Home > Articles

This chapter is from the book

This chapter is from the book

3.6 Design for Processes with Right Half Plane Zeros

When N(s) in Eq. (3.14) has factors of the form (-τs+1) or (τ2s2 – 2τζs + 1), with τ and ζ greater than zero, its inverse is unstable. In this case the IMC controller cannot be formed as given by Eq. (3.15). The integral square error (ISE)7 optimal choice of controller for such cases is to invert that portion of the model which has zeros in the left half plane and add poles at the mirror image of the right half plane zeros (Morari & Zafiriou, 1989). That is, we assume that the model given by Eq. (3.14) can be rewritten as

Equation 3.19a



N-(s) contains only left half plane zeros, none of which have small damping ratios.


N+(s) contains only right half plane zeros, and can be written as

Equation 3.19b


Notice that the gain of N+(s) is one.

Before designing the IMC controller, we strongly recommend that the model be put in time constant form (i.e., the numerator and denominator are factored into products of the form (±τs + 1), (τ2s2 ± 2τζs + 1) so that it is easy to form N+(s) and N-(s). The MATLAB functions tcf and tfn provided with IMCTUNE were developed specifically to put transfer functions into time constant form, and to facilitate their manipulation in this form. There are also other software programs that can be used to accomplish the desired factorization as described in Section 3.9.

The ISE optimal IMC controller for Eq. (3.19a) is

Equation 3.20


where the zeros of N+(–s) are all in the left half plane and are the mirror images of the zeros of N+(s). r = relative order of N(s)/D(s) as before.

The choice of controller given by Eq. (3.20) results in a loop response given by

Equation 3.21


The loop response given in Eq. (3.21) is optimal in an ISE sense for a filter time constant of zero, and is suboptimal for finite . Also, when is zero, the loop transfer function given by Eq. (3.21) is called all-pass, since the magnitude of the frequency response is one over all frequencies.

Example 3.3 One Right Half Plane Zero

The process model is

Equation 3.22a


Putting Eq. (3.22a) in time constant form yields

Equation 3.22b


The IMC controller is

Equation 3.22c


The resulting loop response is

Equation 3.22d


Figure 3.6 compares the step response of the ISE optimal loop transmission given by Eq. (3.21) with = 0 to step responses of suboptimal responses obtained by increasing and decreasing the controller time constant. Notice that the faster response obtained with pq(s) = (–s + 1)/(.5s + 1) comes at the expense of a more negative initial response. Thus, for this simple example, the ISE optimal response is also qualitatively the best compromise between a more sluggish response and a faster response with a more negative initial response.

03fig06.gifFigure 3.6. Response of processes with one right half plane zero.

The transfer function given by the controller of Eq. (3.22c) does not result in an optimal response to a step setpoint change unless is zero. We could get closer to an optimal transfer function by selecting the IMC controller as

Equation 3.22e


The controller given by Eq. (3.22e) was obtained by forcing the coefficient of the linear term of its expanded denominator to be one, which is the same as the linear term in the denominator of Eq. (3.22c) when is zero. The loop response then becomes that given by Eq. (3.22f).

Equation 3.22f


The cubic and quadratic terms in s in the denominator of Eq. (3.22f) are small relative to its linear term so that Eq. (3.22f) approaches the optimal transfer function given by Eq. (3.22d) with equal to zero. Notice, however, that Eq. (3.22e) is valid only for ≤ .5. For larger values of the filter time constant, Eq. (3.22e) is not stable.

The approach used to obtain the controller given by Eq. (3.22e) can be used to develop nearly optimal controllers for arbitrary nonminimum8 phase processes. However, such controllers will generally be useful only for small filter time constants.

Example 3.4 Two Right Half Plane Zeros

When the initial process response to a step is in a direction opposite to that of the final steady state, as in the previous example, the process is said to exhibit an inverse response. Processes with an odd number of right half plane zeros exhibit inverse responses. Processes with an even number of right half plane zeros do not have inverse responses, since the initial value at time zero plus is always in the direction of the steady-state9 as shown in Figure 3.7 for the loop response given by

Equation 3.23


03fig07.gifFigure 3.7. Response of processes with two right half plane zeros.

with τ= .5, 1, and 2.

Notice that the ISE optimal response in Figure 3.7 is again that which also gives the qualitatively best compromise between a sluggish response and a response with too much initial overshoot.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020