprime if they have of roots of n s i. The most useful generalizations of these concepts for a MIMO transfer function G s turn out to be as follows.
These values are called the transmission zeros of G s. We shall return to this later. It turns out that also in the MIMO case the total num- ber of poles with the appropriate multiplicities gives us the dimension of a minimal realization.
The monic P s. The integer r is called the rank of P s and is the maximum family. Informally, the order of a nonzero minor of P s. All minors of order larger than r are equal to zero, gcd set of roots is the intersection of the sets and therefore there are no divisors of order larger than r.
Lemma Proof of Lemma The Smith form of the real polynomial ma- trix A square real polynomial matrix U s is called unimodular if its inverse is also a polynomial matrix. A matrix is unimodular if and only if its Note. Recall the determinant is a nonzero constant independent of s. The matrices Theorem The Smith-McMillan form of the real ra- Note. All the common factors Note. For scalar rational matrices in the entries of However, we Attention!
Unfortunately, when the zero and characteristic polynomials have common roots, the corresponding poles and zeros do not appear in det G s. However, even in this case we may use We shall see in Lecture 19 that this is possible if G s is a transfer matrix for which we have a minimal realization.
The rank Note. Smith-McMillan factorization in Theorem This indicates that the columns of G z dependent. Moreover, using This reasoning allows us to state the following property of transmission zeros. For every transmission zero z 0 of the Note. For such systems, the blocking property of transmission Ck for which G s zeros is somewhat trivial. Contrary to what is advertised, the func- tions eig tf and tzero tf do not necessarily compute the poles and trans- mission zeros of the transfer function tf.
Derive the transmission- blocking property P It also introduces the notion of system inverse and its connection to poles and zeros. Order of Minimal Realizations 4. System Inverse 5. Existence of an Inverse 6. Poles and Zeros of an Inverse 7. Section 3. This matrix is used to introduce a notion of zeros for state-space realizations. The roots of z P s are called the invariant zeros of the state- Dr s of P s. The Invariant zeros also have a blocking property.
This reasoning allows us to state the following property of invariant zeros. Property P In this case, the rank of P s also drops, making z 0 an invariant zero of LTI. Recall that the Theorem Therefore the inclusions in We saw in Lecture 8 that asymptotic stability of LTI is equiv- alent to all eigenvalues of A having strictly negative real parts. The transfer function of For systems. Note that the this system, we observe strict inclusions in We saw in Section LTI We say that the system Attention!
The system The left equality in From the properties of matrix inverses, we know that In this case, we say that the system has a left or a right inverse. These names are inspired by the Attention! When the initial conditions are and not by the nonzero, these are generally not equal.
However, if both the system and its inverse connections in are asymptotically stable, then the effect of initial conditions disappears and we still Figure Cascade interconnections.
The system in Figure Alternatively, for every input y to This shows that the state-space system This leads to the following result. We shall see in Exercise We use a contradiction argument to prove necessity. Assume that the system has an Note. Therefore all entries of always proper. Assume that the system with transfer matrix G s is invertible.
Note A system whose transmission Note In view of P Such a control law is shown in Figure The same desired transfer function can also be achieved with a feedback controller Note. It turns out that, in general, the closed- Note. We shall loop controller is more robust with respect to modeling uncertainty.
Open-loop control. Feedback controller that achieves a closed-loop transfer function Q s from r to y. For the open-loop design, we obtain the following transfer functions from r to y, Attention! The function tzero sys computes the invariant zeros of the state-space system sys. Contrary to what is advertised, it does not necessarily compute the transmission zeros of the system cf.
You may verify this by trying this function on the system in Examples The functions tzero minreal sys and eig minreal sys return the transmission zeros and poles of the transfer function of the system sys, which can either be in state-space or transfer function form. Exam- ple The function inv sys computes the inverse of the system sys. When sys is a state-space model, inv returns a state-space model, and when sys is a transfer function, it returns a transfer function.
Hint: Use the eigenvector tests. Verify that the transfer function of the controller in the dashed box in Fig- ure Optimal Regulation 3. Feedback Invariants 4. Feedback Invariants in Optimal Control 5. Optimal State Feedback 6. Note the problem. The measured output y t corresponds to the signal s that can be measured Figure Measured outputs are typically and are therefore available for control.
The controlled output z t corresponds to the signal s that one would like to Note. Controlled make as small as possible in the shortest possible time. At other times one may have be viewed as design parameters. Many other options are possible. However, decreasing the energy of the controlled output will require a large control signal, and a small control signal will lead to large con- trolled outputs.
A functional maps functions in we say that a functional this case signals, i. In fact, Attention! To keep the feedback invariant in Proposition The following has been proved. Assume that there exists a symmetric solution P to the algebraic Ric- Note.
Asymptotic cati equation See Example This is especially important when the units used for the different components of u and z make the values for these variables numerically very different from each other. We shall pursue this further in Section Use the feedback invariant in Exercise The Hamiltonian Matrix 2. Domain of the Riccati Operator 3. Stable Subspaces 4. Stable Subspace of the Hamiltonian Matrix 5. To study the solutions of this equation, it is convenient to expand the last term in the left-hand side of Then the following properties hold.
In general the ARE has multiple 2. P is a symmetric matrix. To prove statement 1, we left-multiply To prove statement 2, we just look at the top n rows of the matrix equation Therefore Exercise Therefore it must actually be identically zero.
See Properties Stable subspaces. Then Exercise From P We thus conclude that the 2n eigenvalues of H are distributed symmetrically with respect to the imaginary axis.
To check that we actually have n eigenvalues with a negative real part and another n with a positive real part, we need to make sure that H has no eigenvalues over the imaginary axis. This point is addressed by the following result. In this case, Lemma Since the corresponding eigenvalues do not have negative real parts, this contradicts the stabilizability and detectability assumptions. Therefore, we conclude from property P Combining Lemma H is in the domain of the Riccati operator, 2.
P is symmetric , Note It is insightful to interpret the results of Theorem The need for A, G to be detectable can be intuitively understood by the fact that if the system had unstable modes that did not appear in z, it could be possible to make JLQR very small, even though the state x might be exploding.
Prove Properties P Hint: Transform M into its Jordan normal form. LQR Design Example 5. Cheap Control Case 6. The Loop-shaping Design Method review 9. Throughout this whole lecture we assume that Attention! We shall see in Example State feedback open-loop gain. We saw in the Lecture 20 that under appropriate stabilizability and detectability as- sumptions, the LQR control results in a closed-loop system that is asymptotically stable. LQR controllers also have desirable properties in the frequency domain.
For the LQR criterion in This is represented graphically in criterion. Nyquist plot for a LQR state feedback controller. If the process gain is multiplied by a constant 0.
This corresponds to a negative gain margin of 20 log A small sensitivity function is desirable for good disturbance rejection. Gener- inequality shows that ally, this is especially important at low frequencies. Generally, this is especially important at low frequencies. A small complementary sensitivity function is desirable for good noise rejec- tion. Generally, this is especially important at high frequencies.
To some extent, this limits the controlled outputs that should be placed in z. Loop shaping some upper bounds on the magnitude of the sensitivity function and its comple- consists of designing the controller to meet mentary. A brief review of this We discuss next a few rules that allow us to perform loop shaping using LQR. Low-frequency open-loop gain. To understand the implications sys. However, this is often achieved at the expense of a slower response.
In such cases, one may actually add dynamics to more accurately shape L s. We will see in Section Aircraft roll angle dynamics. This parameter allows us to move the whole Larger values for this parameter result in a magnitude Bode plot up and down. Bode plots for the open-loop gain of the LQR controllers in Example This parameter allows us to control the This parameter allows us to control the speed of the response. Closed-loop step responses for the LQR controllers in Example In Figure Nyquist plots for the open-loop gain of the LQR controllers in Example This limiting case is called cheap control and it turns out that whether or not the above conjectures are true depends on the transmission zeros of the system.
To proceed, we should consider the square and nonsquare cases separately. Two conclusions can be drawn. Therefore q closed-loop poles converge to Note.
This property Attention! This means that in general one wants to avoid transmission zeros from the of LQR resembles a similar property of the control input u to the controlled output z, especially slow transmission zeros that will root locus, except that attract the poles of the closed loop. This result can Theorem This result shows a fundamental limitation due to unstable transmission zeros.
It shows that when there are transmission zeros from the input u to the con- trolled output z, it is not possible to reduce the energy of z arbitrarily, even if one is willing to spend much control energy. Taking limits on both sides of The command sigma sys draws the norm- Bode plot of the system sys.
For scalar transfer functions, this command plots the usual magnitude Bode plot, but for MIMO transfer matrices, it plots the norm of the transfer matrix versus the frequency. The command nyquist sys draws the Nyquist plot of the system sys. Especially when there are poles very close to the imaginary axis e. The Nyquist criterion is used to investi- gate the stability of the negative-feedback connection in Figure It allows one to compute the number of unstable i.
This leads to a closed curve that is always symmetric with respect to Note. Negative feedback. For the multivariable Nyquist criteria, we count encirclements around the count gives ENC. The basic idea behind loop shaping is to convert the desired speci- method is covered extensively, e. The review in this section is focused on the SISO case, so it does not address the state feedback case for systems with more than one state. However, we shall see in Lecture 23 that we Figure Closed-loop system.
Larger phase margins generally correspond to a smaller overshoot for the step response of the closed-loop system. For the closed-loop system in Figure This shaping can be achieved appropriate gain, lead, using three basic tools. However, this does 1. Proportional gain. Multiplying the controller by a constant k moves the mag- correspond to additions in the nitude Bode plot up and down, without changing its phase.
Lead compensation. A lead compensator also 3. Lag compensation. A lag compensator also increases the phase, so it can decrease the phase margin. To avoid this, one should only introduce lag compensation away from the cross-over frequency. Show that A square matrix S is called cf. Certainty Equivalence 2. Optimal Set Point Control 7. LTR Design Example 9. This approach is usually known as certainty equivalence and leads to the architecture in Figure In this lecture we consider the problem of con- structing state estimates for use in certainty equivalence controllers.
Certainty equivalence controller. Since neither d nor n are known, solving This leads to state estimators that respond fast to changes in the output y. This leads to state estimators that respond cautiously slowly to unexpected changes in the measured output. We shall see in Section The reader may recall that we had Then the MEE estimator for The results in Lecture 21 provide conditions for the existence of an appropriate solution to the dual ARE See Theorem Part 1 is a straightforward application of Theorem To do this, we rewrite Although H0 Proposition Here, by feedback invariant we Proof of Proposition Moreover, Note 14, p.
In this context, A large Q corresponds to little measurement noise and leads to state estimators that respond fast to changes in the measured output. A large R corresponds to small disturbances and leads to state estimates that respond cautiously slowly to unexpected changes in the measured output.
However, for processes that do not satisfying Selecting Note. The following items should be kept in mind regarding Theorem When the process has zeros in the right half-plane, loop-gain recovery will gen- erally work only up to the frequencies of the nonminimum-phase zeros. When the zeros are in the left half-plane but close to the axis, the closed loop will not be very robust with respect to uncertainty in the position of the zeros.
This is because the controller will attempt to cancel these zeros. Such equilibrium point xeq , u eq must satisfy the equation Note. In this case, the system of equa- tions This By the zero-blocking property, one Theorem We shall solutions. The last terms on each equation cancel because of As seen in Exercise Linear quadratic set point control with state feedback. Closed-loop transfer matrices.
To determine the transfer matrix from the reference r to the control input u, we use the diagram in Figure Section This which corresponds to the control architecture shown in Figure When the Closed-loop transfer matrices. As discussed in Section This command returns the optimal estimator gain L, the solution P to the corre- sponding algebraic Riccati equation, and a state-space model est for the estimator.
For the LQG state estimators, we used the parameters for the loop transfer recovery theorem Theorem Bode plots of the open-loop gain and closed-loop step response for the LQR controllers in Example Moreover, at high frequencies the output feedback controllers exhibit much faster and better! Prove the following. Hint: Use the eigenvector test. Verify that a solution to Note that in this case the process has an integrator.
This parameterization is subse- quently used as the basis for a control design method based on numerical optimization. Properties 3. Q Parameterization 4. Suppose, however, that instead of We can rewrite This means that, for every input signal v Attention! In the to the interconnection of CLTI with It is available free to adopters of the text.
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